Operator Theory in Harmonic and Non-commutative Analysis: 23rd International Workshop in Operator Theory and its Applications, Sydney, July 2012

Operator TheoryAdvances and Applications

Joseph A. BallMichael A. DritschelA.F.M. ter ElstPierre PortalDenis PotapovEditors

Operator Theory in Harmonic and Non-commutative Analysis23rd International Workshop in Operator Theory and its Applications, Sydney, July 2012

Operator Theory: Advances and Applications

Founded in 1979 by Israel Gohberg

Volume 240

Joseph A. Ball (Blacksburg, VA, USA) Harry Dym (Rehovot, Israel) Marinus A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Wien, Austria) Christiane Tretter (Bern, Switzerland)

Vadim Adamyan (Odessa, Ukraine)

Albrecht Böttcher (Chemnitz, Germany) B. Malcolm Brown (Cardiff, UK) Raul Curto (Iowa, IA, USA) Fritz Gesztesy (Columbia, MO, USA) Pavel Kurasov (Stockholm, Sweden) Leonid E. Lerer (Haifa, Israel) Vern Paulsen (Houston, TX, USA) Mihai Putinar (Santa Barbara, CA, USA) Leiba Rodman (Williamsburg, VA, USA) Ilya M. Spitkovsky (Williamsburg, VA, USA)

Lewis A. Coburn (Buffalo, NY, USA) Ciprian Foias (College Station, TX, USA) J.William Helton (San Diego, CA, USA) Thomas Kailath (Stanford, CA, USA) Peter Lancaster (Calgary, Canada) Peter D. Lax (New York, NY, USA) Donald Sarason (Berkeley, CA, USA) Bernd Silbermann (Chemnitz, Germany) Harold Widom (Santa Cruz, CA, USA)

Associate Editors: Honorary and Advisory Editorial Board:

Editors:

Wolfgang Arendt (Ulm, Germany)

Subseries Linear Operators and Linear Systems Subseries editors: Daniel Alpay (Beer Sheva, Israel) Birgit Jacob (Wuppertal, Germany) André C.M. Ran (Amsterdam, The Netherlands) Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze (Potsdam, Germany) Michael Demuth (Clausthal, Germany) Jerome A. Goldstein (Memphis, TN, USA) Nobuyuki Tose (Yokohama, Japan) Ingo Witt (Göttingen, Germany)

Editors

Operator Theory

Joseph A. Ball • Michael A. Dritschel • A.F.M. ter Elst

in Harmonicand Non- ommutative Analysis

Pierre Portal • Denis Potapov

c

23rd International Workshop in Operator Theoryand its Applications, Sydney, July 2012

ISBN 978-3-319-06265-5 ISBN 978-3-319-06266-2 (eBook) DOI 10.1007/978-3-319-06266- Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: Mathematics Subject Classification (2 010): 30E05, 30H20, 34B24, 34C25, 34K13, 34L05, 34L40, 35Q58,

© Springer 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer Basel is part of Springer Science+Business Media (www.birkhauser-science.com)

ISSN 0255- ISSN 2 -0156 296 4878 (electronic)

2

42A45, 42B37, 46L53, 47A10, 47A13, 47A20, 47A40, 47A48, 47A55, 47A60, 47A75, 47B07, 47B10, 47B20,47B33, 47B35, 47B38, 47B40, 47D06, 47L20, 60H15

International Publishing Switzerland

2014942552

Editors

Department of Mathematics

Department of Mathematics

University of Newcastle upon Tyne

Joseph A. BallDepartment of MathematicsVirginia Polytechnic InstituteBlacksburg, VA, USA

Michael A. Dritschel

Newcastle upon Tyne, UK

A.F.M. ter Elst

University of AucklandAucklan

Mathematical Sciences InstitutePierre Portal

The Australian National UniversityCanberra, ACT, Australia

University of New South Wales

Denis PotapovSchool of Mathematics and Statistics

Sydney, NSW, Australia

d, New ZealandUniversité Lille 1Villeneuve d’Ascq, Franceand

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

A. AmentaTent Spaces over Metric Measure Spaces under Doublingand Related Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

P. Auscher and S. StahlhutRemarks on Functional Calculus for Perturbed First-orderDirac Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

W. Bauer,C. Herrera Yanez and N. Vasilevski(m,λ)-Berezin Transform and Approximation of Operators onWeighted Bergman Spaces over the Unit Ball . . . . . . . . . . . . . . . . . . . . . . . 45

C.C. Cowen, S. Jung and E. KoNormal and Cohyponormal Weighted CompositionOperators on H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

R.E. Curto, I.S. Hwang and W.Y. LeeA Subnormal Toeplitz Completion Problem . . . . . . . . . . . . . . . . . . . . . . . . . 87

S. Dey and K.J. HariaGeneralized Repeated Interaction Model and Transfer Functions . . . . 111

F. Gesztesy and R. WeikardSome Remarks on the Spectral Problem Underlying theCamassa–Holm Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

G. GodefroyRemarks on Spaces of Compact Operators between ReflexiveBanach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

B. JefferiesHarmonic Analysis and Stochastic Partial Differential Equations:The Stochastic Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

vi Contents

S. Patnaik and G. WeissSubideals of Operators – A Survey and Introductionto Subideal-Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

W.J. RickerMultipliers and Lp-operator Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

A. SkripkaTaylor Approximations of Operator Functions . . . . . . . . . . . . . . . . . . . . . . . 243

Introduction

The XXIII International Workshop on Operator Theory and its Applications(IWOTA 2012) was held at the University of New South Wales (Sydney, Aus-tralia) from 16 July to 20 July 2012. With 140 participants from all parts of theworld, and 22 plenary speakers representing many different branches of operatortheory, the meeting was a great success. Building on the strengths of Australianmathematical analysis, the meeting focused on the role of operator theory in har-monic and non-commutative analysis. Other themes were also well represented,from pure operator theory in Banach spaces through to engineering applications.The meeting certainly demonstrated the unity within the diversity of the field withdiscussions highlighting many connections between different branches of operatortheory. It was financially supported by the Australian Mathematical Sciences Insti-tute (AMSI), the University of New South Wales, the Centre for Mathematics andits Applications of the Australian National University, and the National ScienceFoundation of the USA. This support was vital and is gratefully acknowledged.

This volume contains the proceedings of the conference. It reflects the qualityand the diversity of the research presented at IWOTA 2012. Each paper has beencarefully refereed and has only been accepted if it meets the standards of thejournal Integral Equations and Operator Theory. We are very thankful to theauthors and the referees for their contributions.

The editors:

Joe Ball, Michael Dritschel, Tom ter Elst,Pierre Portal, and Denis Potapov.

Operator Theory:Advances and Applications, Vol. 240, 1–29c©

Tent Spaces over Metric Measure Spacesunder Doubling and Related Assumptions

Alex Amenta

Abstract. In this article, we define the Coifman–Meyer–Stein tent spacesT p,q,α(X) associated with an arbitrary metric measure space (X, d, μ) un-der minimal geometric assumptions. While gradually strengthening our geo-metric assumptions, we prove duality, interpolation, and change of aperturetheorems for the tent spaces. Because of the inherent technicalities in dealingwith abstract metric measure spaces, most proofs are presented in full detail.

Mathematics Subject Classification (2010). 42B35.

Keywords. Duality, change of aperture, complex interpolation, volume dou-bling, Hardy–Littlewood maximal operator.

1. Introduction

The purpose of this article is to indicate how the theory of tent spaces, as devel-oped by Coifman, Meyer, and Stein for Euclidean space in [7], can be extendedto more general metric measure spaces. Let X denote the metric measure spaceunder consideration. If X is doubling, then the methods of [7] seem at first tocarry over without much modification. However, there are some technicalities tobe considered, even in this context. This is already apparent in the proof of theatomic decomposition given in [17].

Further still, there is an issue with the proof of the main interpolation resultof [7] (see Remark 3.20 below). Alternate proofs of the interpolation result havesince appeared in the literature – see for example [12], [4], [6], and [14] – butthese proofs are given in the Euclidean context, and no indication is given of theirgeneral applicability. In fact, the methods of [12] and [4] can be used to obtain apartial interpolation result under weaker assumptions than doubling. This resultrelies on some tent space duality; we show in Section 3.2 that this holds once weassume that the uncentred Hardy–Littlewood maximal operator is of strong type(r, r) for all r > 1.1

Supported by the Australian Research Council Discovery grant DP120103692.1This fact is already implicit in [7].

2014 Springer International Publishing Switzerland

2 A. Amenta

Finally, we consider the problem of proving the change of aperture result whenX is doubling. The proof in [7] implicitly uses a geometric property of X whichwe term (NI), or ‘nice intersections’. This property is independent of doubling,but holds for many doubling spaces which appear in applications – in particular,all complete Riemannian manifolds have ‘nice intersections’. We provide a proofwhich does not require this assumption.

2. Spatial assumptions

Throughout this article, we implicitly assume that (X, d, μ) is a metric measurespace; that is, (X, d) is a metric space and μ is a Borel measure on X . The ballcentred at x ∈ X of radius r > 0 is the set

B(x, r) := {y ∈ X : d(x, y) < r},and we write V (x, r) := μ(B(x, r)) for the volume of this set. We assume that thevolume function V (x, r) is finite2 and positive; one can show that V is automati-cally measurable on X × R+.

There are four geometric assumptions which we isolate for future reference:

(Proper): a subset S ⊂ X is compact if and only if it is both closed and bounded,and the volume function V (x, r) is lower semicontinuous as a function of(x, r);3

(HL): the uncentred Hardy–Littlewood maximal operator M, defined for measur-able functions f on X by

M(f)(x) := supB�x

1

μ(B)

∫B

|f(y)| dμ(y) (1)

where the supremum is taken over all balls B containing x, is of strong type(r, r) for all r > 1;

(Doubling): there exists a constant C > 0 such that for all x ∈ X and r > 0,

V (x, 2r) ≤ CV (x, r);

(NI): for all α, β > 0 there exists a positive constant cα,β > 0 such that for allr > 0 and for all x, y ∈ X with d(x, y) < αr,

μ(B(x, αr) ∩B(y, βr))

V (x, αr)≥ cα,β .

We do not assume that X satisfies any of these assumptions unless men-tioned otherwise. However, readers are advised to take (X, d, μ) to be a completeRiemannian manifold with its geodesic distance and Riemannian volume if theyare not interested in such technicalities.

2Since X is a metric space, this implies that μ is σ-finite.3Note that this is a strengthening of the usual definition of a proper metric space, as the usual

definition does not involve a measure. We have abused notation by using the word ‘proper’ inthis way, as it is convenient in this context.

Tent Spaces over Metric Measure Spaces 3

It is well known that doubling implies (HL). However, the converse is nottrue. See for example [10] and [18], where it is shown that (HL) is true for R2

with the Gaussian measure. We will only consider (NI) along with doubling, sowe remark that doubling does not imply (NI): one can see this by taking R2

(now with Lebesgue measure) and removing an open strip.4 One can show that allcomplete doubling length spaces – in particular, all complete doubling Riemannianmanifolds – satisfy (NI).

3. The basic tent space theory

3.1. Initial definitions and consequences

Let X+ denote the ‘upper half-space’ X×R+, equipped with the product measuredμ(y) dt/t and the product topology. Since X and R+ are metric spaces, withR+ separable, the Borel σ-algebra on X+ is equal to the product of the Borelσ-algebras on X and R+, and so the product measure on X+ is Borel (see [5,Lemma 6.4.2(i)]).

We say that a subset C ⊂ X+ is cylindrical if it is contained in a cylinder:that is, if there exists x ∈ X and a, b, r > 0 such that C ⊂ B(x, r) × (a, b).Note that cylindricity is equivalent to boundedness when X+ is equipped with anappropriate metric, and that compact subsets of X+ are cylindrical.

Cones and tents are defined as usual: for each x ∈ X and α > 0, the cone ofaperture α with vertex x is the set

Γα(x) := {(y, t) ∈ X+ : y ∈ B(x, αt)}.For any subset F ⊂ X we write

Γα(F ) :=⋃x∈F

Γα(x).

For any subset O ⊂ X , the tent of aperture α over O is defined to be the set

Tα(O) := (Γα(Oc))c.

Writing

FO(y, t) :=dist(y,Oc)

t= t−1 inf

x∈Ocd(y, x),

one can check that Tα(O) = F−1O ((α,∞)). Since FO is continuous (due to the

continuity of dist(·, Oc)), we find that tents over open sets are measurable, and soit follows that cones over closed sets are also measurable. We remark that tents(resp. cones) over non-open (resp. non-closed) sets may not be measurable.

Let F ⊂ X be such that O := F c has finite measure. Given γ ∈ (0, 1), wesay that a point x ∈ X has global γ-density with respect to F if for all balls B

4One could instead remove an open bounded region with sufficiently regular boundary, for ex-ample an open square. This yields a connected example.

4 A. Amenta

containing x,μ(B ∩ F )

μ(B)≥ γ.

We denote the set of all such points by F ∗γ , and define O∗

γ := (F ∗γ )

c. An importantfact here is the equality

O∗γ = {x ∈ X :M(1O)(x) > 1− γ},

where 1O is the indicator function of O. We emphasise that M denotes the un-centred maximal operator. When O is open (i.e., when F is closed), this showsthat O ⊂ O∗

γ and hence that F ∗γ ⊂ F . Furthermore, the function M(1O) is lower

semicontinuous whenever 1O is locally integrable (which is always true, since weassumed O has finite measure), which implies that F ∗

γ is closed (hence measur-able) and that O∗

γ is open (hence also measurable). Note that if X is doubling,then since M is of weak-type (1, 1), we have that

μ(O∗γ) �γ,X μ(O).

Remark 3.1. In our definition of points of γ-density, we used balls containingx rather than balls centred at x (as is usually done). This is done in order toavoid using the centred maximal function, which may not be measurable withoutassuming continuity of the volume function V (x, r).

Here we find it convenient to introduce the notion of the α-shadow of a subsetof X+. For a subset C ⊂ X+, we define the α-shadow of C to be the set

Sα(C) := {x ∈ X : Γα(x) ∩C �= ∅}.Shadows are always open, for if A ⊂ X+ is any subset, and if x ∈ Sα(A), then thereexists a point (z, tz) ∈ Γα(x)∩A, and one can easily show that B(x, αtz − d(x, z))is contained in Sα(A).

The starting point of the tent space theory is the definition of the operatorsAα

q and Cαq . For q ∈ (0,∞), the former is usually defined for measurable functions

f on Rn+1+ (with values in R or C, depending on context) by

Aαq (f)(x)

q :=

∫∫Γα(x)

|f(y, t)|q dλ(y) dt

tn+1

where x ∈ Rn and λ is the Lebesgue measure. There are four reasonable ways togeneralise this definition to our possibly non-doubling metric measure space X :5

these take the form

Aαq (f)(x)

q :=

∫∫Γα(x)

|f(y, t)|q dμ(y)

V (a,bt)

dt

t

where a ∈ {x, y} and b ∈ {1, α}. In all of these definitions, if a function f on X+

is supported on a subset C ⊂ X+, then Aαq (f) is supported on Sα(C); we will use

this fact repeatedly in what follows. Measurability of Aαq (f)(x) in x when a = y

follows from Lemma A.6 in the Appendix; the choice a = x can be taken care

5We do not claim that these are the only reasonable generalisations.


of with a straightforward modification of this lemma. The choice a = x, b = 1appears in [2, 17], and the choice a = y, b = 1 appears in [15, §3]. These definitionsall lead to equivalent tent spaces when X is doubling. We will take a = y, b = αin our definition, as it leads to the following fundamental technique, which workswith no geometric assumptions on X .

Lemma 3.2 (Averaging trick). Let α > 0, and suppose Φ is a nonnegative measur-able function on X+. Then∫

X

∫∫Γα(x)

Φ(y, t)dμ(y)

V (y, αt)

dt

tdμ(x) =

∫∫X+

Φ(y, t) dμ(y)dt

t.

Proof. This is a straightforward application of the Fubini–Tonelli theorem, whichwe present explicitly due to its importance in what follows:∫

X

∫∫Γα(x)

Φ(y, t)dμ(y)

V (y, αt)

dt

tdμ(x)

=

∫X

∫ ∞

0

∫X

1B(x,αt)(y)Φ(y, t)dμ(y)

V (y, αt)

dt

tdμ(x)

=

∫ ∞

0

∫X

∫X

1B(y,αt)(x) dμ(x)Φ(y, t)dμ(y)

V (y, αt)

dt

t

=

∫ ∞

0

∫X

V (y, αt)

V (y, αt)Φ(y, t) dμ(y)

dt

t

=

∫∫X+

Φ(y, t) dμ(y)dt

t. �

We will also need the following lemma in order to prove that our tent spacesare complete. Here we need to make some geometric assumptions.

Lemma 3.3. Let X be proper or doubling. Let p, q, α > 0, let K ⊂ X+ be cylindrical,and suppose f is a measurable function on X+. Then∥∥Aα

q (1Kf)∥∥Lp(X)

� ‖f‖Lq(K) �∥∥Aα

q (f)∥∥Lp(X)

, (2)

with implicit constants depending on p, q, α, and K.

Proof. WriteK ⊂ B(x, r) × (a, b) =: C

for some x ∈ X and a, b, r > 0. We claim that there exist constants c0, c1 > 0 suchthat for all (y, t) ∈ C,

c0 ≤ V (y, αt) ≤ c1.

If X is proper, this is an immediate consequence of the lower semicontinuity ofthe ball volume function (recall that we are assuming this whenever we assume

X is proper) and the compactness of the closed cylinder B(x, r) × [a, b]. If X isdoubling, then we argue as follows. Since V (y, αt) is increasing in t, we have that

min(y,t)∈C

V (y, αt) ≥ miny∈B(x,r)

V (y, αa)

6 A. Amenta

and

max(y,t)∈C

V (y, αt) ≤ maxy∈B(x,r)

V (y, αb).

By the argument in the proof of Lemma A.4 (in particular, by (16)), there existsc0 > 0 such that

miny∈B(x,r)

V (y, αa) ≥ c0.

Furthermore, since

V (y, αb) ≤ V (x, αb + r)

for all y ∈ B(x, r), we have that

maxy∈B(x,r)

V (y, αb) ≤ V (x, αb + r) =: c1,

proving the claim.

To prove the first estimate of (2), write∥∥Aαq (1Kf)

∥∥Lp(X)

=

⎛⎝∫Sα(K)

(∫∫Γα(x)

1K(y, t)|f(y, t)|q dμ(y)

V (y, αt)

dt

t

) pq

dμ(x)

⎞⎠1p

�c0,q

(∫Sα(K)

(∫∫K

|f(y, t)|q dμ(y) dtt

) pq

dμ(x)

) 1p

�K ‖f‖Lq(K) .

To prove the second estimate, first choose finitely many points (xn)Nn=1 such that

B(x, r) ⊂N⋃

n=1

B(xn, αa/2)

using either compactness of B(x, r) (in the proper case) or doubling.6 We thenhave (∫∫

K


) 1q

�c1

(∫∫K

N∑n=1

1B(xn,αa/2)(y)|f(y, t)|qdμ(y)

V (y, αt)

dt

t

) 1q

�X,q

N∑n=1

(∫∫K


V (y, αt)

dt

t

) 1q

.

6In the doubling case, this is a consequence of what is usually called ‘geometric doubling’. Aproof that this follows from the doubling condition can be found in [8, §III.1].


If x, y ∈ B(xn, αa/2), then d(x, y) < αa < αt (since t > a), and so∫∫K


V (y, αt)

dt

t≤∫∫

Γα(x)

|f(y, t)|q dμ(y)

V (y, αt)

dt

t. (3)

When p ≥ q, we use Holder’s inequality along with (3) to write

N∑n=1

(∫∫K


V (y, αt)

dt

t

) 1q

=

N∑n=1

(1

V (xn, αa/2)

∫B(xn,αa/2)∫∫

K


V (y, αt)

dt

tdμ(x))

1q

≤N∑

n=1

(1

V (xn, αa/2)

∫B(xn,αa/2)(∫∫

K


V (y, αt)

dt

t

) pq

dμ(x))1p

≤N∑

n=1

⎛⎜⎝ 1

V (xn, αa/2)

∫B(xn,αa/2)

(∫∫Γα(x)

|f(y, t)|q dμ(y)

V (y, αt)

dt

t

) pq

dμ(x)

⎞⎟⎠1p

�K,p

∥∥Aαq (f)

∥∥Lp(X)

,

completing the proof in this case. When p < q, the situation can be handled usingMinkowski’s inequality as follows. Using p/q < 1, we have(

1

V (xn, αa/2)

∫B(xn,αa/2)

∫∫K


V (y, αt)

dt

tdμ(x)

) 1q

≤(

1

V (xn, αa/2)

( ∫B(xn,αa/2)(∫∫

K


V (y, αt)

dt

t

) pq

dμ(x)

) qp) 1

q

≤ C

(1

V (xn, αa/2)

∫B(xn,αa/2)(∫∫

K


V (y, αt)

dt

t

) pq

dμ(x)

) 1p

,

8 A. Amenta

where

C = C(p, q, α,K) = maxn

(V (xn, αa/2)

1p− 1

q

).

We can then proceed as in the case where p ≥ q. �As usual, with α > 0 and p, q ∈ (0,∞), we define the tent space (quasi-)norm

of a measurable function f on X+ by

‖f‖Tp,q,α(X) :=∥∥Aα

q (f)∥∥Lp(X)

,

and the tent space T p,q,α(X) to be the (quasi-)normed vector space consisting ofall such f (defined almost everywhere) for which this quantity is finite.

Remark 3.4. One can define the tent space as either a real or complex vectorspace, according to one’s own preference. We will implicitly work in the complexsetting (so our functions will always be C-valued). Apart from complex interpo-lation, which demands that we consider complex Banach spaces, the difference isimmaterial.

Proposition 3.5. Let X be proper or doubling. For all p, q, α ∈ (0,∞), the tent spaceT p,q,α(X) is complete and contains Lq

c(X+) (the space of functions f ∈ Lq(X+)

with cylindrical support) as a dense subspace.

Proof. Let (fn)n∈N be a Cauchy sequence in T p,q,α(X). Then by Lemma 3.3, forevery cylindrical subset K ⊂ X+ the sequence (1Kfn)n∈N is Cauchy in Lq(K).We thus obtain a limit

fK := limn→∞ 1Kfn ∈ Lq(K)

for each K. If K1 and K2 are two cylindrical subsets of X+, then fK1 |K1∩K2 =fK2 |K1∩K2 , so by making use of an increasing sequence {Km}m∈N of cylindricalsubsets of X+ whose union is X+ (for example, we could take Km := B(x,m) ×(1/m,m) for some x ∈ X) we obtain a function f ∈ Lq

loc(X+) with f |Km = fKm

for each m ∈ N.7 This is our candidate limit for the sequence (fn)n∈N. To see thatf lies in T p,q,α(X), write for any m,n ∈ N

‖1Kmf‖Tp,q,α(X) �p,q ‖1Km(f − fn)‖Tp,q,α(X) + ‖1Kmfn‖Tp,q,α(X)

≤ Cp,q,α,X,m ‖f − fn‖Lq(Km) + ‖fn‖Tp,q,α(X) ,

the (p, q)-dependence in the first estimate being relevant only for p < 1 or q < 1,and the second estimate coming from Lemma 3.3. Since the sequence (fn)n∈N

converges to 1Kmf in Lq(Km) and is Cauchy in T p,q,α(X), we have that

‖1Kmf‖Tp,q,α(X) � supn∈N

‖fn‖Tp,q,α(X)

uniformly in m. Hence ‖f‖Tp,q,α(X) is finite. We now claim that for all ε > 0 there

exists m ∈ N such that for all sufficiently large n ∈ N, we have∥∥1Kcm(fn − f)

∥∥Tp,q,α(X)

≤ ε.

7We interpret ‘locally integrable on X+’ as meaning ‘integrable on all cylinders’, rather than‘integrable on all compact sets’.


Indeed, since the sequence (fn)n∈N is Cauchy in T p,q,α(X), there exists N ∈ Nsuch that for all n, n′ ≥ N we have ‖fn − fn′‖Tp,q,α(X) < ε/2. Furthermore, since

limm→∞

∥∥1Kcm(fN − f)

∥∥Tp,q,α(X)

= 0

by the Dominated Convergence Theorem, we can choose m such that∥∥1Kcm(fN − f)

∥∥Tp,q,α(X)

< ε/2.

Then for all n ≥ N ,∥∥1Kcm(fn − f)

∥∥Tp,q,α(X)

�p,q

∥∥1Kcm(fn − fN)

∥∥Tp,q,α(X)

+∥∥1Kc

m(fN − f)

∥∥Tp,q,α(X)

≤ ‖fn − fN‖Tp,q,α(X) +∥∥1Kc

m(fN − f)

∥∥Tp,q,α(X)

< ε,

proving the claim. Finally, by the previous remark, for all ε > 0 we can find msuch that for all sufficiently large n ∈ N we have

‖fn − f‖Tp,q,α(X) �p,q ‖1Km(fn − f)‖Tp,q,α(X) +∥∥1Kc

m(fn − f)

∥∥Tp,q,α(X)

< ‖1Km(fn − f)‖Tp,q,α(X) + ε

≤ C(p, q, α,X,m) ‖fn − f‖Lq(Km) + ε.

Taking the limit of both sides as n→∞, we find that limn→∞ fn = f in T p,q,α(X),and therefore T p,q,α(X) is complete. To see that Lq

c(X+) is dense in T p,q,α(X),

simply write f ∈ T p,q,α(X) as the pointwise limit

f = limn→∞1Knf.

By the Dominated Convergence Theorem, this convergence holds in T p,q,α(X).�

We note that Lemma 3.2 implies that in the case where p = q, we haveT p,p,α(X) = Lp(X+) for all α > 0.

In the same way as Lemma 3.2, we can prove the analogue of [7, Lemma 1].

Lemma 3.6 (First integration lemma). For any nonnegative measurable functionΦ on X+, with F a measurable subset of X and α > 0,∫

F

∫∫Γα(x)

Φ(y, t) dμ(y) dt dμ(x) ≤∫∫

Γα(F )

Φ(y, t)V (y, αt) dμ(y) dt.

Remark 3.7. There is one clear disadvantage of our choice of tent space norm: itis no longer clear that

‖·‖Tp,q,α(X) ≤ ‖·‖Tp,q,β(X) (4)

when α < β. In fact, this may not even be true for general non-doubling spaces.This is no great loss, since for doubling spaces we can revert to the ‘original’ tentspace norm (with a = x and b = 1) at the cost of a constant depending only onX , and for this choice of norm (4) is immediate.

10 A. Amenta

In order to define the tent spaces T∞,q,α(X), we need to introduce the oper-ator Cαq . For measurable functions f on X+, we define

Cαq (f)(x) := supB�x

(1

μ(B)

∫∫Tα(B)


) 1q

,

where the supremum is taken over all balls containing x. Since Cαq (f) is lowersemicontinuous (see Lemma A.7), Cαq (f) is measurable. We define the (quasi-)norm

‖·‖T∞,q,α(X) for functions f on X+ by

‖f‖T∞,q,α(X) :=∥∥Cαq (f)∥∥L∞(X)

,

and the tent space T∞,q,α(X) as the (quasi-)normed vector space of measurablefunctions f onX+, defined almost everywhere, for which ‖f‖T∞,q,α(X) is finite. The

proof that T∞,q,α(X) is a (quasi-)Banach space is similar to that of Proposition3.5 once we have established the following analogue of Lemma 3.3.

Lemma 3.8. Let q, α > 0, let K ⊂ X+ be cylindrical, and suppose f is a measurablefunction on X+. Then

‖f‖Lq(K) � ‖f‖T∞,q,α(X) , (5)

with implicit constant depending only on α, q, and K (but not otherwise on X).Furthermore, if X is proper or doubling, then we also have

‖1Kf‖T∞,q,α(X) � ‖f‖Lq(K) ,

again with implicit constant depending only on α, q, and K.

Proof. We use Lemma A.4. To prove the first estimate, for each ε > 0 we canchoose a ball Bε such that Tα(Bε) ⊃ K and μ(Bε) < β1(K) + ε. Then

‖f‖Lq(K) ≤∥∥1Tα(Bε)f

∥∥Lq(X+)

= μ(Bε)1q μ(Bε)

− 1q

∥∥1Tα(Bε)f∥∥Lq(X+)

≤ (β1(K) + ε)!q ‖f‖T∞,q,α(X) .

In the final line we used that μ(Bε) > 0 to conclude thatμ(Bε)

−1/q∥∥1Tα(Bε)f

∥∥Lq(X+)

is less than the essential supremum of Cαq (f). Sinceε > 0 was arbitrary, we have the first estimate. For the second estimate, assumingthat X is proper or doubling, observe that

‖1Kf‖T∞,q,α(X) ≤ supB⊂X

(1

μ(B)

∫∫Tα(B)∩K


) 1q

≤(

1

β0(K)

∫∫K

|f(y, t)|q dμ(y)dtt

) 1q

= β0(K)−1q ‖f‖Lq(K) ,

completing the proof. �


Remark 3.9. In this section we did not impose any geometric conditions on ourspace X besides our standing assumptions on the measure μ and the propernessassumption (in the absence of doubling). Thus we have defined the tent spaceT p,q,α(X) in considerable generality. However, what we have defined is a global tentspace, and so this concept may not be inherently useful when X is non-doubling.Instead, our interest is to determine precisely where geometric assumptions areneeded in the tent space theory.

3.2. Duality, the vector-valued approach, and complex interpolation

3.2.1. Midpoint results. The geometric assumption (HL) from Section 2 nowcomes into play. For r ≥ 1, we denote the Holder conjugate of r by r′ := r/(r− 1)with r′ =∞ when r = 1.

Proposition 3.10. Suppose that X is either proper or doubling, and satisfies as-sumption (HL). Then for p, q ∈ (1,∞) and α > 0, the pairing

〈f, g〉 :=∫∫

X+

f(y, t)g(y, t)dμ(y)dt

t(f ∈ T p,q,α(X), g ∈ T p′,q′,α(X))

realises T p′,q′,α(X) as the Banach space dual of T p,q,α(X), up to equivalence ofnorms.

This is proved in the same way as in [7]. We provide the details in the interestof self-containment.

Proof. We first remark that if p = q, the duality statement is a trivial consequenceof the equality T p,p,α(X) = Lp(X+). In general, suppose f ∈ T p,q,α(X) and

g ∈ T p′,q′,α(X). Then by the averaging trick and Holder’s inequality, we have

|〈f, g〉| ≤∫X

∫∫Γα(x)

|f(y, t)g(y, t)| dμ(y)

V (y, αt)

dt

tdμ(x)

≤∫X

Aαq (f)(x)Aα

q′ (g)(x) dμ(x)

≤ ‖f‖Tp,q,α(X) ‖g‖Tp′,q′,α(X) . (6)

Thus every g ∈ T p′,q′,α(X) induces a bounded linear functional on T p,q,α(X)

via the pairing 〈·, ·〉, and so T p′,q′,α(X) ⊂ (T p,q,α(X))∗. Conversely, suppose� ∈ (T p,q,α(X))∗. If K ⊂ X+ is cylindrical, then by the properness or doublingassumption, we can invoke Lemma 3.3 to show that � induces a bounded lin-ear functional �K ∈ (Lq(K))∗, which can in turn be identified with a function

gK ∈ Lq′(K). By covering X+ with an increasing sequence of cylindrical subsets,

we thus obtain a function g ∈ Lq′loc(X

+) such that g|K = gK for all cylindricalK ⊂ X+. If f ∈ Lq(X+) is cylindrically supported, then we have∫∫

X+


t=

∫∫supp f

f(y, t)gsupp f (y, t) dμ(y)dt

t

= �supp f (f) = �(f),

(7)

12 A. Amenta

recalling that f ∈ T p,q,α(X) by Lemma 3.3. Since the cylindrically supportedLq(X+) functions are dense in T p,q,α(X), the representation (7) of �(f) in termsof g is valid for all f ∈ T p,q,α(X) by dominated convergence and the inequality

(6), provided we show that g is in T p′,q′,α(X). Now suppose p < q. We will show

that g lies in T p′,q′,α(X), thus showing directly that (T p,q,α(X))∗ is contained

in T p′,q′,α(X). It suffices to show this for gK , where K ⊂ X+ is an arbitrarycylindrical subset, provided we obtain an estimate which is uniform in K. Weestimate

‖gK‖q′

Tp′,q′ ,α(X)=∥∥∥Aα

q′(gK)q′∥∥∥

Lp′/q′ (X)

by duality. Let ψ ∈ L(p′/q′)′(X) be nonnegative, with ‖ψ‖L(p′/q′)′ (X) ≤ 1. Then by

the Fubini–Tonelli theorem,∫X

Aαq′ (gK)(x)q

′ψ(x) dμ(x)

=

∫X

∫∫X+

1B(y,αt)(x)|gK(y, t)|q′ dμ(y)

V (y, αt)

dt

tψ(x) dμ(x)

=

∫ ∞

0

∫X

1

V (y, αt)

∫B(y,αt)

ψ(x) dμ(x) |gK (y, t)|q′ dμ(y) dtt

=

∫∫X+

Mαtψ(y)|gK(y, t)|q′ dμ(y) dtt,

where Ms is the averaging operator defined for y ∈ X and s > 0 by

Msψ(y) :=1

V (y, s)

∫B(y,s)

ψ(x) dμ(x).

Thus we can write formally∫X

Aαq′ (gK)(x)q

′ψ(x) dμ(x) = 〈fψ, g〉, (8)

where we define

fψ(y, t) :=

{Mαtψ(y)gK(y, t)

q′/2gK(y, t)(q

′/2)−1 when gK(y, t) �= 0,

0 when gK(y, t) = 0,

noting that gK(y, t)(q′/2)−1 is not defined when gK(y, t) = 0 and q′ < 2. However,

the equality (8) is not valid until we show that fψ lies in T p,q,α(X). To this end,estimate

Aαq (fψ) ≤

(∫∫Γα(x)

Mαtψ(y)q|gK(y, t)|q(q′−1) dμ(y)

V (y, αt)

dt

t

) 1q

≤(∫∫

Γα(x)

Mψ(x)q|gK(y, t)|q′ dμ(y)

V (y, αt)

dt

t

) 1q

=Mψ(x)Aαq′ (gK)(x)q

′/q.


Taking r such that 1/p = 1/r + 1/(p′/q′)′ and using (HL), we then have∥∥Aαq (fψ)

∥∥Lp(X)

≤∥∥∥(Mψ)Aα

q′ (gK)q′/q∥∥∥Lp(X)

≤ ‖Mψ‖L(p′/q′)′ (X)

∥∥∥Aαq′(gK)q

′/q∥∥∥Lr(X)

�X ‖ψ‖L(p′/q′)′ (X)

∥∥Aαq′ (gK)

∥∥q′/qLrq′/q(X)

≤ ∥∥Aαq′(gK)

∥∥q′/qLrq′/q(X)

.

One can show that rq′/q = p′, and so fψ is in T p,q,α(X) by Lemma 3.3. By (8),taking the supremum over all ψ under consideration, we can write

‖gK‖q′

Tp′,q′,α(X)≤ ‖�‖ ‖fψ‖Tp,q,α(X)

�X ‖�‖ ‖gK‖q′/q

Tp′,q′,α(X),

and consequently, using that ‖gK‖Tp′,q′ ,α(X) <∞,

‖gK‖Tp′,q′,α(X) �X ‖�‖ .Since this estimate is independent of K, we have shown that g ∈ T p′,q′,α(X),

and therefore that (T p,q,α(X))∗ is contained in T p′,q′,α(X). This completes theproof when p < q. To prove the statement for p > q, it suffices to show that thetent space T p′,q′,α(X) is reflexive. Thanks to the Eberlein–Smulian theorem (see[1, Corollary 1.6.4]), this is equivalent to showing that every bounded sequence

in T p′,q′,α(X) has a weakly convergent subsequence. Let {fn}n∈N be a sequence

in T p′,q′,α(X) with ‖fn‖Tp′,q′ ,α(X) ≤ 1 for all n ∈ N. Then by Lemma 3.3, for

all cylindrical K ⊂ X+ the sequence {fn}n∈N is bounded in Lq′(K), and so by

reflexivity of Lq′(K) we can find a subsequence {fnj}j∈N which converges weakly in

Lq′(K). We will show that this subsequence also converges weakly in T p′,q′,α(X).

Let � ∈ (T p′,q′,α(X))∗. Since p′ < q′, we have already shown that there exists afunction g ∈ T p,q,α(X) such that �(f) = 〈f, g〉. For every ε > 0, we can find acylindrical set Kε ⊂ X+ such that

‖g − 1Kεg‖Tp,q,α(X) ≤ ε.

Thus for all i, j ∈ N and for all ε > 0 we have

�(fni)− �(fnj )

= 〈fni − fnj ,1Kεg〉+ 〈fni − fnj , g − 1Kεg〉≤ 〈fni − fnj ,1Kεg〉+ (‖fni‖Tp′,q′ ,α(X) +

∥∥fnj

∥∥Tp′,q′,α(X)

) ‖g − 1Kεg‖Tp,q,α

≤ 〈fni − fnj ,1Kεg〉+ 2ε.

As i, j → ∞, the first term on the right-hand side above tends to 0, and sowe conclude that {fnj}n∈N converges weakly in T p′,q′,α(X). This completes theproof. �

14 A. Amenta

Remark 3.11. As mentioned earlier, property (HL) is weaker than doubling, butthis is still a strong assumption. We note that for Proposition 3.10 to hold for agiven pair (p, q), the uncentred Hardy–Littlewood maximal operator need only beof strong type ((p′/q′)′, (p′/q′)′). Since (p′/q′)′ is increasing in p and decreasing inq, the condition required on X is stronger as p→ 1 and q →∞.

Given Proposition 3.10, we can set up the vector-valued approach to tentspaces (first considered in [12]) using the method of [4]. Fix p ∈ (0,∞), q ∈ (1,∞),and α > 0. For simplicity of notation, write

Lqα(X

+) := Lq

(X+;

dμ(y)

V (y, αt)

dt

t

).

We define an operator Tα : Tp,q,α(X)→ Lp(X ;Lq

α(X+)) from the tent space into

the Lqα(X

+)-valued Lp space on X (see [9, §2] for vector-valued Lebesgue spaces)by setting

Tαf(x)(y, t) := f(y, t)1Γα(x)(y, t).

One can easily check that

‖Tαf‖Lp(X;Lqα(X+)) = ‖f‖Tp,q,α(X) ,

and so the tent space T p,q,α(X) can be identified with its image under Tα inLp(X ;Lq

α(X+)), provided that Tαf is indeed a strongly measurable function of x ∈

X . This can be shown for q ∈ (1,∞) by recourse to Pettis’ measurability theorem[9, §2.1, Theorem 2], which reduces the question to that of weak measurability of

Tαf . To prove weak measurability, suppose g ∈ Lq′α (X); then

〈Tαf(x), g〉 =∫∫

Γα(x)

f(y, t)g(y, t)dμ(y)

V (y, αt)

dt

t,

which is measurable in x by Lemma A.6. Thus Tαf is weakly measurable, andtherefore Tαf is strongly measurable as claimed.

Now assume p, q ∈ (1,∞) and consider the operator Πα, sending X+-valuedfunctions on X to C-valued functions on X+, given by

(ΠαF )(y, t) :=1

V (y, αt)

∫B(y,αt)

F (x)(y, t) dμ(x)

whenever this expression is defined. Using the duality pairing from Proposition3.10 and the duality pairing 〈〈·, ·〉〉 for vector-valued Lp spaces, for f ∈ T p,q,α(X)

and G ∈ Lp′(X ;Lq′

α (X+)) we have

〈〈Tαf,G〉〉 =∫X

∫∫X+

Tαf(x)(y, t)G(x)(y, t)dμ(y)

V (y, αt)

dt

tdμ(x)

=

∫∫X+

f(y, t)

V (y, αt)

∫X

1B(y,αt)(x)G(x)(y, t) dμ(x) dμ(y)dt

t

=

∫∫X+

f(y, t)(ΠαG)(y, t) dμ(y)dt

t

= 〈〈f,ΠαG〉〉.


Thus Πα maps Lp′(X ;Lq′

α (X+)) to T p′,q′,α(X), by virtue of being the adjoint of

Tα. Consequently, the operator Pα := TαΠα is bounded from Lp(X ;Lqα(X

+)) toitself for p, q ∈ (1,∞). A quick computation shows that ΠαTα = I, so that Pα

projects Lp(X ;Lqα(X

+)) onto Tα(Tp,q,α(X)). This shows that Tα(T

p,q,α(X)) is acomplemented subspace of Lp(X ;Lq

α(X+)). This observation leads to the basic

interpolation result for tent spaces. Here [·, ·]θ denotes the complex interpolationfunctor (see [3, Chapter 4]).

Proposition 3.12. Suppose that X is either proper or doubling, and satisfies as-sumption (HL). Then for p0, p1, q0, and q1 in (1,∞), θ ∈ [0, 1], and α > 0, wehave (up to equivalence of norms)

[T p0,q0,α(X), T p1,q1,α(X)]θ = T p,q,α(X),

where 1/p = (1− θ)/p0 + θ/p1 and 1/q = (1 − θ)/q0 + θ/q1.

Proof. Recall the identification

T r,s,α(X) ∼= TαTr,s,α(X) ⊂ Lr(X ;Ls

α(X+))

for all r ∈ (0,∞) and s ∈ (1,∞). Since

[Lp0(X ;Lq0α (X+)), Lp1(X ;Lq1

α (X+))]θ = Lp(X ; [Lq0α (X+), Lq1

α (X+)]θ)

= Lp(X ;Lqα(X

+))

applying the standard result on interpolation of complemented subspaces withcommon projections (see [19, Theorem 1.17.1.1]) yields

[T p0,q0,α(X), T p1,q1,α(X)]θ = Lp(X ;Lqα(X

+)) ∩ (T p0,q0,α(X) + T p1,q1,α(X))

= T p,q,α(X). �

Remark 3.13. Since [19, Theorem 1.17.1.1] is true for any interpolation functor(not just complex interpolation), analogues of Proposition 3.12 hold for any in-terpolation functor F for which the spaces Lp(X ;Lq

α(X+)) form an appropriate

interpolation scale. In particular, Proposition 3.12 (appropriately modified) holdsfor real interpolation.

Remark 3.14. Following the first submission of this article, the anonymous refereesuggested a more direct proof of Proposition 3.12, which avoids interpolation ofcomplemented subspaces.

Since Tα acts as an isometry both from T p0,q0,α(X) to Lp0(X ;Lq0α (X+)) and

from T p1,q1,α(X) to Lp1(X ;Lq1α (X+)), if f ∈ [T p0,q0,α(X), T p1,q1,α(X)]θ, then

‖f‖Tp,q,α(X) = ‖Tαf‖Lp(X;Lqα(X+)) ≤ ‖f‖[Tp0,q0,α(X),Tp1,q1,α(X)]θ

due to the exactness of the complex interpolation functor (and similarly for thereal interpolation functor). Hence [T p0,q0,α(X), T p1,q1,α(X)]θ ⊂ T p,q,α(X), and thereverse containment follows by duality. We have chosen to include both proofs fortheir own intrinsic interest.

16 A. Amenta

3.2.2. Endpoint results. We now consider the tent spaces

T 1,q,α(X) and T∞,q,α(X),

and their relation to the rest of the tent space scale. In this section, we prove thefollowing duality result using the method of [7].

Proposition 3.15. Suppose X is doubling, and let α > 0 and q ∈ (1,∞). Then thepairing 〈·, ·〉 of Proposition 3.10 realises T∞,q,α(X) as the Banach space dual ofT 1,q,α(X), up to equivalence of norms.

As in [7], we require a small series of definitions and lemmas to prove thisresult. We define truncated cones for x ∈ X , α, h > 0 by

Γαh(x) := Γα(x) ∩ {(y, t) ∈ X+ : t < h},

and corresponding Lusin operators for q > 0 by

Aαq (f |h)(x) :=

(∫∫Γαh(x)

|f(y, t)|q dμ(y)

V (y, αt)

dt

t

) 1q

.

One can show that Aαq (f |h) is measurable in the same way as for Aα

q (f).

Lemma 3.16. For each measurable function g on X+, each q ∈ [1,∞), and eachM > 0, define

hαg,q,M (x) := sup{h > 0 : Aα

q (g|h)(x) ≤MCαq (g)(x)}for x ∈ X. If X is doubling, then for sufficiently large M (depending on X, q, andα), whenever B ⊂ X is a ball of radius r,

μ{x ∈ B : hαg,q,M (x) ≥ r} �X,α μ(B).

Proof. Let B ⊂ X be a ball of radius r. Applying Lemmas A.5 and 3.6, thedefinition of Cαq , and doubling, we have∫

B

Aαq (g|r)(x)q dμ(x)

=

∫B

∫∫Γαr (x)

1Tα((2α+1)B)(y, t)|g(y, t)|q dμ(y)

V (y, αt)

dt

tdμ(x)

≤∫B

∫∫Γα(x)

1Tα((2α+1)B)(y, t)|g(y, t)|q dμ(y)

V (y, αt)

dt

tdμ(x)

≤∫∫

Tα((2α+1)B)

|g(y, t)|q dμ(y) dtt

≤ μ((2α+ 1)B) infx∈B

Cαq (g)(x)q

�X,α μ(B) infx∈B

Cαq (g)(x)q .


We can estimate∫B

Aαq (g|r)(x)q dμ(x)≥ (M inf

x∈BCαq (g)(x))qμ{x ∈ B : Aα

q (g|r)(x) > M infx∈B

Cαq (g)(x)},and after rearranging and combining with the previous estimate we get

M q

(μ(B)− μ{x ∈ B : Aα

q (g|r)(x) ≤M infx∈B

Cαq (g)(x)})

�X,α μ(B).

More rearranging and straightforward estimating yields

μ{x ∈ B : Aαq (g|r)(x) ≤MCαq (g)(x)} ≥ (1−M−qCX,α)μ(B).

Since hαg,q,M (x) ≥ r if and only ifAα

q (g|r)(x) ≤MCαq (g)(x) asAαq (g|h) is increasing

in h, we can rewrite this as

μ{x ∈ B : hαg,q,M (x) ≥ r} ≥ (1−M−qCX,α)μ(B).

Choosing M > C1/qX,α completes the proof. �

Corollary 3.17. With X, g, q, and α as in the statement of the previous lemma,there exists M = M(X, q, α) such that whenever Φ is a nonnegative measurablefunction on X+, we have∫∫

X+

Φ(y, t)V (y, αt) dμ(y) dt �X,α

∫X

∫∫Γαhαg,q,M

(x)/α(x)

Φ(y, t) dμ(y) dt dμ(x).

Proof. This is a straightforward application of the Fubini–Tonelli theorem alongwith the previous lemma. Taking M sufficiently large, Lemma 3.16 gives∫∫

X+

Φ(y, t)V (y, αt) dμ(y) dt

�X,α

∫∫X+

Φ(y, t)

∫{x∈B(y,αt):hα

g,q,M(x)≥αt}dμ(x) dμ(y) dt

=

∫X

∫ hαg,q,M (x)/α

0

∫B(x,αt)

Φ(y, t) dμ(y) dt dμ(x)

=

∫X

∫∫Γαhαg,q,M

(x)/α(x)

Φ(y, t) dμ(y) dt dμ(x)

as required. �

We are now ready for the proof of the main duality result.

Proof of Proposition 3.15. First suppose f ∈ T 1,q,α(X) and g ∈ T∞,q′,α(X). ByCorollary 3.17, there exists M = M(X, q, α) > 0 such that∫∫

X+

|f(y, t)||g(y, t)| dμ(y) dtt

�X,α

∫X

∫∫Γαh(x)

(x)

|f(y, t)||g(y, t)| dμ(y)

V (y, αt)

dt

tdμ(x),

18 A. Amenta

where h(x) := hαg,q′,M (x)/α. Using Holder’s inequality and the definition of h(x),

we find that ∫X

(∫∫Γαh(x)

(x)

|f(y, t)||g(y, t)| dμ(y)

V (y, αt)

dt

t

)dμ(x)

≤∫X

Aαq (f |h(x))(x)Aα

q′ (g|h(x))(x) dμ(x)

≤M

∫X

Aαq (f)(x)Cαq′ (g)(x) dμ(x)

�X,q,α ‖f‖T 1,q,α(X) ‖g‖T∞,q,α(X) .

Hence every g ∈ T∞,q′,α(X) induces a bounded linear functional on T 1,q,α(X) via

the pairing 〈f, g〉 above, and so T∞,q′,α(X) ⊂ (T 1,q,α(X))∗. Conversely, suppose� ∈ (T 1,q,α(X))∗. Then as in the proof of Proposition 3.10, from � we construct a

function g ∈ Lq′loc(X

+) such that∫∫X+


t= �(f)

for all f ∈ T 1,q,α(X) with cylindrical support. We just need to show that g is in

T∞,q′,α(X). By the definition of the T∞,q′,α(X) norm, it suffices to estimate(1

μ(B)

∫∫Tα(B)

|g(y, t)|q′ dμ(y) dtt

) 1q′

,

where B ⊂ X is an arbitrary ball. For all nonnegative ψ ∈ Lq(Tα(B)) with‖ψ‖Lq(Tα(B)) ≤ 1, using that Sα(Tα(B)) = B we have that

‖ψ‖T 1,q,α(X) =

∫B

Aαq (ψ)(x) dμ(x) ≤ μ(B)1/q

′ ‖ψ‖T q,q,α(X)

= μ(B)1/q′ ‖ψ‖Lq(X+) ≤ μ(B)1/q

′.

In particular, ψ is in T 1,q,α(X), so we can write∫∫Tα(B)

gψ dμdt

t= �(ψ).

Arguing by duality and using the above computation, we then have(1

μ(B)

∫∫Tα(B)

|g(y, t)|q′ dμ(y) dtt

)1/q′

= μ(B)−1/q′ supψ

∫∫Tα(B)

gψ dμdt

t

= μ(B)−1/q′ supψ

�(ψ)

≤ μ(B)−1/q′ ‖�‖ ‖ψ‖T 1,q,α(X)

≤ ‖�‖ ,


where the supremum is taken over all ψ described above. Now taking the supremumover all balls B ⊂ X , we find that

‖g‖T∞,q′,α(X) ≤ ‖�‖ ,which completes the proof that (T 1,q,α(X))∗ ⊂ T∞,q′,α(X). �

Once Proposition 3.15 is established, we can obtain the full scale of inter-polation using the ‘convex reduction’ argument of [4, Theorem 3] and Wolff’sreiteration theorem (see [20] and [13]).

Proposition 3.18. Suppose that X is doubling. Then for p0, p1 ∈ [1,∞] (not bothequal to ∞), q0 and q1 in (1,∞), θ ∈ [0, 1], and α > 0, we have (up to equivalenceof norms)

[T p0,q0,α(X), T p1,q1,α(X)]θ = T p,q,α(X),

where 1/p = (1− θ)/p0 + θ/p1 and 1/q = (1 − θ)/q0 + θ/q1.

Proof. First we will show that

[T 1,q0,α(X), T p1,q1,α(X)]θ ⊃ T p,q,α(X). (9)

Suppose f ∈ T p,q,α(X) is a cylindrically supported simple function. Then thereexists another cylindrically supported simple function g such that f = g2. Then

‖f‖Tp,q,α(X) = ‖g‖2T 2p,2q,α(X) ,

and so g is in T 2p,2q,α(X). By Proposition 3.12 we have the identification

T 2p,2q,α(X) = [T 2,2q0,α(X), T 2p1,2q1,α(X)]θ (10)

up to equivalence of norms, and so by the definition of the complex interpolationfunctor (see Section A.3), there exists for each ε > 0 a function

Gε ∈ F(T 2,2q0,α(X), T 2p1,2q1,α(X))

such that Gε(θ) = g and

‖Gε‖F(T 2,2q0,α(X),T 2p1,2q1,α(X) ≤ (1 + ε) ‖g‖[T 2,2q0,α(X),T 2p1,2q1,α(X)]θ

� (1 + ε) ‖g‖T 2p,2q,α(X) ,

the implicit constant coming from the norm equivalence (10). Define Fε := G2ε.

Then we have

Fε ∈ F(T 1,q0,α(X), T p1,q1,α(X)),

with

‖Fε‖F(T 1,q0,α(X),Tp1,q1,α(X)) = ‖Gε‖2F(T 2,2q0,α(X),T 2p1,2q1,α(X))

� (1 + ε)2 ‖g‖2T 2p,2q,α(X)

= (1 + ε)2 ‖f‖Tp,q,α(X) .

Therefore

‖f‖[T 1,q0,α(X),Tp1,q1,α(X)]θ� ‖f‖Tp,q,α(X) ,

20 A. Amenta

and so the inclusion (9) follows from the fact that cylindrically supported simplefunctions are dense in T p,q,α(X). By the duality theorem [3, Corollary 4.5.2] for in-terpolation (using that T p1,q1,α(X) is reflexive, the inclusion (9), and Propositions3.10 and 3.15, we have

[T p′1,q

′1,α(X), T∞,q′0,α(X)]1−θ ⊂ T p′,q′,α(X).

Therefore we have the containment

[T p0,q0,α(X), T∞,q1,α(X)]θ ⊂ T p,q,α(X). (11)

The reverse containment can be obtained from

[T 1,q0,α(X), T p1,q1,α(X)]θ ⊂ T p,q,α(X) (12)

(for p1, q0, q1 ∈ (1,∞)) by duality. The containment (12) can be obtained asin Remark 3.14, with p0 = 1 not changing the validity of this method.8 Finally,it remains to consider the case when p0 = 1 and p1 = ∞. This is covered byWolff reiteration. Set A1 = T 1,q0,α(X), A2 = T p,q,α(X), A3 = T p+1,q3,α(X), andA4 = T∞,q1,α(X) for an approprate choice of q3.

9 Then for an appropriate indexη, we have [A1, A3]θ/η = A2 and [A2, A4](η−θ)/(1−θ) = A3. Therefore by Wolffreiteration, we have [A1, A4]θ = A2; that is,[T 1,q0,α(X), T∞,q1,α(X)]θ = T p,q,α(X). This completes the proof. �

Remark 3.19. Note that doubling is not explicitly used in the above proof; it isonly required to the extent that it is needed to prove Propositions 3.10 and 3.15 (asProposition 3.12 follows from 3.10). If these propositions could be proven undersome assumptions other than doubling, then it would follow that Proposition 3.18holds under these assumptions.

Remark 3.20. The proof of [7, Lemma 5], which amounts to proving the con-tainment (9), contains a mistake which is seemingly irrepairable without resort-ing to more advanced techniques. This mistake appears on page 323, line -3,when it is stated that “A(fk) is supported in O∗

k − Ok+1” (and in particular,that A(fk) is supported in Oc

k+1). However (reverting to our notation), since

fk := 1T ((Ok)∗γ)\T ((Ok+1)∗γ)f , A12(fk) is supported on

S1(T ((Ok)∗γ) \ T ((Ok+1)

∗γ)) = (Ok)

∗γ

and we cannot conclude that A12(fk) is supported away from Ok+1. Simple one-

dimensional examples can be constructed which show that this is false in general.Hence the containment (9) is not fully proven in [7]; the first valid proof in theEuclidean case that we know of is in [4] (the full range of interpolation is notobtained in [12]).

8We thank the anonymous referee once more for this suggestion.9More precisely, we need to take 1/q3 = (1− 1/p′)/q0 + (1/p′)/q1.


3.3. Change of aperture

Under the doubling assumption, the change of aperture result can be proven with-out assuming (NI) by means of the vector-valued method. The proof is a combi-nation of the techniques of [12] and [4].

Proposition 3.21. Suppose X is doubling. For α, β ∈ (0,∞) and p, q ∈ (0,∞), thetent space (quasi-)norms ‖·‖Tp,q,α(X) and ‖·‖Tp,q,β(X) are equivalent.

Proof. First suppose p, q ∈ (1,∞). Since X is doubling, we can replace our defini-tion of Aα

q with the definition

Aαq (f)(x)

q :=

∫∫Γα(x)

|f(y, t)|q dμ(y)

V (y, t)

dt

t;

using the notation of Section 3.1, this is the definition with a = y and b = 1.Having made this change, the vector-valued approach to tent spaces (see Section3.2) transforms as follows. The tent space T p,q,α(X) now embeds isometrically intoLp(X ;Lq

1(X+)) via the operator Tα defined, as before, by

Tαf(x)(y, t) := f(y, t)1Γα(x)(y, t)

for f ∈ T p,q,α(X). The adjoint of Tα is the operator Πα, now defined by

(ΠαG)(y, t) :=1

V (y, t)

∫B(y,αt)

G(z)(y, t) dμ(z)

for G ∈ Lp(X ;Lq1(X

+)). The composition Pα := TαΠα is then a bounded projec-tion from Lp(X ;Lq

1(X+)) onto TαT

p,q,α(X), and can be written in the form

PαG(x)(y, t) =1Γα(x)(y, t)

V (y, t)

∫B(y,αt)

G(z)(y, t) dμ(z).

For f ∈ T p,q,α(X), we can easily compute

PβTαf(x)(y, t) = Tβf(x)(y, t)V (y,min(α, β)t)

V (y, t). (13)

Without loss of generality, suppose β > α. Then we obviously have

‖·‖Tp,q,α(X) �q,α,β,X ‖·‖Tp,q,β(X)

by Remark 3.7. It remains to show that

‖·‖Tp,q,β(X) �p,q,α,β,X ‖·‖Tp,q,α(X) . (14)

From (13) and doubling, for f ∈ T p,q,α(X) we have that

Tβf(x)(y, t) �X,α PβTαf(x)(y, t),

and so we can write

‖f‖Tp,q,β(X) = ‖Tβf‖Lp(X;Lq1(X

+)) �X,α ‖PβTαf‖Lp(X;Lq1(X

+))

≤ ‖Pβ‖L(Lp(X;Lq1(X

+))) ‖Tαf‖Lp(X;Lq1(X

+)) �p,q,β,X ‖f‖Tp,q,α(X)

22 A. Amenta

since Pβ is a bounded operator on Lp(X ;Lq1(X

+)). This shows (14), and completesthe proof for p, q ∈ (1,∞). Now suppose that at least one of p and q is not in(1,∞), and suppose f ∈ T p,q,α(X) is a cylindrically supported simple function.Choose an integer M such that both Mp and Mq are in (1,∞). Then there existsa cylindrically supported simple function g with gM = f . We then have

‖f‖1/MTp,q,α(X) =∥∥gM∥∥1/M

Tp,q,α(X)= ‖g‖TMp,Mq,α(X)

�p,q,α,β,X ‖g‖TMp,Mq,β(X) = ‖f‖1/MTp,q,β(X)

,

and so the result is true for cylindrically supported simple functions, with animplicit constant which does not depend on the support of such a function. Sincethe cylindrically supported simple functions are dense in T p,q,α(X), the proof iscomplete. �

Remark 3.22. Written more precisely, with p, q ∈ (0,∞) and β < 1, the inequality(14) is of the form

‖·‖Tp,q,1(X) �p,q,X sup(y,t)∈X+

(V (y, t)

V (y, βt)

)M

‖·‖Tp,q,β(X) .

where M is such that Mp,Mq ∈ (1,∞).

3.4. Relations between A and CAgain, this proposition follows from the methods of [7].

Proposition 3.23. Suppose X satisfies (HL), and suppose 0 < q < p < ∞ andα > 0. Then ∥∥Cαq (f)∥∥Lp(X)

�p,q,X

∥∥Aαq (f)

∥∥Lp(X)

.

Proof. Let B⊂X be a ball. Then by the Fubini–Tonelli theorem, using the factthat Sα(Tα(B)) = B,

1

μ(B)

∫∫Tα(B)


=1

μ(B)

∫∫Tα(B)

|f(y, t)|qV (y, αt)

∫B(y,αt)

dμ(x) dμ(y)dt

t

=1

μ(B)

∫X

∫∫Tα(B)

1B(y,αt)(x)|f(y, t)|q dμ(y)

V (y, αt)

dt

tdμ(x)

=1

μ(B)

∫B

∫∫Tα(B)

1B(x,αt)(y)|f(y, t)|q dμ(y)

V (y, αt)

dt

tdμ(x)

≤ 1

μ(B)

∫B

∫∫X+

1B(x,αt)(y)|f(y, t)|q dμ(y)

V (y, αt)

dt

tdμ(x)

=1

μ(B)

∫B

Aαq (f)(x)

q dμ(x).


Now fix x ∈ X and take the supremum of both sides of this inequality over allballs B containing x. We find that

Cαq (f)(x)q ≤M(Aαq (f)

q)(x).

Since p/q > 1, we can apply (HL) to get∥∥Cαq (f)∥∥Lp(X)≤∥∥∥M(Aα

q (f)q)1/q

∥∥∥Lp(X)

=∥∥M(Aα

q (f)q)∥∥1/qLp/q(X)

�p,q,X

∥∥Aαq (f)

q∥∥1/qLp/q(X)

=∥∥Aα

q (f)∥∥Lp(X)

as desired. �

Remark 3.24. If X is doubling, and if p, q ∈ (0,∞), then for α > 0 we also havethat ∥∥Aα

q (f)∥∥Lp(X)

�p,q,X

∥∥Cαq (f)∥∥Lp(X).

This can be proven as in [7, §6], completely analogously to the proofs above.

Appendix: Assorted lemmas and notation

A.1. Tents, cones, and shadows

Lemma A.1. Suppose A and B are subsets of X, with A open, and suppose Tα(A) ⊂Tα(B). Then A ⊂ B.

Proof. Suppose x ∈ A. Then dist(x,Ac) > 0 since A is open, and so dist(x,Ac) >αt for some t > 0. Hence (x, t) ∈ Tα(A) ⊂ Tα(B), so that dist(x,Bc) > αt > 0.Therefore x ∈ B. �

Lemma A.2. Let C ⊂ X+ be cylindrical, and suppose α > 0. Then Sα(C) isbounded.

Proof. Write C ⊂ B(x, r) × (a, b) for some x ∈ X and r, a, b > 0. Then Sα(C) ⊂Sα(B(x, r) × (a, b)), and one can easily show that

Sα(B(x, r) × (a, b)) ⊂ B(x, r + αb),

showing the boundedness of Sα(C). �

Lemma A.3. Let C ⊂ X+, and suppose α > 0. Then Tα(Sα(C)) is the minimalα-tent containing C, in the sense that Tα(S) ⊃ C for some S ⊂ X implies thatTα(Sα(C)) ⊂ Tα(S).

24 A. Amenta

Proof. A straightforward set-theoretic manipulation shows that C is contained inTα(Sα(C)). We need to show that Sα(C) is minimal with respect to this property.Suppose that S ⊂ X is such that C ⊂ Tα(S), and suppose (w, tw) is in Tα(Sα(C)).With the aim of showing that dist(w, Sc) > αtw, suppose that y ∈ Sc. ThenΓα(y) ∩ Tα(S) = ∅, and so Γα(y) ∩ C = ∅ since Tα(S) contains C. Thus y ∈Sα(C)c, and so

d(w, y) ≥ dist(w, Sα(C)c) > αtw

since (w, tw) ∈ Tα(Sα(C)). Taking an infimum over y ∈ Sc, we get that

dist(w, Sc) > αtw,

which says precisely that (w, tw) is in Tα(S). Therefore Tα(Sα(C)) ⊂ Tα(S) asdesired. �

Lemma A.4. For a cylindrical subset K ⊂ X+, define

β0(K) := infB⊂X

{μ(B) : Tα(B) ∩K �= ∅} andβ1(K) := infB⊂X

{μ(B) : Tα(B) ⊃ K},

with both infima taken over the set of balls B in X. Then β1(K) is positive, andif X is proper or doubling, then β0(K) is also positive.

Proof. We first prove that β0 := β0(K) is positive, assuming that X is proper ordoubling. Write

K ⊂ C := B(x0, r0)× [a0, b0]

for some x0 ∈ X and a0, b0, r0 > 0. If B is a ball such that Tα(B) ∩K �= ∅, thenwe must have Tα(B) ∩ C �= ∅, and so we can estimate

β0 ≥ infB⊂X

{μ(B) : Tα(B) ∩ C �= ∅}.

Note that if B = B(c(B), r(B)) is a ball with c(B) ∈ B(x0, r0), then Tα(B)∩C �= ∅ if and only if r(B) ≥ αa0. Defining

I(x) := inf{V (x, r) : r > 0, Tα(B(x, r)) ∩ C �= ∅}for x ∈ X , we thus see that I(x) = V (x, αa0) when x ∈ B(x0, r0), and so I|

B(x0,r0)

is lower semicontinuous as long as the volume function is lower semicontinuous.Now suppose B = B(y, ρ) is any ball with Tα(B) ∩ C �= ∅. Let (z, tz) be a pointin Tα(B) ∩ C. We claim that the ball

B := B

(z,

1

2(ρ− d(z, y) + αtz)

)is contained in B, centred in B(x0, r0), and is such that Tα(B) ∩ C �= ∅. The

second fact is obvious: (z, tz) ∈ C implies z ∈ B(x0, r0). For the first fact, observe


that

B ⊂ B(y, d(z, y) + (ρ− d(z, y) + αtz)/2)

= B(y, (ρ+ d(z, y) + αtz)/2)

⊂ B(y, (ρ+ (ρ− αtz) + αtz)/2)

= B(y, ρ),

since (z, tz) ∈ Tα(B) implies that d(z, y) < ρ − αtz. Finally, we have (z, tz) ∈Tα(B): since c(B) = z, we just need to show that tz < r(B)/α. Indeed, we have

r(B)

α=

1

2

(ρ− d(z, y)

α+ tz

),

and tz < (ρ− d(z, y))/α as above. The previous paragraph shows that

infx∈X

I(x) ≥ infx∈B(x0,r0)

I(x),

and so we are reduced to showing that the right-hand side of this inequality ispositive, since β0 ≥ infx∈X I(x).

If X is proper: Since B(x0, r0) is compact and I|B(x0,r0)

is lower semicontinuous,

I|B(x0,r0)attains its infimum on B(x0, r0). That is,

infx∈B(x0,r0)

I(x) = minx∈B(x0,r0)

Ix > 0, (15)

by positivity of the ball volume function.

If X is doubling: Since I(x) = V (x, αa0) when x ∈ B(x0, r0), we can write

infx∈B(x0,r0)

I(x) ≥ infx∈B(x0,r0)

V (x, ε),

where ε = min(αa0, 3r0). If x ∈ B(x0, r0), then B(x0, r0) ⊂ B(x, 2r0) ⊂B(x, 3r0), and so since 3r0/ε ≥ 1,

V (x0, r0) ≤ V (x, 3r0) = V (x, ε(3r0/ε))

�X V (x, ε).

Hence V (x, ε) �X V (x0, r0), and therefore

infx∈B(x0,r0)

V (x, ε) � V (x0, r0) > 0 (16)

as desired.

We now prove that β1 = β1(K) is positive. Recall from Lemma A.3 that if Tα(B) ⊃K, then Tα(B) ⊃ Tα(Sα(K)). Since shadows are open, Lemma A.1 tells us thatB ⊃ Sα(K). Hence μ(B) ≥ μ(Sα(K)), and so

β1 ≥ μ(Sα(K)) > 0

by positivity of the ball volume function.10 �10If Sα(K) is a ball, then β1(K) = μ(Sα(K)).

26 A. Amenta

Lemma A.5. Let B be an open ball in X of radius r. Then for all x ∈ B, thetruncated cone Γα

r (x) is contained in Tα((2α+ 1)B).

Proof. Suppose (y, t) ∈ Γαr (x) and z ∈ ((2α + 1)B)c, so that d(y, x) < αt < αr

and d(c(B), z) ≥ (2α+ 1)r. Then by the triangle inequality

d(y, z) ≥ d(c(B), z) − d(c(B), x) − d(x, y)

> (2α+ 1)r − r − αr

= αr

> αt,

so that dist(y, ((2α+ 1)B)c) > αt, which yields (y, t) ∈ Tα((2α+ 1)B). �

A.2. Measurability

We assume (X, d, μ) has the implicit assumptions from Section 2.

Lemma A.6. Let α > 0, and suppose Φ is a non-negative measurable function onX+. Then the function

g : x �→∫∫

Γα(x)

Φ(y, t) dμ(y)dt

t

is μ-measurable.

We present two proofs of this lemma: one uses an abstract measurabilityresult, while the other is elementary (and in fact stronger, proving that g is notonly measurable but lower semicontinuous).

First proof. By [16, Theorem 3.1], it suffices to show that the function

F (x, (y, t)) := 1B(y,αt)(x)Φ(y, t)

is measurable on X ×X+. For ε > 0, define

fε(x, (y, t)) :=dist(x,B(y, αt))

dist(x,B(y, αt)) + dist(x,B(y, αt+ ε)c).

Then fε(x, (y, t)) is continuous in x, and converges pointwise to 1B(y,αt)(x) asε→ 0. Hence

F (x, (y, t)) = limε→0

fε(x, (y, t))Φ(y, t) =: limε→0

Fε(x, (y, t)),

and therefore it suffices to show that each Fε(x, (y, t)) is measurable on X ×X+.Since Fε is continuous in x and measurable in (y, t), Fε is measurable on X×X+,11

and the proof is complete. �

11See [11, Theorem 1], which tells us that Fε is Lusin measurable; this implies Borel measurabilityon X ×X+.


Second proof. For all x ∈ X and ε > 0, define the vertically translated cone

Γαε (x) := {(y, t) ∈ X+ : (y, t− ε) ∈ Γα(x)} ⊂ Γα(x).

If y ∈ B(x, αε), then is it easy to show that Γαε (x) ⊂ Γα(y): indeed, if (z, t) ∈

Γαε (x), then d(z, x) < α(t− ε), and so

d(z, y) ≤ d(z, x) + d(x, y) < α(t− ε) + αε = αt.

For all x ∈ X and ε > 0, define

gε(x) :=

∫∫Γαε (x)

Φ(y, t) dμ(y)dt

t.

For each x ∈ X , as ε ↘ 0, we have gε(x) ↗ g(x) by monotone convergence. Fixλ > 0, and suppose that g(x) > λ. Then there exists ε(x) such that gε(x)(x) > λ.If y ∈ B(x, αε(x)), then by the previous paragraph we have

g(y) ≥ gε(x)(x) > λ.

Therefore g is lower semicontinuous, and thus measurable. �

Lemma A.7. Let f be a measurable function on X+, q ∈ (0,∞), and α > 0. ThenCαq (f) is lower semicontinuous.

Proof. Let λ > 0, and suppose x ∈ X is such that Cαq (f)(x) > λ. Then there existsa ball B � x such that

1

μ(B)

∫∫Tα(B)


> λq.

Hence for any z ∈ B, we have Cαq (f)(z) > λ, and so the set {x ∈ X : Cαq (f)(x) > λ}is open. �

A.3. Interpolation

Here we fix some notation involving complex interpolation.

An interpolation pair is a pair (B0, B1) of complex Banach spaces whichadmit embeddings into a single complex Hausdorff topological vector space. Tosuch a pair we can associate the Banach space B0 +B1, endowed with the norm

‖x‖B0+B1:= inf{‖x0‖B0

+ ‖x1‖B1: x0 ∈ B0, x1 ∈ B1, x = x0 + x1}.

We can then consider the space F(B0, B1) of functions f from the closed strip

S = {z ∈ C : 0 ≤ Re(z) ≤ 1}into the Banach space B0 +B1, such that

• f is analytic on the interior of S and continuous on S,• f(z) ∈ Bj whenever Re(z) = j (j ∈ {0, 1}), and• the traces fj := f |Re z=j (j ∈ {0, 1}) are continuous maps into Bj whichvanish at infinity.

28 A. Amenta

The space F(B0, B1) is a Banach space when endowed with the norm

‖f‖F(B0,B1):= max

(sup

Re z=0‖f(z)‖B0

, supRe z=1

‖f(z)‖B1

).

We define the complex interpolation space [B0, B1]θ for θ ∈ [0, 1] to be the subspaceof B0 +B1 defined by

[B0, B1]θ := {f(θ) : f ∈ F(B0, B1)}endowed with the norm

‖x‖[B0,B1]θ:= inf

f(θ)=x‖f‖F(B0,B1)

.

Acknowledgements

We thank Pierre Portal and Pascal Auscher for their comments and suggestions,particularly regarding the proofs of Lemmas 3.3 and A.6. We further thank LashiBandara, Li Chen, Mikko Kemppainen and Yi Huang for discussions on this work,as well as the participants of the Workshop in Harmonic Analysis and Geometryat the Australian National University for their interest and suggestions. Finally,we thank the referee for their detailed comments.

References

[1] F. Albiac and N.J. Kalton, Topics in Banach space theory, Graduate Texts in Math-ematics, vol. 233, Springer, New York, 2006.

[2] P. Auscher, A. McIntosh, and E. Russ, Hardy spaces of differential forms on Rie-mannian manifolds, J. Geom. Anal. 18 (2008), 192–248.

[3] J. Bergh and J. Lofstrom, Interpolation spaces, Grundlehren der mathematischenWissenschaften, vol. 223, Springer-Verlag, Berlin, 1976.

[4] A. Bernal, Some results on complex interpolation of T pq spaces, Interpolation spaces

and related topics (Ramat-Gan), Israel Mathematical Conference Proceedings, vol. 5,1992, pp. 1–10.

[5] V.I. Bogachev, Measure theory, vol. 2, Springer-Verlag, 2007.

[6] W.S. Cohn and I.E. Verbitsky, Factorization of tent spaces and Hankel operators, J.Funct. Anal. 175 (2000), 308–329.

[7] R.R. Coifman, Y. Meyer, and E.M. Stein, Some new function spaces and their ap-plications to harmonic analysis, J. Funct. Anal. 62 (1985), 304–335.

[8] R.R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains es-paces homogenes, Lecture Notes in Mathematics, vol. 242, Springer-Verlag, Berlin,1971.

[9] J. Diestel and J.J. Uhl, Jr., Vector measures, Mathematical Surveys, vol. 15, Amer-ican Mathematical Society, Providence, 1977.

[10] L. Forzani, R. Scotto, P. Sjogren, and W. Urbina, On the Lp-boundedness of thenon-centred Gaussian Hardy–Littlewood maximal function, Proc. Amer. Math. Soc.130 (2002), no. 1, 73–79.


[11] K. Gowrisankaran, Measurability of functions in product spaces, Proc. Amer. Math.Soc. 31 (1972), no. 2, 485–488.

[12] E. Harboure, J. Torrea, and B. Viviani, A vector-valued approach to tent spaces, J.Anal. Math. 56 (1991), 125–140.

[13] S. Janson, P. Nilsson, and J. Peetre, Notes on Wolff’s note on interpolation spaces,Proc. London Math. Soc. s3-48 (1984), no. 2, 283–299, with appendix by MishaZafran.

[14] M. Kemppainen, The vector-valued tent spaces T 1 and T∞, J. Austral. Math. Soc.(to appear), arxiv:1105.0261.

[15] J. Maas, J. van Neerven, and P. Portal, Conical square functions and non-tangentialmaximal functions with respect to the Gaussian measure, Publ. Mat. 55 (2011), 313–341.

[16] L. Mattner, Product measurability, parameter integrals, and a Fubini counterexample,Enseign. Math. 45 (1999), 271–279.

[17] E. Russ, The atomic decomposition for tent spaces on spaces of homogeneous type,CMA/AMSI research symposium “Asymptotic geometric analysis, harmonic analysisand related topics” (Canberra) (A. McIntosh and P. Portal, eds.), Proceedings of theCentre for Mathematics and its Applications, vol. 42, 2007, pp. 125–135.

[18] P. Sjogren, A remark on the maximal function for measures in Rn, Amer. J. Math.105 (1983), no. 5, 1231–1233.

[19] H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Company, Am-sterdam, 1978.

[20] T.H. Wolff, A note on interpolation spaces, Harmonic Analysis (Minneapolis, Minn.,1981), Lecture Notes in Mathematics, vol. 908, Springer, Berlin, 1982, pp. 199–204.

Alex AmentaMathematical Sciences InstituteAustralian National UniversityActon ACT 0200e-mail: [email protected]


Remarks on Functional Calculus forPerturbed First-order Dirac Operators

Pascal Auscher and Sebastian Stahlhut

Abstract. We make some remarks on earlier works on R-bisectoriality in Lp ofperturbed first-order differential operators by Hytonen, McIntosh and Portal.They have shown that this is equivalent to bounded holomorphic functionalcalculus in Lp for p in any open interval when suitable hypotheses are made.Hytonen and McIntosh then showed that R-bisectoriality in Lp at one valueof p can be extrapolated in a neighborhood of p. We give a different proof ofthis extrapolation and observe that the Hytonen-McIntosh proof has impacton the splitting of the space into kernel and range.

Mathematics Subject Classification (2010). Primary 47A60; Secondary 42B37,47F05.

Keywords. Differential operators with bounded measurable coefficients, ex-trapolation of norm inequalities, R-bisectorial operators, coercivity condi-tions, kernel/range decomposition.

1. Introduction

Recall that an unbounded operator A on a Banach space X is called bisectorial ofangle ω ∈ [0, π/2) if it is closed, its spectrum is contained in the closure of

Sω := {z ∈ C; | arg(±z)| < ω},and one has the resolvent estimate

‖(I + λA)−1‖L(X ) ≤ Cω′ ∀ λ /∈ Sω′ , ∀ ω′ > ω.

Assuming reflexivity of X , this implies that the domain is dense and also the factthat the null space and the closure of the range split. More precisely, we say that

the operator A kernel/range decomposes if X = N(A) ⊕ R(A) (⊕ means that thesum is topological). Here N(A) denotes the kernel or null space and R(A) its range,while the domain is denoted by D(A). Bisectoriality in a reflexive space is stableunder taking adjoints.


32 P. Auscher and S. Stahlhut

For any bisectorial operator, one can define a calculus of bounded operatorsby the Cauchy integral formula,

ψ(A) :=1

2πi

∫∂Sω′

ψ(λ)

(I − 1

λA)−1

dλ

λ,

ψ ∈ Ψ(Sω′′) :={φ ∈ H∞(Sω′′) : φ ∈ O

(inf(|z|, |z−1|)α), α > 0

},

with ω′′ > ω′ > ω and where H∞(Sω′′) is the space of bounded holomorphicfunctions in Sω′′ . If this calculus may be boundedly extended to all ψ ∈ H∞(Sω′′)for all ω′′ > ω, then A is said to have an H∞-calculus of angle ω.

Assume X = Lq of some σ-finite measure space and q ∈ (1,∞). A closedoperator A is called R-bisectorial of angle ω if its spectrum is contained in Sw andfor all ω′ > ω, there exists a constant C > 0 such that∥∥∥∥∥∥∥

⎛⎝ k∑j=1

|(I + λjA)−1uj|2⎞⎠1/2

∥∥∥∥∥∥∥q

≤ C

∥∥∥∥∥∥∥⎛⎝ k∑

j=1

|uj|2⎞⎠1/2

∥∥∥∥∥∥∥q

for all k ∈ N, λ1, . . . , λk /∈ Sω′ and u1, . . . , uk ∈ Lq. This is the so-called R-boundedness criterion applied to the resolvent family. Note that the definitionimplies that A is bisectorial. This notion can be defined on any Banach space butwe do not need this here.

In [10] and [11], the equivalence between bounded H∞-calculus and R-bi-sectoriality is studied for some perturbed first-order Hodge–Dirac and Dirac typebisectorial operators in Lp spaces (earlier work on such operators appears in [1]).For general bisectorial operators in subspaces of Lp, p ∈ (1,∞), it is known thatthe former implies the latter [14, Theorem 5.3], but the converse is not known.For these first-order operators, the converse holds provided the R-bisectorialityin Lp is assumed for all p in a given open subinterval of (1,∞), not just for onefixed value of p ∈ (1,∞). Subsequently, in [9], the R-bisectoriality in Lp for thesefirst-order operators is shown to be stable under perturbation of p, allowing toapply [10] and [11] and complete the study. The main part of the argument usesan extrapolation “a la” Calderon and Zygmund, by real methods. Here, we wishto observe that this part can be obtained with an extrapolation “a la” Sneıbergusing complex function theory. Nevertheless, the argument in [9] is useful to obtainfurther characterizations of R-bisectoriality in Lp in terms of kernel/range decom-position. Indeed, we shall see that for the first-order operators in Lp consideredin [9], this property remains true by perturbation of p in the same interval as forperturbation of R-bisectoriality.

Our plan is to first review properties of perturbed Dirac type operators atsome abstract level of generality. Then we consider the first-order differential op-erators of [11, 9]. We next show the Sneıberg extrapolation for (R-)bisectoriality

Functional Calculus for Perturbed First-order Dirac Operators 33

of such operators and conclude for the equivalence of R-bisectoriality and H∞-calculus. We then show that H∞-calculus, R-bisectoriality, bisectoriality hold si-multaneously to kernel/range decomposition on a certain open interval. We in-terpret this with the motivating example coming from a second-order differentialoperator in divergence form, showing that this interval agrees with an intervalstudied in [2].

2. Abstract results

In this section, we assume without mentioning the following: X is a reflexive com-plex Banach space. The duality between X and its dual X ∗ is denoted 〈u∗, u〉 andis anti-linear in u∗ and linear in u. Next, D is a closed, densely defined operator onX and B is a bounded operator on X . We state a first proposition on propertiesof BD,DB and their duals under various hypotheses.

Proposition 2.1.

1. BD with D(BD) = D(D) is densely defined. Its adjoint (BD)∗ is closed, andD((BD)∗) = {u ∈ X ; B∗u ∈ D(D∗)} = D(D∗B∗) with (BD)∗ = D∗B∗.

2. Assume that ‖Bu‖ � ‖u‖ for all u ∈ R(D). Then,

(i) B|R(D) : R(D)→ R(BD) is an isomorphism.

(ii) BD and D∗B∗ are both densely defined and closed.(iii) DB|R(D) and BD|R(BD) are similar under conjugation by B|R(D).

3. Assume that ‖Bu‖ � ‖u‖ for all u ∈ R(D) and X = N(D) ⊕ R(D). ThenN(D) = N(BD).

4. Assume that ‖Bu‖ � ‖u‖ for all u ∈ R(D) and X = N(D)⊕ R(BD). Then,

(i) X = N(DB)⊕ R(D).(ii) R(DB) = R(D).

5. Assume that ‖Bu‖ � ‖u‖ for all u ∈ R(D) and X = N(D)⊕ R(BD). Then,

(i) X = N(D∗B∗)⊕ R(D∗).(ii)

(R(BD)

)∗= R(D∗) in the duality 〈 , 〉, with comparable norms.

(iii) ‖B∗u∗‖ � ‖u∗‖ for all u∗ ∈ R(D∗), hence B∗|R(D∗) : R(D∗)→ R(B∗D∗)

is an isomorphism.(iv) (DB)∗ = B∗D∗.(v) D∗B∗|R(D∗) and B∗D∗|R(B∗D∗) are similar under conjugation by

B∗|R(D∗).

(vi) R(B∗D∗) =(R(D)

)∗in the duality 〈 , 〉, with comparable norms.

(vii) D∗B∗|R(D∗) is the adjoint of BD|R(BD) in the duality 〈 , 〉.(viii) B∗D∗|R(B∗D∗) is the adjoint to DB|R(D) in the duality 〈 , 〉.

Proof. We skip the elementary proofs of (1) and (2) except for (2iii). See theproof of [8, Lemma 4.1] where this is explicitly stated on a Hilbert space. Thereflexivity of X is used to deduce that D∗B∗ = (BD)∗ is densely defined. We next


show (2iii). Note that R(D) is an invariant subspace for DB. Let β = B|R(D)

. If

u ∈ D(BD|R(BD)) = R(BD) ∩ D(BD) = R(BD) ∩ D(D), then β−1u ∈ R(D) ∩D(Dβ) = R(D) ∩ D(DB) = D(DB|R(D)) and

BDu = βDu = β(DB)(β−1u).

We now prove (3). Clearly N(D) ⊂ N(BD). Conversely, let u ∈ N(BD). From

X = N(D) ⊕ R(D) write u = v + w with v ∈ N(D) and w ∈ R(D). It follows that

Du = Dw and 0 = BDu = BDw. As B|R(D) : R(D)→ R(BD) is an isomorphism,

we have w = 0. Hence, u = v ∈ N(D).

We next prove (4). We know that DB is closed. Its null space is N(DB) ={u ∈ X ; Bu ∈ N(D)}.

Let us first show (i), namely that X = N(DB)⊕R(D). As X = N(D)⊕R(BD)

by assumption, the projection P1 on R(BD) along N(D) is bounded on X . Let

u ∈ X . As P1Bu ∈ R(BD), there exists v ∈ R(D) such that P1Bu = Bv and‖v‖ � ‖Bv‖ = ‖P1Bu‖ � ‖u‖. Since Bu = (I − P1)Bu+ P1Bu and (I − P1)Bu ∈N(D), we have B(u − v) ∈ N(D), that is u − v ∈ N(DB). It follows that u =

u− v + v ∈ N(DB) + R(D) with ‖v‖+ ‖u− v‖ � ‖u‖.Next, we see that R(DB) = R(D). Indeed, the inclusion R(DB) ⊆ R(D) is

trivial. For the other direction, if v ∈ R(D), then one can find u ∈ D(D) such that

v = Du. Using X = N(D) ⊕ R(BD), one can select u ∈ R(BD) = BR(D) and

write u = Bw with w ∈ R(D). Hence v = DBw ∈ R(DB).

We turn to the proof of (5). Item (i) is proved as Lemma 6.2 in [11]. To see (ii),

we observe that if u∗ ∈ R(D∗), then R(BD) � u �→ 〈u∗, u〉 is a continuous linear

functional. Conversely, if � ∈ (R(BD))∗, then by the Hahn–Banach theorem, there

is u∗ ∈ X ∗ such that �(u) = 〈u∗, u〉 for all u ∈ R(BD). Write u∗ = v∗ + w∗ with

v∗ ∈ N(D∗B∗) and w∗ ∈ R(D∗) by (i). Since 〈v∗, u〉 = 0 for all u ∈ R(BD), we

have �(u) = 〈w∗, u〉 for all u ∈ R(BD) with w∗ ∈ R(D∗).To see (iii), consider again β = B|R(D). Let u

∗ ∈ R(D∗), u ∈ R(D). Then

〈B∗u∗, u〉 = 〈u∗, Bu〉 = 〈u∗, βu〉.

Using (ii), we have proved B∗|R(D∗) = β∗ and the conclusion follows.

To see item (iv), we remark that combining (iii) and item (2) applied toB∗D∗, we have (B∗D∗)∗ = DB, hence (DB)∗ = B∗D∗ by reflexivity.

Item (v) follows from item (iii) as for item (2iii).

Item (vi) follows from R(D) = β−1R(BD) with β as above:(R(D)

)∗=

β∗(

R(BD))∗

and we conclude using item (ii) and B∗|R(D∗) = β∗.

Item (vii) follows from the dualities (BD)∗ = D∗B∗ and(R(BD)

)∗= R(D∗).


To prove item (viii), we recall that DB|R(D)

= β−1BD|R(BD)

β. Thus using

what precedes,(DB|R(D)

)∗= β∗(BD|R(BD)

)∗(β∗)−1 = B∗(D∗B∗|R(D∗))(β

∗)−1

= B∗D∗|R(B∗D∗). �

Remark 2.2. Note that the property ‖Bu‖ � ‖u‖ for all u ∈ R(D) alone does not

seem to imply ‖B∗u∗‖ � ‖u∗‖ for all u∗ ∈ R(D∗). Hence the situation for BD andB∗D∗ is not completely symmetric without further hypotheses.

Here is an easy way to check the assumptions above from kernel/range de-composition assumptions.

Corollary 2.3. Assume that ‖Bu‖ � ‖u‖ for all u ∈ R(D). If D and BD ker-

nel/range decompose, then X = N(D)⊕R(BD). In particular, this holds if D andBD are bisectorial.

Proof. By Proposition 2.1, (3), N(D) = N(BD). We conclude from X = N(BD)⊕R(BD). �

Corollary 2.4. Assume that ‖Bu‖ � ‖u‖ for all u ∈ R(D) and that D kernel/rangedecomposes. If BD kernel/range decomposes, so does DB. If BD is bisectorial,so is DB, with the same angle as BD. The same holds if R-bisectorial replacesbisectorial everywhere when X = Lp.

Proof. The statement about kernel/range decomposition is a consequence of Corol-lary 2.3 and Proposition 2.1, item (4). Assume next that BD is bisectorial andlet us show that DB is bisectorial. By Proposition 2.1, item (2), DB|R(D) and

BD|R(BD) are similar, thus DB|R(D) is bisectorial. Trivially DB|N(DB) = 0 is also

bisectorial. As X = N(DB)⊕R(D) by Corollary 2.3 and Proposition 2.1, item (4),we conclude that DB is bisectorial in X .

The proof for R-bisectoriality is similar. �

Remark 2.5. The converse, DB (R-)bisectorial implies BD (R-)bisectorial, seemsunclear under the above assumptions on B and D, even if X is reflexive which weassumed. So it appears that the theory is not completely symmetric for BD andfor DB under such assumptions.

Corollary 2.6. Assume that D kernel/range decomposes. The following are equiv-alent:

1. ‖Bu‖ � ‖u‖ for all u ∈ R(D) and BD bisectorial in X .

2. ‖B∗u∗‖ � ‖u∗‖ for all u∗ ∈ R(D∗) and B∗D∗ bisectorial in X ∗.

Moreover the angles are the same. If either of them holds, then DB and D∗B∗ arealso bisectorial, with the same angle. The same holds with R-bisectorial replacingbisectorial everywhere if X is an Lp space with σ-finite measure and 1 < p <∞.


Proof. It is enough to assume (1) by symmetry (recall that we assume X reflexive).

That ‖B∗u∗‖ � ‖u∗‖ for all u∗ ∈ R(D∗) follows from Corollary 2.3 and Proposition2.1, item (5). Next, as B∗D∗ = (DB)∗ by Proposition 2.1, item (5), and as DBis bisectorial by Corollary 2.4, B∗D∗ is also bisectorial by general theory. Thisproves the equivalence. Checking details, one sees that the angles are the same.Bisectoriality of DB and D∗B∗ are already used in the proofs. The proof is thesame for R-bisectoriality, which is stable under taking adjoints on reflexive Lp

space with σ-finite measure (see [15, Corollary 2.11]). �

3. First-order constant coefficients differential systems

Assume now that D is a first-order differential operator on Rn acting on functionsvalued in CN whose symbol satisfies the conditions (D0), (D1) and (D2) in [9]. Wedo not assume that D is self-adjoint. Let 1 < q <∞ and Dq(D) = {u ∈ Lq ; Du ∈Lq} with Lq := Lq(Rn;CN ) and Dq = D on Dq(D). We keep using the notationD instead of Dq for simplicity. The followings properties have been shown in [11].

1. D is a R-bisectorial operator with H∞-calculus in Lq.2. Lq = Nq(D)⊕ Rq(D).

3. Nq(D) and Rq(D), 1 < q <∞, are complex interpolation families.4. D has the coercivity condition

‖∇u‖q � ‖Du‖q for all u ∈ Dq(D) ∩ Rq(D) ⊂W 1,q.

Here, we use the notation ∇u for ∇⊗ u.5. The same properties hold for D∗.

Let us add one more property.

Proposition 3.1. Let t > 0. The spaces Dq(D), 1 < q <∞, equipped with the norm|||f |||q,t := ‖f‖q + t‖Df‖q, form a complex interpolation family. The same holdsfor D∗.

Proof. Since D is bisectorial in Lq, we have ‖(I + itD)−1u‖q ≤ C‖u‖q with C in-dependent of t. [To be precise, we should write Dq for q and use that the resolventsare compatible for different values of q, that is, the resolvents for different q agreeon the intersection of the Lq’s.] Thus (I + itD)−1 : (Lq, ‖ ‖q) → (Dq(D), ||| |||q,t)is an isomorphism with uniform bounds with respect to t:

‖u‖q ≤ ‖(I + itD)−1u‖q + t‖D(I + itD)−1u‖q ≤ (2C + 1)‖u‖q.The conclusion follows by the fonctoriality of complex interpolation. �

4. Perturbed first-order differential systems

Let B ∈ L∞(Rn;L(CN )). Identified with the operator of multiplication by B(x),B ∈ L(Lq) for all q. Its adjoint B∗ has the same property. With D as before,


introduce the set of coercivity of B

I(BD) = {q ∈ (1,∞) ; ‖Bu‖q � ‖u‖q for all u ∈ Rq(D)}.By density, we may replace Rq(D) by its closure. For q ∈ I(BD), B|Rq(D) :

Rq(D)→ Rq(BD) is an isomorphism. Let

bq = inf

(‖Bu‖q‖u‖q ; u ∈ Rq(D), u �= 0

)> 0.

Lemma 4.1. The set I(BD) of coercivity of B is open.

Proof. We have for all 1 < q < ∞, ‖Bu‖q ≤ ‖B‖∞‖u‖q. Thus, the bounded map

B : Rq(D)→ Lq is bounded below by bq for each q ∈ I(BD). Using that Rq(D) and

Lq are complex interpolation families, the result follows from a result of Sneıberg[16] (see also Kalton–Mitrea [12]). �Remark 4.2. If B is invertible in L∞(Rn;L(CN )), then B is invertible in L(Lq) andits inverse is the operator of multiplication by B−1. In this case, I(BD) = (1,∞).

For further use, let us recall the statement of Sneıberg (concerning lowerbound) and Kalton–Mitrea (concerning invertibility even in the quasi-Banachcase).

Proposition 4.3. Let (Xs) and (Ys) be two complex interpolation families of Banachspaces for 0 < s < 1. Let T be an operator with C = sup0<s<1 ‖T ‖L(Xs,Ys) < ∞.Assume that for s0 ∈ (0, 1) and δ > 0, ‖Tu‖Ys0

≥ δ‖u‖Xs0for all u ∈ Xs0 .

Then, there is an interval J around s0 whose length is bounded below by a numberdepending on C, δ, s0 on which ‖Tu‖Ys ≥ δ

2‖u‖Xs for all u ∈ Xs and all s ∈ J .If, moreover, T is invertible in L(Xs0 , Ys0) with lower bound δ then there is aninterval J whose length is bounded below by a number depending on C, δ, s0 onwhich T is invertible with ‖Tu‖Ys ≥ δ

2‖u‖Xs for all u ∈ Xs and all s ∈ J .

Our point is that the lower bound on the size of J is universal for complexfamilies. See Proposition 4.5 below. Define two more sets related to the opera-tor BD:

B(BD) = {q ∈ I(BD) ; BD bisectorial in Lq}R(BD) = {q ∈ I(BD) ; BD R−bisectorial in Lq}

Note that these are subsets of I(BD). We can define the analogous sets for B∗D∗.

Proposition 4.4. Let 1 < p < ∞. Then p ∈ R(BD) (resp. B(BD)) if and only ifp′ ∈ R(B∗D∗) (resp. B(B∗D∗)).

Proof. This is Corollary 2.6. �Our next results are the new observation of this paper, simplifying the ap-

proach of [9].

Proposition 4.5. The sets B(BD) and R(BD) are open.


Proof. Let us consider the openness of B(BD) first. We know that for all p ∈I(BD), BD is densely defined and closed on Lp from Proposition 2.1, item (2).Fix q ∈ B(BD). Let ω be the angle of bisectoriality in Lq and ω < μ < π/2. LetCμ = supλ/∈Sμ

‖(I+λBD)−1‖L(Lq). Fix λ /∈ Sμ. Then ‖λBD(I+λBD)−1‖L(Lq) ≤Cμ + 1. Thus for all u ∈ Dq(D),

‖(I + λBD)u‖q ≥ (2Cμ)−1‖u‖q + (2Cμ + 2)−1|λ|‖BDu‖q

≥ (2Cμ)−1‖u‖q + (2Cμ + 2)−1bq|λ|‖Du‖q

≥ δ|||u|||q,|λ|with δ = inf((2Cμ)

−1, (2Cμ + 2)−1bq) > 0. Also

‖(I + λBD)u‖q ≤ C|||u|||q,|λ|with C = sup(1, ‖B‖∞). Applying Proposition 4.3 thanks to Proposition 3.1, weobtain an open interval J around q contained in I(BD) such that for all λ /∈ Sμ

and p ∈ J , (I + λBD)−1 is bounded on Lp with bound 2/δ.The proof for perturbation of R-bisectoriality is basically the same, with Cμ

being the R-bound of (I + λBD)−1, that is the best constant in the inequality∥∥∥∥∥∥∥⎛⎝ k∑

j=1

|(I + λjBD)−1uj|2⎞⎠1/2

∥∥∥∥∥∥∥q

≤ C

∥∥∥∥∥∥∥⎛⎝ k∑

j=1

|uj|2⎞⎠1/2

∥∥∥∥∥∥∥q

for all k ∈ N, λ1, . . . , λk /∈ Sμ and u1, . . . , uk ∈ Lq. One works in the sumsLq⊕· · ·⊕Lq equipped with the norm of the right-hand side and Dq(D)⊕· · ·⊕Dq(D)equipped with ∥∥∥∥∥∥∥

⎛⎝ k∑j=1

|uj|2⎞⎠1/2

∥∥∥∥∥∥∥q

+

∥∥∥∥∥∥∥⎛⎝ k∑

j=1

|λj |2|Duj |2⎞⎠1/2

∥∥∥∥∥∥∥q

.

To obtain the R-lower bound (replacing δ), one linearizes using the Kahane-Kintchine inequality with the Rademacher functions⎛⎝ k∑

j=1

|uj|2⎞⎠1/2

∼⎛⎝∫ 1

0

∣∣∣∣ k∑j=1

rj(t)uj

∣∣∣∣q dt⎞⎠1/q

,

valid for any q ∈ (1,∞) (see, for example, [15] and follow the argument above).Details are left to the reader. �

Remark 4.6. The sets B(BD) and R(BD) may not be intervals. They are (pos-sibly empty) intervals when restricted to each connected component of I(BD)because (R-)bisectoriality interpolates in Lp scales. See [13, Corollary 3.9] for aproof concerning R-sectoriality. In particular, if I(BD) = (1,∞) these sets are(possibly empty) open intervals.


Theorem 4.7. For p ∈ I(BD), the following assertions are equivalent:

(i) BD is R-bisectorial in Lp.(ii) BD is bisectorial and has an H∞-calculus in Lp.

Moreover, the angles in (i) and (ii) are the same. Furthermore, if one of the items

holds, then they hold as well for DB, and also for B∗D∗ and D∗B∗ in Lp′.

Proof. The implication (ii)⇒ (i) is a general fact proved in [14]. Assume converselythat (i) holds. Then, there is an interval (p1, p2) around p for which (i) holds withthe same angle by Proposition 4.5. Note also that (2) and (3) of Proposition 2.1

apply with X = Lq for each q ∈ (p1, p2). Hence, B∗ has a lower bound on Rq′(D∗).

We may apply Corollary 8.17 of [11], which states that D∗B∗ satisfies (ii) on Lq′ .By duality, we conclude that BD satisfies (ii) in Lq.

The last part of the statement now follows from Corollary 2.6. �

Remark 4.8. As p ∈ R(BD) if and only if p′ ∈ R(B∗D∗), Proposition 4.5 and The-orem 4.7 can be compared to Theorem 2.5 of [9] for the stability of R-bisectorialityand the equivalence withH∞-calculus. The argument here is much easier and fairlygeneral once we have Proposition 3.1. However, the argument in [9] is useful sinceit contains a quantitative estimate on how far one can move from p. We come backto this below. Recall that the motivation of [9, Theorem 2.5], thus reproved here,is to complete the theory developed in [11].

5. Relation to kernel/range decomposition

For a closed unbounded operator A on a reflexive Banach space X , recall that Akernel/range decomposes if X = N(A)⊕ R(A) and that it is implied by bisectori-ality. The converse is not true (the shift on �2(Z) is invertible, so the kernel/rangedecomposition is trivial, but it is not bisectorial as its spectrum is the unit cir-cle). For the class of BD operators in the previous section, we shall show that aconverse holds.

For a set A ⊆ (1,∞), let A′ = {q′ ; q ∈ A}.Consider D and B as in Section 4. Recall that p ∈ R(BD) if and only if

p′ ∈ R(B∗D∗). That is, R(B∗D∗)′ = R(BD). Recall also that R(BD) ⊆ I(BD),hence R(BD) ⊆ I(B∗D∗)′ as well.

Assume p0 ∈ R(BD) and let I0 be the connected component of I(BD) ∩I(B∗D∗)′ that contains p0. It is an open interval.

Let

B0(BD) = {q ∈ I0 ; BD bisectorial in Lq}R0(BD) = {q ∈ I0 ; BD R−bisectorial in Lq}H0(BD) = {q ∈ I0 ; BD bisectorial in Lq with H∞−calculus}S0(BD) = ccp0{q ∈ I0 ; BD kernel/range decomposes in Lq}

The notation ccp0 means the connected component that contains p0.


Theorem 5.1. Assume R0(BD) is not empty. Then, the four sets above are equalopen intervals.

Proof. It is clear that H0(BD) ⊆ R0(BD) ⊆ B0(BD). By Proposition 4.5 and thediscussion in Remark 4.6, R0(BD) and B0(BD) are open subintervals of I0. ByTheorem 4.7, we also know that H0(BD) = R0(BD).

As bisectoriality implies kernel/range decomposition, B0(BD) is containedin the set {q ∈ I0 ; BD kernel/range decomposes in Lq}. As B0(BD) contains p0,we have B0(BD) ⊆ S0(BD). Thus it remains to show that S0(BD) ⊆ R0(BD),which is done in the next results. �

For 1 < p <∞, let p∗, p∗ be the upper and lower Sobolev exponents: p∗ = npn−p

if p < n and p∗ =∞ if p ≥ n, while p∗ = npn+p .

Lemma 5.2. Let p ∈ R0(BD). Then BD|Rq(BD) is R-bisectorial (in Rq(BD)) for

q ∈ I0 ∩ (p∗, p∗).

Proof. The (non-trivial) argument to extrapolateR-bisectoriality at p toR-bisecto-riality at any q ∈ I0 ∩ (p∗, p) is exactly what is proved in Sections 3 and 4 of [9],taken away the arguments related to kernel/range decomposition which are notassumed here. We next provide the argument for q ∈ I0 ∩ (p, p∗). By symmetryof the assumptions, we obtain that B∗D∗|Rq′ (B∗D∗) is R-bisectorial. By duality

of R-bisectoriality in subspaces of reflexive Lebesgue spaces and Proposition 2.1,item (5), DB|Rq(D) is R-bisectorial. By Proposition 2.1, item (2), this implies that

BD|Rq(BD)

is R-bisectorial. �

Corollary 5.3. S0(BD) ⊆ R0(BD).

Proof. The set {q ∈ I0 ; BD kernel/range decomposes in Lq} is open (this wasobserved in [9], again as a consequence of Sneıberg’s result). Thus, as a connectedcomponent, S0(BD) is an open interval. Write R0(BD) = (r−, r+) and S0(BD) =(s−, s+) and recall that (r−, r+) ⊆ (s−, s+). Assume s− < r−. One can find p, qwith q ∈ I0 ∩ (p∗, p) and s− < q ≤ r− < p < r+. By the previous lemma, we have

that BD|Rq(BD) is R-bisectorial in Rq(BD). Also BD|Nq(BD) = 0 is R-bisectorial.

As q ∈ S0(BD) = (s−, s+), we have Lq = Rq(BD) ⊕ Nq(BD). Hence, BD isR-bisectorial in Lq. This is a contradiction as q /∈ R0(BD). Thus r− ≤ s−. Theargument to obtain s+ ≤ r+ is similar. �

Remark 5.4. It was observed and heavily used in [9] that for a given p, Lp bound-edness of the resolvent of BD self-improves to off-diagonal estimates. Thus, theset of those p ∈ I0 for which one has such estimates in addition to bisectorialityin Lp is equal to B0(BD) as well.


6. Self-adjoint D and accretive B

The operators D and B are still as in Section 4. In addition, assume that D isself-adjoint on L2 and that B is strictly accretive in R2(D), that is for some κ > 0,

Re〈u,Bu〉 ≥ κ‖u‖22, ∀u ∈ R2(D).

Then, B and B∗ have lower bound κ on R2(D) and R2(D∗) = R2(D). In thiscase, BD and DB = (B∗D)∗ (replacing B by B∗) are bisectorial operators in L2.Moreover, using that B is multiplication and D a coercive first-order differentialoperator with constant coefficients, [8, Theorem 3.1] (see [4] for a direct proof)shows that BD and DB have H∞-calculus in L2. Thus, Theorem 5.1 applies andone has the

Theorem 6.1. There exists an open interval I(BD)=(q−(BD),q+(BD))⊆ (1,∞),containing 2, with the following dichotomy: H∞-calculus, R-bisectoriality, bisec-toriality and kernel/range decomposition hold for BD in Lp if p ∈ I(BD) andall fail if p = q±(BD) except, may be, when q±(BD) is already an endpointof the set I0 of Section 5 with p0 = 2. The same property holds for DB withI(DB) = I(BD). The same property holds for B∗D and DB∗ in the dual intervalI(DB∗) = I(B∗D) = (I(BD))′.

In applications, one tries to find an interval of p for bisectoriality, which isthe easiest property to check.

The example that motivated the study of perturbed Dirac operators is thefollowing setup, introduced in [7] and exploited in [8] to reprove the Kato squareroot theorem obtained in [5] for second-order operators and in [6] for systems. LetA ∈ L∞(Rn;L(Cm ⊗ Cn)) satisfy∫

Rn

∇u(x) · A(x)∇u(x)dx � ‖∇u‖22, (1)

for all u ∈W 1,2(Rn;Cm). Then BD, with B =

(I 00 A

)andD =

(0 −div∇ 0

), has

a bounded H∞-calculus in Lp(Rn;Cm⊕ [Cm⊗Cn]) for all p ∈ (q−(BD), q+(BD)),with angle at most equal to the accretivity angle of A.

Let us finish with the interpretation of the kernel/range decomposition in

this particular example. As BD =

(0 −div

A∇ 0

), we see that

Np(BD) = {u = (0, g) ∈ Lp(Rn;Cm ⊕ [Cm ⊗ Cn]) ; divg = 0}and

Rp(BD) = {u = (f, g) ∈ Lp(Rn;Cm ⊕ [Cm ⊗ Cn]) ; g = A∇h, h ∈ W 1,p(Rn;Cm)},where W 1,p(Rn;Cm) is the homogeneous Sobolev space. Thus,

Lp(Rn;Cm ⊕ [Cm ⊗ Cn]) = Np(BD)⊕ Rp(BD) (2)


is equivalent to the Hodge splitting adapted to A for vector fields

Lp(Rn;Cm ⊗ Cn) = Np(div)⊕A∇W 1,p(Rn;Cm). (3)

Writing details for DB instead we arrive at the equivalence between

Lp(Rn;Cm ⊕ [Cm ⊗ Cn]) = Np(DB)⊕ Rp(DB) (4)

and a second Hodge splitting adapted to A for vector fields

Lp(Rn;Cm ⊗ Cn) = Np(divA)⊕∇W 1,p(Rn;Cm). (5)

As q±(BD) = q±(DB), we obtain that (3) and (5) hold for p ∈ (q−(BD), q+(BD))and fail at the endpoints.

Let L = −divA∇. It was shown in [2, Corollary 4.24] that (5) holds if andonly if p ∈ (q+(L

∗)′, q+(L)), where the number q+(L) is defined as the supremumof those p > 2 for which t1/2∇e−tL is uniformly bounded on Lp for t > 0 (Strictlyspeaking, this is done when m = 1, and Section 7.2 in [2] gives an account ofthe extension to systems). As a consequence, we have shown that q+(BD) =q+(DB) = q+(L) and q−(BD) = q−(DB) = q+(L

∗)′.In the previous example, the matrix B is block-diagonal. If B is a full matrix,

then DB and BD happen to be in relation with a second-order system in Rn+1+

as first shown in [3]. Their study brought new information to the boundary valueproblems associated to such systems when p = 2. Details when p �= 2 will appearin the forthcoming PhD thesis of the second author.

Acknowledgment

This work is part of the forthcoming PhD thesis of the second author. The firstauthor thanks the organizers of the IWOTA 2012 conference in Sydney for a stim-ulating environment. Both authors were partially supported by the ANR project“Harmonic Analysis at its Boundaries”, ANR-12-BS01-0013-01.

References

[1] Sergey S. Ajiev. Extrapolation of the functional calculus of generalized Dirac opera-tors and related embedding and Littlewood–Paley-type theorems. I. J. Aust. Math.Soc., 83(3):297–326, 2007.

[2] Pascal Auscher. On necessary and sufficient conditions for Lp-estimates of Riesztransforms associated to elliptic operators on Rn and related estimates. Mem. Amer.Math. Soc., 186(871):xviii+75, 2007.

[3] Pascal Auscher, Andreas Axelsson, and Alan McIntosh. Solvability of elliptic systemswith square integrable boundary data. Ark. Mat. 48 (2010), 253–287.

[4] Pascal Auscher, Andreas Axelsson, and Alan McIntosh. On a quadratic estimaterelated to the Kato conjecture and boundary value problems. Contemp. Math.,505:105–129, 2010.

[5] Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and PhilippeTchamitchian. The solution of the Kato square root problem for second order el-liptic operators on Rn. Ann. of Math. (2) 156, 2 (2002), 633–654.


[6] Pascal Auscher, Steve Hofmann, Alan McIntosh, and Philippe Tchamitchian. TheKato square root problem for higher order elliptic operators and systems on Rn. J.Evol. Equ., 1(4):361–385, 2001. Dedicated to the memory of Tosio Kato.

[7] Pascal Auscher, Alan McIntosh and Andrea Nahmod. The square root problem ofKato in one dimension, and first order elliptic systems. Indiana Univ. Math. J. 46,3 (1997), 659–695.

[8] Andreas Axelsson, Stephen Keith, and Alan McIntosh. Quadratic estimates andfunctional calculi of perturbed Dirac operators. Invent. Math., 163(3):455–497, 2006.

[9] Tuomas Hytonen and Alan McIntosh. Stability in p of the H∞-calculus of first-order systems in Lp. The AMSI-ANU Workshop on Spectral Theory and HarmonicAnalysis, 167–181, Proc. Centre Math. Appl. Austral. Nat. Univ., 44, Austral. Nat.Univ., Canberra, 2010.

[10] Tuomas Hytonen, Alan McIntosh, and Pierre Portal. Kato’s square root problem inBanach spaces. J. Funct. Anal., 254(3):675–726, 2008.

[11] Tuomas Hytonen, Alan McIntosh, and Pierre Portal. Holomorphic functional calcu-lus of Hodge–Dirac operators in Lp. J. Evol. Equ., 11 (2011), 71–105.

[12] Nigel Kalton and Marius Mitrea. Stability results on interpolation scales of quasi-Banach spaces and applications. Trans. Amer. Math. Soc., 350(10):3903–3922, 1998.

[13] Nigel Kalton, Peer Kunstmann and Lutz Weis. Perturbation and interpolation the-orems for the H∞-calculus with applications to differential operators. Math. Ann.336 (2006), no. 4, 747–801.

[14] Nigel Kalton and Lutz Weis. The H∞-calculus and sums of closed operators. Math.Ann. 321 (2001), no. 2, 319–345.

[15] Peer Kunstmann and Lutz Weis. Maximal Lp-regularity for parabolic equations,Fourier multiplier theorems and H∞-functional calculus. Functional analytic meth-ods for evolution equations, 65–311, Lecture Notes in Math., 1855, Springer, Berlin,2004.

[16] I. Ja. Sneıberg. Spectral properties of linear operators in interpolation families ofBanach spaces. Mat. Issled., 9(2(32)):214–229, 254–255, 1974.

Pascal Auscher and Sebastian StahlhutUniv. Paris-Sudlaboratoire de MathematiquesUMR 8628 du CNRSF-91405 Orsaye-mail: [email protected]

[email protected]


(m,λ)-Berezin Transform andApproximation of Operators on WeightedBergman Spaces over the Unit Ball

Wolfram Bauer, Crispin Herrera Yanez and Nikolai Vasilevski

Abstract. We establish various results on norm approximations of boundedlinear operators acting on the weighted Bergman space A2

λ(Bn) over the unit

ball by means of Toeplitz operators with bounded measurable symbols. Themain tool here is the so-called (m,λ)-Berezin transform defined and studiedin the paper. In a sense, this is a further development of the ideas and resultsof [6, 7, 9] to the case of operators acting on A2

λ(Bn).

Mathematics Subject Classification (2010). Primary 47B35; Secondary 30H20,30E05.

Keywords. Toeplitz operator; unit ball; (m,λ)-Berezin transform; norm ap-proximation.

1. Introduction

Let Bn be the open Euclidean unit ball in Cn, and let

dvλ(z) = cλ(1− |z|2)λdv(z), (1.1)

be a family of standard weighted probability measures on Bn, where the weightparameter fulfills λ > −1, the normalizing constant cλ is given in (2.1) below, anddv is the standard volume form on Bn. In this paper we consider the family ofweighted Bergman spaces A2

λ(Bn) over Bn which consist of all complex analytic

functions that are square integrable with respect to dvλ. As is well known A2λ(B

n)forms a closed subspace of L2(Bn, dvλ) and has the structure of a reproducingkernel Hilbert space. We denote by Bλ the (orthogonal) Bergman projection ofL2(Bn, dvλ) onto A2

λ(Bn).

The first named author has been supported by an “Emmy-Noether scholarship” of DFG

(Deutsche Forschungsgemeinschaft).The third named author has been partially supported by CONACYT Project 102800, Mexico.


46 W. Bauer, C. Herrera Yanez and N. Vasilevski

Given an essentially bounded measurable function a ∈ L∞(Bn), we write Ta

for the Toeplitz operator with symbol a, which acts on A2λ(B

n) as Taf = Bλ(af).That is, the Toeplitz operator is defined as the compression of a multiplicationoperator on L2(Bn, dvλ) onto the Bergman space. For simplicity we suppress thedependence on λ in the notation Ta.

Due to their simple structure Toeplitz operators form an important, tractableand intensively studied subclass in the algebra L(A2

λ(Bn)) of all bounded linear

operators on A2λ(B

n). Moreover, this subclass is dense in L(A2λ(B

n)) with respectto the strong operator topology (shortly: SOT). In the case of the unweightedBergman space (i.e., if λ = 0) and for a more general type of domains this propertywas established in [4]. However, the proof in [4] almost literally generalizes to theweighted case of λ ∈ (−1,∞).

A further challenging task is to characterize the closure of the above subclassof Toeplitz operators in the norm topology; or, in other words, to characterize theoperators S ∈ L(A2

λ(Bn)) that can be approximated in norm by Toeplitz operators.

One of the tools, that proves to be useful and efficient here, is the so-calledm-Berezin transform. The m-Berezin transform for any bounded linear operatoracting on the unweighted Bergman space over the unit disk was defined in [9] andgoes back to the work of Berezin (cf. [1]). For the unweighted Bergman space overthe unit ball it was defined in [7], again for all bounded linear operators. The caseof the weighted Bergman space has not been covered so far.

The (k, α)-Berezin transform for complex-valued regular measures (and inparticular for bounded measurable symbols) on the weighted p-Bergman spaceover the unit ball was defined and studied in detail in [6]. This definition doesnot depend on the value of a weight parameter characterizing the space, andis not applicable to any bounded linear operator. Note in this context that anoperator, being perfectly defined and bounded on a weighted Bergman space withthe specific weight parameter, may easily have no sense for other values of theweight parameter. Thus any definition of the Berezin transform applicable forbounded linear operator acting on the specific weighted Bergman must forciblyinclude the value of a weight parameter in its definition.

Following the recipe in [7, proof of Proposition 2.1] we define in (3.3) the(m,λ)-Berezin transform for general bounded operators acting on the weightedBergman space A2

λ(Bn). Note that our definition restricted to measures (more

precisely: restricted to Toeplitz operators with measure symbols) does not coincidewith the one in [6], which, being defined for the weighted Bergman space, does notdepend on a weight parameter. Hence a further extension of the approach of [6]to general bounded linear operators forcibly involves some modifications.

Note that our (m,λ)-Berezin transform of a Toeplitz operator Ta acting onA2

λ(Bn) coincides with the (standard) (0, λ + m)-Berezin transform for Ta now

considered on the weighted Bergman space A2λ+m(Bn). Consequently the Berezin

quantization procedure together with its correspondence principle suggest thatthe limit of the (m,λ)-Berezin transform as m → ∞ may serve (and it does!) as

(m,λ)-Berezin Transform 47

a good approximation tool for certain classes of operators acting on the initialBergman space.

The aim of this paper is to establish various results on norm approximationsvia the (m,λ)-Berezin transform. More precisely, we describe conditions underwhich a bounded linear operator S (Theorem 4.7 and Proposition 4.9) can beapproximated in norm by Toeplitz operators whose symbols are bounded functionsthat are explicitly given as the (m,λ)-Berezin transforms of the initial operator S.In Appendix we remark that the approximation results of [6] remain valid for ourdefinition of the (m,λ)-Berezin transform.

We would like to point out that these results generalize ideas and theorems in[6, 7, 9] and, in particular, form an essential tool in the explicit description of thenorm closure of the set of Toeplitz operators with bounded radial symbols and theC∗-algebra generated by such Toeplitz operators, being considered as a subset ofL(A2

λ(Bn)). Such applications and further aspects of the approximation procedure

in the case of the so-called radial operators will be the subject of a forthcomingpaper (cf. [2]).

2. Preliminaries

Let Bn :={z ∈ Cn : |z|2 := |z1|2 + · · · + |zn|2 < 1

}be the open unit ball in Cn

equipped with the standard weighted measure (1.1), where λ > −1 is fixed. Herecλ is given by

cλ :=Γ(n+ λ+ 1)

πnΓ(λ+ 1), (2.1)

so that vλ(Bn) = 1. We write L2(Bn, dvλ) for the Hilbert space of all functionsthat are square-integrable with respect to dvλ. The corresponding norm and innerproduct are denoted by ‖ · ‖λ and 〈·, ·〉λ, respectively.

Let Z+ := {0, 1, . . .} be the set of non-negative integers. With α ∈ Zn+ we use

the standard notations zα := zα11 · · · zαn

n , α! := α1! · · ·αn! and |α| := α1+ · · ·+αn.As is well known, for all α ∈ Zn

+ we have

‖wα‖λ =

√α!Γ(n+ λ+ 1)

Γ(n+ |α|+ λ+ 1, (2.2)

and we write [eα := wα‖wα‖−1λ : α ∈ Zn

+] for the standard orthonormal basis

of A2λ(B

n).

The Bergman (orthogonal) projection Bλ from L2(Bn, dvλ) onto A2λ(B

n) canbe expressed as an integral operator in the explicit form[

Bλϕ](z) =

∫Bn

ϕ(w)

(1− 〈z, w〉)n+λ+1dvλ(w) with ϕ ∈ L2(Bn, dvλ),


where 〈z, w〉 := z1w1 + · · · + znwn denotes the Euclidean inner product on Cn.The reproducing kernel of the Bergman space A2

λ(Bn) is given by

Kλz (w) =

1

(1− 〈w, z〉)n+λ+1=

∞∑|α|=0

Γ(n+ |α|+ λ+ 1)

α!Γ(n+ λ+ 1)zαwα. (2.3)

We frequently use the normalized version of the Bergman kernel and write

kλz (w) =Kλ

z (w)

‖Kλz ‖λ

=(1 − |z|2)n+λ+1

2

(1 − 〈w, z〉)n+λ+1.

By φz(w) we denote a biholomorphism of Bn that interchanges 0 and z. Moreprecisely, we choose the explicit form of φz(w) given, for example, in [12, p.5] suchthat φ0(w) = −w. Recall [12, p.37] that the complex Jacobian det(φ′

z) of φz hasthe form

det(φ′z(w)) = (−1)n (1− |z|2)n+1

2

(1− 〈w, z〉)n+1= (−1)nk0z(w). (2.4)

It is standard that the kernel Kλz transforms under the biholomorphisms φu as

Kλz (w) = kλu(z)K

λφu(z)

(φu(w)

)kλu(w). (2.5)

Given z ∈ Bn, we introduce the unitary operator Uz on A2λ(B

n) which acts as theweighted composition operator(

Uzf)(w) :=

(1− |z|2)n+λ+12

(1− 〈w, z〉)n+λ+1(f ◦ φz)(w)

= kλz (w) · f ◦ φz(w). (2.6)

It is easy to check that Uz is self-adjoint and so U2z = I. Since φ0 induces a

reflection at the origin we have(U0f

)(w) = f(−w).

For a fixed z ∈ Bn we define an automorphism on the algebra L(A2λ(B

n)) ofall bounded operator on A2

λ(Bn) by

L(A2λ(B

n)) � S �−→ Sz := UzSUz ∈ L

(A2λ(B

n)). (2.7)

In particular, if S = Ta is a Toeplitz operator then (Ta)z = Ta◦φz .Throughout the paper and as a convention we will denote by C a positive

constant appearing in various estimates and whose value may change from placeto place.

3. The (m,λ)-Berezin transform

Recall that the m-Berezin transform for the unweighted Bergman space over theunit disk and over the unit ball were defined in [9] and [7], respectively. In thecase where λ �= 0 the notion of the (k, α)-Berezin transform for measures on theweighted p-Bergman space over Bn was introduced in [6].


A generalization of the concept of the m-Berezin transform to an arbitrarybounded operator on the Bergman space A2

λ(Bn) requires a modification of the

definition in [6]. We will follow the recipe in [7] and first introduce some notation.Put

Cm,α :=

(m

|α|)(−1)|α| |α|!

α1! · · ·αn!=

(−1)|α|α!

m!

(m− |α|)! , (3.1)

so thatm∑

|α|=0

Cm,α zαwα = (1− 〈z, w〉)m . (3.2)

Definition 3.1. We define the (m,λ)-Berezin transform of S ∈ L(A2λ(B

n)) by

(Bm,λS) (z) :=cλ+m

cλ

m∑|α|=0

Cm,α

⟨Szw

α, wα⟩λ. (3.3)

Note that a direct application of the Cauchy–Schwarz inequality gives thefollowing pointwise estimate∣∣ (Bm,λS) (z)

∣∣ ≤ ‖S‖cλ+m

cλ

m∑|α|=0

∣∣Cm,α

∣∣‖wα‖2λ =: C(λ,m, n) ‖S‖,

where the constant C(λ,m, n) > 0 is independent of z ∈ Bn. That is, Bm,λS is abounded function on Bn with

‖Bm,λS‖∞ ≤ C(λ,m, n) ‖S‖. (3.4)

As usual we define the (m,λ)-Berezin transform of a function a ∈ L∞(Bn) by

Bm,λ(a)(z) := (Bm,λTa) (z) =cλ+m

cλ

m∑|α|=0

Cm,α

⟨(a ◦ φz)w

α, wα⟩λ

=cλ+m

cλ

∫Bn

(a ◦ φz)(w) cλ(1− |w|2)λ+mdv(w)

=

∫Bn

(a ◦ φz)(w) dvλ+m(w).

(3.5)

As was mentioned earlier Definition 3.1 is different from the one in [6], where the

(m,λ)-Berezin transform Bm,λ for finite, complex-valued, regular measures ν onBn was introduced. In fact, in the special case of ν := advλ with a ∈ L∞(Bn) thelatter one gives

Bm,λ(ν)(z) =

∫Bn

(a ◦ φz)(w) dvm(w),

which, differently from (3.5), is independent of the weight parameter λ. This seemsto be inadequate as the initial data (measures and, more generally, operators) aredefined on the specific weighted Bergman space A2

λ(Bn).

The next two propositions give alternative formulas for the (m,λ)-Berezintransform that, from time to time, are more suitable to work with. Note that


the formula of the second proposition, in the particular case when n = 1 andλ = 0, coincides with the definition of the m-Berezin transform on the unit diskby Suarez [9].

We study then the properties of the (m,λ)-Berezin transform which will beused both in this and in a subsequent paper. The majority of our proofs usearguments similar to the ones of the unweighted case [7].

Proposition 3.2. Let S ∈ L (A2λ(B

n)), m ≥ 0 and z ∈ Bn. Then

(Bm,λS) (z) =cλ+m

cλ

(1− |z|2)m+λ+n+1

×∫Bn

∫Bn

(1− 〈u,w〉)m Km+λz (u)Km+λ

z (w)S∗Kλw(u)dvλ(u)dvλ(w).

Proof. We have

(Bm,λS)(z) =cλ+m

cλ

m∑|α|=0

Cm,α

⟨Szw

α, wα⟩λ

=cλ+m

cλ

m∑|α|=0

Cm,α

∫Bn

S(φαz k

λz

)(w)φα

z (w)kλz (w)dvλ(w) (3.6)

=cλ+m

cλ

m∑|α|=0

Cm,α

∫Bn

∫Bn

φαz (u)k

λz (u)φ

αz (w)k

λz (w)S

∗Kλw(u)dvλ(u)dvλ(w).

In the last equality we use that

S(φαz k

λz

)(w) = 〈S (φα

z kλz

),Kλ

w〉λ = 〈φαz k

λz , S

∗Kλw〉λ.

Then, by (3.2) and (2.5), the expression (3.6) equals to

cλ+m

cλ

∫Bn

∫Bn

(1− 〈φz(u), φz(w)〉

)mkλz (u)k

λz (w)S

∗Kλw(u)dvλ(u)dvλ(w)

=cλ+m

cλ

∫Bn

∫Bn

(kλz (u)k

λz (w)

Kλw(u)

) mλ+n+1

kλz (u)kλz (w)S

∗Kλw(u)dvλ(u)dvλ(w)

=cλ+m

cλ

(1− |z|2)m+λ+n+1×

×∫Bn

∫Bn

(1− 〈u,w〉)mKm+λ

z (u)Km+λz (w)S∗Kλ

w(u)dvλ(u)dvλ(w),

which finishes the proof. �


n)), m ≥ 0 and z ∈ Bn. Then

(Bm,λS) (z) =cλ+m

cλ

(1− |z|2)m+λ+n+1

m∑|α|=0

Cm,α

⟨S(wαKm+λ

z ), wαKm+λz

⟩λ.

(3.7)


Proof. We have∫Bn

∫Bn

(1− 〈u,w〉)mKm+λ

z (u)Km+λz (w)S∗Kλ

w(u)dvλ(u)dvλ(w)

=

m∑|α|=0

Cm,α

∫Bn

∫Bn

uαwαKm+λz (u)Km+λ

z (w)S∗Kλw(u)dvλ(u)dvλ(w)

=

m∑|α|=0

Cm,α

∫Bn

S(uαKm+λz )(w)wαKm+λ

z (w)dvλ(w).

Thus the result follows from Proposition 3.2. �

Lemma 3.4. Given z, w ∈ Bn, the automorphism U := φφw(z) ◦ φw ◦ φz of Bn

extends to a unitary transformation of Cn, and

UzUw = [(−1)n det U ]n+λ+1n+1 · VUUφw(z),

where the operator VU is given by(VUf

)(u) := f

(Uu).Proof. Since U in an automorphism of the unit ball having 0 as a fixed point itfollows by the Cartan theorem that U acts by multiplication on a unitary matrix.This matrix will also be denoted by U , i.e., U(u) = Uu.

Differentiating the equality φφw(z) ◦ U = φw ◦ φz we have

φ′φw(z)(U(u))U ′(u) = φ′

w(φz(u))φ′z(u),

which implies

(−1)nk0φw(z)(Uu) det U = (−1)nk0w(φz(u)) · (−1)nk0z(u).

As kλz = (k0z)n+λ+1n+1 the application of identity (2.6) together with the last formula

gives

(UzUwf)(u) = kλz (u) · kλw(φz(u)) · (f ◦ φw ◦ φz)(u)

= (det U)n+λ+1n+1 · (−1)n(n+λ+1)

n+1 kλφw(z)(Uu) · (f ◦ φφw(z) ◦ U)(u)= [(−1)n det U ]n+λ+1

n+1 · (VUUφw(z)f)(u).

Note that [(−1)n det U ]n+λ+1n+1 is a complex number of modulus one. �

Theorem 3.5. Let S∈L(A2λ(B

n)), m≥0 and z∈Bn. Then Bm,λSz=(Bm,λS)◦φz.


Proof. By definition

(Bm,λSz)(0) =cλ+m

cλ

m∑|α|=0

Cm,α

⟨U0SzU0w

α, wα⟩λ

=cλ+m

cλ

m∑|α|=0

Cm,α

⟨Sz(−w)α, (−w)α

⟩λ

=cλ+m

cλ

m∑|α|=0

Cm,α

⟨Szw

α, wα⟩λ= Bm,λS(z) = (Bm,λS) ◦ φz(0).

For any η ∈ Bn, by Proposition 3.2 and Lemma 3.4 we have

(Bm,λSz) ◦ φη(0) = Bm,λ((Sz)η)(0)

=cλ+m

cλ

∫Bn

∫Bn

(1− 〈u,w〉)m ((Sz)η)∗Kλw(u)dvλ(u)dvλ(w)

=cλ+m

cλ

∫Bn

∫Bn

(1− 〈u,w〉)m UηUzS∗UzUηKλw(u)dvλ(u)dvλ(w)

=cλ+m

cλ

∫Bn

∫Bn

(1− 〈u,w〉)m VUUφz(η)S∗Uφz(η)V

∗UKλ

w(u)dvλ(u)dvλ(w)

= Bm,λSφz(η)(0),

where VU is the unitary operator of Lemma 3.4. This implies the statement. �

The next two lemmas are preparatory for the proof of Proposition 3.8, whichplays a crucial role and states the commutativity of the (m,λ)-Berezin transformsfor different values of the parameter m.

Lemma 3.6. Let S ∈ L (A2λ(B

n))and m, j ≥ 0. If |S∗Kλ

z (w)| ≤ C for any w ∈ Bn,then

Bm,λBj,λS = Bj,λBm,λS.

Proof. Due to Theorem 3.5, we only need to check that(Bm,λBj,λS)(0) = (Bj,λBm,λS)(0). Property (3.5), Proposition 3.2, and Fubini’stheorem imply that

Bm,λ(Bj,λS)(0) = Bm,λ(TBj,λS)(0) = cm+λ

∫Bn

Bj,λS(z)(1− |z|2)m+λdv(z)

=

∫Bn

cm+λcj+λ

cλ(1 − |z|2)m+j+2λ+n+1

×∫Bn

∫Bn

(1 − 〈u,w〉)jKj+λz (u)Kj+λ

z (w)S∗Kλw(u)dvλ(u)dvλ(w)dv(z)

=

∫Bn

∫Bn

cm+λcj+λ

cλFm,j(u,w)S∗Kλ

w(u)dvλ(u)dvλ(w),


where the function Fm,j(u,w) in the integrand is defined by

Fm,j(u,w) := (1−〈u,w〉)j∫Bn

(1−|z|2)m+j+2λ+n+1Kj+λz (u)Kj+λ

z (w)dv(z). (3.8)

Observe that (3.8) can be represented as a finite sum

Fm,j(u,w) =

l∑i=1

Hi(u)Gi(w)

for certain holomorphic functions Hi and Gi. By [3, Lemma 10], it is sufficient toshow that Fm,j(w,w) = Fj,m(w,w), where w ∈ Bn, which can be easily verifiedby a direct calculation

Fm,j(w,w) = (1− |w|2)j∫Bn

(1− |z|2)m+j+2λ+n+1|Kj+λz (w)|2dv(z)

= (1− |w|2)j∫Bn

(1− |φw(z)|2)m+j+2λ+n+1|Kj+λw (φw(z))|2|k0w(z)|2dv(z)

= (1− |w|2)m∫Bn

(1− |z|2)m+j+2λ+n+1|Km+λz (w)|2dv(z)

= Fj,m(w,w).

In the second equality we have changed variables using (2.4). �

By S1 = S1(A2λ(B

n)) denote the set of all trace class operators acting onA2

λ(Bn). Given A ∈ S1, we write tr[A] for its trace, and recall that the trace norm

of A is given by

‖A‖S1 := tr[√

A∗A].

Given f, g ∈ A2λ(B

n), the rank-one-operator f⊗g, acting on A2λ(B

n) by the formula(f ⊗ g)h = 〈h, g〉λf obviously belongs to S1. Furthermore,

‖f ⊗ g‖S1 = ‖f‖λ · ‖g‖λand tr [f ⊗ g] = 〈f, g〉λ. Recall as well that if A ∈ S1 has rank m, then one hasthe inequality

‖A‖S1 ≤√m (tr [A∗A])

12 .

Lemma 3.7. For any S ∈ L (A2λ(B

n)), there exist sequences {Sα}, satisfying the

property

|S∗αK

λz (w)| ≤ C(α), (3.9)

such that Bm,λSα point-wisely converges to Bm,λS.

Proof. Let H∞ = H∞(Bn) denote the algebra of bounded holomorphic functionson Bn. Both, the density of H∞ in A2

λ(Bn) and the density of finite rank operators

in the ideal K of compact operators on L(A2λ(B

n)), imply that the set

F :=

{ l∑j=1

fj ⊗ gj : fj , gj ∈ H∞}


is norm-dense in K. At the same time K is dense in L(A2λ(B

n)) with respect to thestrong operator topology. Thus, for each S ∈ L (A2

λ(Bn))there exists a sequence

{Sα} of finite rank operators

Sα =

l(α)∑j=1

fα,j ⊗ gα,j

converging strongly to S. The representation (3.7) shows that Bm,λSα convergesto Bm,λS point-wise. To finish the proof we estimate

|S∗αK

λz (w)| =

∣∣∣ l(α)∑j=1

(gα,j ⊗ fα,j)Kλz (w)

∣∣∣ = ∣∣∣ l(α)∑j=1

⟨Kλ

z (w), fα,j(w)⟩λgα,j(w)

∣∣∣≤

l(α)∑j=1

|fα,j(z)||gα,j(w)| ≤l(α)∑j=1

‖fα,j‖∞‖gα,j‖∞ = C(α)

uniformly for z ∈ Bn. �


n))and m, j ≥ 0, then Bm,λBj,λS = Bj,λBm,λS.

Proof. Let S ∈ L (A2λ(B

n)). According to Lemma 3.7 there exists a sequence {Sα}

of operators that satisfy (3.9) and the point-wise convergence Bm,λSα → Bm,λSholds. Lemma 3.6 implies that(

Bm,λBj,λSα

)(z) =

(Bj,λBm,λSα

)(z). (3.10)

By representation (3.5),(Bm,λBj,λSα

)(z) =

∫Bn

(Bj,λSα) ◦ φz(u)dvm+λ(u).

As the sequence {Sα} converges to S strongly we have by construction

‖(Bj,λSα) ◦ φz‖∞ = ‖Bj,λSα‖∞ ≤ ‖Bj,λ‖ · ‖Sα‖ ≤ C(j, λ) · ‖S‖.Furthermore (Bj,λSα) ◦ φz(u) converges to (Bj,λS) ◦ φz(u). As a consequence thefunctions (Bm,λBj,λSα)(z) and (Bj,λBm,λSα)(z) converge to (Bm,λBj,λS)(z) and(Bj,λBm,λS)(z), respectively. Passing to the limit in (3.10) finishes the proof. �

Corollary 3.9. For all λ > −1 and m ∈ Z+ the (m,λ)-Berezin transform is one-to-one on bounded operators on A2

λ(Bn).

Proof. Since Bm,λ restricted to functions coincides with the usual Berezin trans-form on A2

λ+m(Bn) (cf. (3.5)) it is one-to-one on functions (on Toeplitz operators).

Now assume that S ∈ L(A2λ(B

n)) such that Bm,λS ≡ 0. From Proposition 3.8 weobtain that

0 = B0,λBm,λS = Bm,λB0,λS

and therefore B0,λS ≡ 0. Since B0,λ is known to be one-to-one on bounded oper-ators over A2

λ(Bn) we conclude that S = 0, which finishes the proof. �


Recall that the pseudo-hyperbolic metric on the unit ball is defined as

ρ(z, w) := |φz(w)| = |φw(z)|.As is well known ρ(·, ·) is invariant under the automorphisms φu of Bn. The nextresult shows the Lipschitz continuity of B0,λS with respect to this metric.

Theorem 3.10. Let S ∈ L(A2λ(B

n)). Then there exists a constant C(n, λ) > 0 suchthat ∣∣(B0,λS

)(z)− (B0,λS

)(w)∣∣ ≤ C(n, λ)‖S‖ ρ(z, w).

Proof. By definition and the above-mentioned properties of trace class operatorswe have∣∣(B0,λS

)(z)− (B0,λS

)(w)∣∣ = |〈Sz1, 1〉λ − 〈Sw1, 1〉λ|= |tr [Sz(1⊗ 1)]− tr [Sw(1⊗ 1)]|= |tr [Sz(1⊗ 1)− SUw(1 ⊗ 1)Uw]|= |tr [Sz(1⊗ 1)− SUz(UzUw1⊗ UzUw1)Uz]| = (∗).

By Lemma 3.4, we estimate

(∗) < ‖Sz‖‖1⊗ 1− Uφw(z)1⊗ Uφw(z)1‖S1

≤√2‖Sz‖

(2− 2|⟨1, kλφw(z)

⟩λ|2)1/2

= 2‖S‖[1− (1− |φw(z)|2)n+λ+1]1/2

≤ C(n, λ)‖S‖|φw(z)|,which according to the definition of the pseudo-hyperbolic metric shows the result.

�

Now representation (3.5) yields

Corollary 3.11. Let S ∈ L(A2λ(B

n)), and a := B0,λS ∈ L∞(Bn). Then

limm→∞ ‖Bm,λ(a)− a‖∞ = 0.

Proof. Let ε > 0 and choose δ > 0 with |a(z)−a(w)| < ε whenever z, w ∈ Bn withρ(z, w) < δ. If w ∈ Bn and m ∈ N, we have according to (3.5)

|Bm,λ(a)(w) − a(w)|

≤ cλ+m

∫Bn

∣∣a ◦ φw(z)− a ◦ φw(0)∣∣(1 − |z|2)λ+mdv(z)

≤ cλ+m

{∫0≤|z|<δ

+

∫1>|z|≥δ

}∣∣a ◦ φw(z)− a ◦ φw(0)∣∣(1− |z|2)λ+mdv(z).

Since ρ(·, ·) is invariant under the automorphisms φw and ρ(z, 0) < |z| (see, forexample, [12, page 28]), we have ρ(φw(z), φz(0)) = ρ(z, 0) < δ in the first integral,


and therefore by the Lipschitz continuity of a

cλ+m

∫0≤|z|<δ

∣∣a ◦ φw(z)− a ◦ φw(0)∣∣(1 − |z|2)λ+mdv(z) < ε. (3.11)

Now, we estimate the second integral above.

cλ+m

∫1>|z|≥δ

∣∣a ◦ φw(z)− a ◦ φw(0)∣∣(1− |z|2)λ+mdv(z)

≤ 2cλ+m‖a‖∞∫1>|z|≥δ

(1− |z|2)λ+mdv(z)

≤ 2cλ+m‖a‖∞(1− δ)λ+mvol(Bn).

(3.12)

Since the normalizing constant cλ+m has at most polynomial growth as m → ∞(see the definition (2.1) and [5, Formula 8.328.2]) it is clear that the right-hand sideconverges to zero as m → ∞. The assertion follows by combining the estimates(3.11) and (3.12). �

4. Approximation by Toeplitz operators

We start this section with a technical statement which is taken from [8, Proposition1.4.10] and also stated as Lemma 3.1 in [7].

Lemma 4.1. Suppose a < 1 and a+ b < n+ 1. Then

supz∈Bn

∫Bn

dv(w)

(1− |w|2)a |1− 〈w, z〉|b <∞.

Let 1 < q <∞ and p be the conjugate exponent of q. Note that the inequality

q = 1 +1

p− 1<

n+ 2(1 + λ)

n+ 1 + λ= 1 +

1 + λ

n+ 1 + λ=: R (4.1)

is equivalent to

p > 2 +n

1 + λ.

In what follows we use the standard weighted Lp-norm ‖ · ‖p,λ defined by

‖f‖p,λ =

(∫Bn

|f(z)|pdvλ(z)) 1

p

.

Lemma 4.2. Let S ∈ L (A2λ(B

n)), p > 2 + n

1+λ , and put h(z) = (1− |z|2)−a with

a =(1 + λ)(n+ 1 + λ)

n+ 2(1 + λ)=

1 + λ

R.

Then there exists C(n, p, λ) > 0 such that∫Bn

|(SKλz )(w)|h(w)dvλ(w) ≤ C(n, p, λ)‖Sz1‖p,λh(z), (4.2)


for all z ∈ Bn, and∫Bn

|(SKλz )(w)|h(z)dvλ(z) ≤ C(n, p, λ)‖S∗

w1‖p,λh(w), (4.3)

for all w ∈ Bn.

Proof. Given z ∈ Bn, the equality

Uz1 = (1− |z|2)n+λ+12 Kλ

z

implies that

SKλz =

1

(1− |z|2)n+λ+12

SUz1

=1

(1− |z|2)n+λ+12

UzSz1 = (Sz1 ◦ φz)Kλz .

We change the variable u = φz(w) and apply the Holder inequality∫Bn

|(SKλz )(w)|

(1− |w|2)a dvλ(w) = cλ

∫Bn

|Sz1 ◦ φz(w)||Kλz (w)|(1 − |w|2)λ

(1− |w|2)a dv(w)

=1

(1− |z|2)a∫Bn

|Sz1(u)||1− 〈u, z〉|n+λ+1−2a(1− |u|2)a dvλ(u)

≤ ‖Sz1‖p,λ(1− |z|2)a

(cλ

∫Bn

dv(u)

(1− |u|2)aq−λ|1− 〈u, z〉|(n+λ+1−2a)q

)1/q

.

According to (4.1) we have aq − λ < 1 and aq − λ + (n + λ + 1 − 2a)q < n + 1.Hence inequality (4.2) follows from Lemma 4.1.

The second inequality (4.3) follows from (4.2) after replacing S by S∗, inter-changing w and z, and making use of(

S∗Kλw

)(z) =

⟨S∗Kλ

w,Kλz

⟩λ=⟨Kλ

w, SKλz

⟩λ= SKλ

z (w), (4.4)

which holds for all z, w ∈ Bn. �

Lemma 4.3. Let S ∈ L(A2λ(B

n)) and p > 2 + n1+λ . Then

‖S‖ ≤ C(n, p, λ)(supz∈Bn

‖Sz1‖p,λ)1/2(

supz∈Bn

‖S∗z1‖p,λ

)1/2,

where C(n, p, λ) is the constant of Lemma 4.2.

Proof. By (4.4) we have that

(Sf)(w) =⟨Sf,Kλ

w

⟩λ=

∫Bn

f(z)(S∗Kλw)(z)dvλ(z) =

∫Bn

f(z)(SKλz )(w)dvλ(z),

for f ∈ A2λ(B

n) and w ∈ Bn. Now Lemma 4.2 and the Schur theorem (see, forexample, [11, Corollary 3.2.3]) imply the result. �


Lemma 4.4. Let {Sm} be a bounded sequence in L(A2λ(B

n)) withlimm→∞ ‖B0,λSm‖∞ = 0. Then

supz∈Bn

|⟨(Sm)z1, f⟩λ| → 0

as m→∞ for any f ∈ A2λ(B

n), and

supz∈Bn

|(Sm)z1(·)| → 0 (4.5)

uniformly on compact subsets of Bn as m→∞.

Proof. To prove the first statement it is sufficient to check that for each multi-index k

supz∈Bn

∣∣⟨(Sm)z1, wk⟩λ

∣∣→ 0 as m→∞.

Using (2.3) we calculate

B0,λSm(φz(u)) = B0,λ(Sm)z(u) = (1− |u|2)n+λ+1⟨(Sm)zK

λu ,K

λu

⟩λ

= (1− |u|2)n+λ+1∞∑

|α|=0

∞∑|β|=0

Γ(n+ |α|+ λ+ 1)

α!Γ(n+ λ+ 1)

Γ(n+ |β|+ λ+ 1)

β!Γ(n+ λ+ 1)

× ⟨(Sm)zwα, wβ

⟩λuαuβ.

Given a multi-index k and r ∈ (0, 1), we obtain by using (2.1)∫|u|<r

B0,λSm(φz(u))uk

(1− |u|2)n+λ+1dvλ(u)

=

∞∑|α|=0

∞∑|β|=0

Γ(n+ |α|+ λ+ 1)

α!Γ(n+ λ+ 1)

Γ(n+ |β|+ λ+ 1)

β!Γ(n+ λ+ 1)

⟨(Sm)zw

α, wβ⟩λ

×∫|u|<r

uα+kuβdvλ(u)

=

∞∑|α|=0

Γ(n+ |α|+ λ+ 1)

α!Γ(n+ λ+ 1)

Γ(n+ |α|+ |k|+ λ+ 1)

(α+ k)!Γ(n+ λ+ 1)

⟨(Sm)zw

α, wα+k⟩λ

×∫|u|<r

|uα+k|2dvλ(u)

=

∞∑|α|=0

Γ(n+ |α|+ λ+ 1)

α!Γ(n+ λ+ 1)

Γ(n+ |α|+ |k|+ λ+ 1)

(α+ k)!πnΓ(λ+ 1)

⟨(Sm)zw

α, wα+k⟩λ

×∫|u|<r

|uα+k|2(1 − |u|2)λdv(u).

Passing in the last integral to the polar coordinates u = sξ, where s ∈ R+ andξ ∈ S2n−1, and making use of the next formulas (here dS is the surface measure


on S2n−1 = ∂Bn) ∫Bn

f(u)dv(u) =

∫ 1

0

s2n−1dr

∫S2n−1

f(sξ)dS(ξ),∫S2n−1

|ξm|2dS(ξ) = 2πnm!

(n− 1 + |m|)!the last expression is equal to

∞∑|α|=0

Γ(n+ |α|+ λ+ 1)

α!Γ(n+ λ+ 1)

Γ(n+ |α|+ |k|+ λ+ 1)

Γ(λ+ 1)Γ(n+ |α|+ |k|)

× ⟨(Sm)zwα, wα+k

⟩λ2

∫ r

0

s2n+2|α|+2|k|−1(1 − s2)λds

=

∞∑|α|=0

Γ(n+ |α|+ λ+ 1)

α!Γ(n+ λ+ 1)

Γ(n+ |α|+ |k|+ λ+ 1)

Γ(λ+ 1)Γ(n+ |α|+ |k|)

× ⟨(Sm)zwα, wα+k

⟩λ

∫ r2

0

sn+|α|+|k|−1(1− s)λds

=∞∑

|α|=0

Γ(n+ |α|+ λ+ 1)

α!Γ(n+ λ+ 1)

⟨(Sm)zw

α, wα+k⟩λIr2(n+ |α|+ |k|, λ+ 1)

=⟨(Sm)z1, w

k⟩λIr2(n+ |k|, λ+ 1)

+∞∑

|α|=1

Γ(n+ |α|+ λ+ 1)

α!Γ(n+ λ+ 1)

⟨(Sm)zw

α, wα+k⟩λIr2(n+ |α|+ |k|, λ+ 1).

Here the function Ix(a, b) is defined in the standard way (see, for example, [5,Formula 8.392],

Ix(a, b) =Γ(a+ b)

Γ(a)Γ(b)

∫ x

0

ta−1(1 − t)b−1dt.

Then we have

|⟨(Sm)z1, wk⟩λ| ≤ 1

Ir2(n+ |k|, λ+ 1)

∣∣∣∣∣∫|u|<r

B0,λSm(φz(u))uk

(1− |u|2)n+λ+1dvλ(u)

∣∣∣∣∣+

∣∣∣∣∣∣∞∑

|α|=1

Γ(n+ |α|+ λ+ 1)

α!Γ(n+ λ+ 1)

⟨(Sm)zw

α, wα+k⟩λ

Ir2(n+ |α|+ |k|, λ+ 1)

Ir2(n+ |k|, λ+ 1)

∣∣∣∣∣∣≤ 1

Ir2(n+ |k|, λ+ 1)‖B0,λSm‖∞ cλ

∫|u|<r

|uk|(1− |u|2)n+1

dv(u)

+

∞∑|α|=1

Γ(n+ |α|+ λ+ 1)

α!Γ(n+ λ+ 1)‖(Sm)z‖‖wα‖λ‖wα+k‖λ Ir2(n+ |α|+ |k|, λ+ 1)

Ir2(n+ |k|, λ+ 1)


≤ ‖B0,λSm‖∞ cλIr2(n+ |k|, λ+ 1)

∫|u|<r

|uk|(1− |u|2)n+1

dv(u)

+ C

∞∑|α|=1

Ir2(n+ |α|+ |k|, λ+ 1)

Ir2(n+ |k|, λ+ 1)= I +Σ,

where C > 0 is a constant independent of m and z. In the last estimate we usedthe boundedness of the sequence {Sm} and the inequality

Γ(n+ |α|+ λ+ 1)

α!Γ(n+ λ+ 1)‖wα‖λ‖wα+k‖λ ≤ 1,

which easily follows from (2.2). The first summand I above tends to zero asm→∞due to the assumptions of the lemma. We estimate now the series in the secondsummand Σ. By [5, Formula 8.328.2]

lim|α|→∞

Γ(n+ |α|+ |k|+ λ+ 1)

Γ(n+ |α|+ |k|)1

(n+ |α|+ |k|)λ+1= 1,

and thus there exists C > 0 such that for all α ∈ Zn+

Γ(n+ |α|+ |k|+ λ+ 1)

Γ(n+ |α|+ |k|)1

(n+ |α|+ |k|)λ+1< C.

Then

Σ1 :=

∞∑|α|=1

Ir2(n+ |α|+ |k|, λ+ 1)

Ir2(n+ |k|, λ+ 1)

=Γ(n+ |k|)Γ(λ+ 1)

Γ(n+ |k|+ λ+ 1)

(∫ r2

0

tn+|k|−1(1− t)λdt

)−1

×∞∑

|α|=1

Γ(n+ |α|+ |k|+ λ+ 1)

Γ(n+ |α|+ |k|)Γ(λ+ 1)

∫ r2

0

tn+|α|+|k|−1(1 − t)λdt

≤ CΓ(n+ |k|)

Γ(n+ |k|+ λ+ 1)

(∫ r2

0

tn+|k|−1(1− t)λdt

)−1

×∞∑

|α|=1

(n+ |α|+ |k|)λ+1

∫ r2

0

tn+|α|+|k|−1(1 − t)λdt.

Estimating the multiple (1− t)λ in both integrals

(1− r2)λ ≤ (1 − t)λ ≤ 1, for λ ≥ 0,

1 ≤ (1− t)λ ≤ (1− r2)λ, for λ ∈ (−1, 0),


we come to the following inequality

Σ1 ≤ CΓ(n+ |k|+ 1)

Γ(n+ |k|+ λ+ 1)(1− r2)−|λ|

∞∑|α|=1

(n+ |α|+ |k|)λr2|α|

= CΓ(n+ |k|+ 1)

Γ(n+ |k|+ λ+ 1)(1− r2)−|λ|

∞∑m=1

(m+ n− 1

n

)(n+m+ |k|)λ r2m.

The power series in r in the last line has radius of convergence equal to 1 andvanishes at 0. Thus the value of Σ becomes small if one takes r sufficiently closedto 0.

Both above estimates, on I and on Σ, are independent of z ∈ Bn, whichproves the first statement of the lemma. In order to prove the second statementwe use the series representation (2.3) and again (2.2)∣∣(Sm)z1(u)

∣∣ = ∣∣⟨(Sm)z1,Kλu

⟩λ

∣∣≤

∞∑|α|=0

Γ(n+ |α|+ λ+ 1)

α!Γ(n+ λ+ 1)

∣∣⟨(Sm)z1, wα⟩λ

∣∣ · |uα|

≤l−1∑

|α|=0

Γ(n+ |α|+ λ+ 1)

α!Γ(n+ λ+ 1)


∣∣+

∞∑|α|=l

Γ(n+ |α|+ λ+ 1)

α!Γ(n+ λ+ 1)‖Sm‖‖wα‖λ|uα|

≤l−1∑

|α|=0

Γ(n+ |α|+ λ+ 1)

α!Γ(n+ λ+ 1)


∣∣+ C

∞∑|α|=l

(Γ(n+ |α|+ λ+ 1)

α!Γ(n+ λ+ 1)

) 12

|uα|

= Σ1 + Σ2.

To estimate Σ2 we use the Cauchy–Schwarz inequality,

Σ2 ≤ C∞∑j=l

(Γ(n+ j + λ+ 1)

j!Γ(n+ λ+ 1)

) 12 ∑|α|=j

[j!

α!

] 12

|uα|

≤ C

∞∑j=l

(Γ(n+ j + λ+ 1)

j!Γ(n+ λ+ 1)

) 12

⎛⎝∑|α|=j

j!

α!|uα|2

⎞⎠12⎛⎝∑

|α|=j

1

⎞⎠12

= C

∞∑j=l

(Γ(n+ j + λ+ 1)

j!Γ(n+ λ+ 1)

) 12((n+ j − 1)!

j!(n− 1)!

) 12

⎛⎝∑|α|=j

j!

α!|uα|2

⎞⎠ 12

.


If we assume that |u| ≤ r < 1 and use the multi-nomial theorem for the expressionin the last brackets ∑

|α|=j

j!

α!|uα|2 = |u|2j,

we finally have

Σ2 ≤ C

∞∑j=l

(Γ(n+ j + λ+ 1)

j!Γ(n+ λ+ 1)

) 12((n+ j − 1)!

j!(n− 1)!

) 12

rj .

By choosing l sufficiently large we can make Σ2 as small as needed. Note that

by the first statement of the lemma the expression Σ1, with l being already fixed,uniformly tends to zero as m→∞. This ends the proof. �

Lemma 4.5. Let {Sm} be a sequence in L(A2λ(B

n)) such that ‖B0,λSm‖∞ → 0 asm→∞. Assume that for some p > 2 + n

1+λ the following inequalities hold

supz∈Bn

‖(Sm)z1‖p,λ ≤ C and supz∈Bn

‖(S∗m)z1‖p,λ ≤ C, (4.6)

where C > 0 is independent of m. Then Sm → 0 as m → ∞ in the L(A2λ(B

n))-norm.

Proof. By Lemma 4.3 and (4.5) we have

‖Sm‖ ≤ C(n, p, λ)

(supz∈Bn

‖(Sm)z1‖p,λ)1/2(

supz∈Bn

‖(S∗m)z1‖p,λ

)1/2

≤ C(n, p, λ).

Then, for 2 + n1+λ < s < p, Holder’s inequality gives

supz∈Bn

‖(Sm)z1‖ss,λ

≤ supz∈Bn

∫|w|>r

|(Sm)z1(w)|sdvλ(w) + supz∈Bn

∫|w|≤r

|(Sm)z1(w)|sdvλ(w)

≤ supz∈Bn

‖(Sm)z1‖sp,λ(∫

|w|>r

dvλ(w)

)1− sp

+ supz∈Bn

∫|w|≤r

|(Sm)z1(w)|sdvλ(w),

where, by (4.5), the second term tends to 0 as m→∞. By the first inequality in(4.6), the first term above can be made arbitrarily small by taking r sufficientlyclose to 1. Finally Lemma 4.3 and the second inequality in (4.6) yield

‖Sm‖ ≤ C(n, s, λ)

(supz∈Bn

‖(Sm)z1‖s,λ)1/2 (

supz∈Bn

‖(S∗m)z1‖s,λ

)1/2

→ 0,

as m→∞ proving the statement of the lemma. �

Corollary 4.6. Let S ∈ L(A2λ(B

n)) such that for some p > 2 + n1+λ ,

supz∈Bn

‖Sz1− (TBm,λS)z1‖p,λ ≤ C and supz∈Bn

‖S∗z1− (TBm,λS∗)z1‖p,λ ≤ C,

(4.7)


where C > 0 is independent of m. Then TBm,λS → S as m → ∞ in L(A2λ(B

n))-norm.

Proof. We set Sm = S − TBm,λS . By Proposition 3.8 we have

B0,λSm = B0,λS −B0,λ(TBm,λS) = B0,λS −Bm,λB0,λS,

which, by Corollary 3.11, uniformly tends to 0 as m → ∞, i.e., ‖B0,λSm‖∞ → 0.An application of Lemma 4.5 finishes the proof. �Theorem 4.7. Let S ∈ L(A2

λ(Bn)). If there is p > 2 + n

1+λ such that

supz∈Bn

‖T(Bm,λS)◦φz1‖p,λ ≤ C and sup

z∈Bn

‖T ∗(Bm,λS)◦φz

1‖p,λ ≤ C, (4.8)

where C > 0 is independent of m. Then TBm,λS → S as m→∞ in the L(A2λ(B

n))-norm.

Proof. We prove first thatsupz∈Bn

‖Sz1‖p,λ <∞. (4.9)

The equality T(Bm,λS)◦φz= (TBm,λS)z , together with Lemma 4.3, implies

‖TBm,λS‖ ≤ C(n, p, λ)(supz∈Bn

‖T(Bm,λS)◦φz1‖p,λ

)1/2(supz∈Bn

‖T ∗(Bm,λS)◦φz

1‖p,λ)1/2

,

< C, (4.10)

where C is independent of m. Let Sm = S−TBm,λS , then by the arguments in theproof of Corollary 4.6 we have

limm→∞ ‖B0,λSm‖∞ = 0.

According to (4.10) the sequence {Sm} is bounded; thus taking a polynomial fwith ‖f‖q,λ = 1 we obtain from Lemma 4.4 that

supz∈Bn

|〈(Sm)z1, f〉λ| → 0, as m→∞.

Then for any z0 ∈ Bn and any ε > 0, there is (a sufficiently large) m such that

|〈Sz01, f〉λ| ≤ supz∈Bn

|〈(Sm)z1, f〉λ|+ |〈(TBm,λS)z01, f〉λ| ≤ ε+ C,

with C being independent of m and z0. This proves (4.9). Further, the equality

T ∗(Bm,λS)◦φz

= TBm,λSz= TBm,λS∗

z= T(Bm,λS∗)◦φz

,

together with (4.8) and (4.9), implies (4.7), and Corollary 4.6 finishes the proof.�

Another approach to approximation theorems involves the invariant Lapla-cian and its application to the (m,λ)-Berezin transform. Recall that the invariant

Laplacian Δ on Bn, defined for u ∈ C2(Bn) and z ∈ Bn, is given by(Δu)(z) := Δ(u ◦ φz)(0), where Δ := 4

n∑j=1

∂2

∂zj∂zj.


Here ∂∂zj

and ∂∂zj

denote the Cauchy–Riemann operators with respect to the com-

plex coordinate zj , j = 1, . . . , n, and Δ is the standard Laplacian on Cn ∼= R2n.Let S ∈ L(A2

λ(Bn)) and m ∈ Z+, then we wish to calculate the function

ΔBm,λS ∈ C∞(Bn).

Note that in the case λ = 0 and n = 1 this calculation was done in Proposition2.4. of [9]. According to Theorem 3.5 we have

Δ[Bm,λS

](z) = Δ

(Bm,λS ◦ φz

)(0) = Δ

(Bm,λSz

)(0)

and therefore we can assume that z = 0. We intend to use the form of Bm,λSin Proposition 3.2. We apply Δ to the z-dependent part of Bm,λS in the integralrepresentation given there. Hence we have to evaluate the derivative

Δ

[(1− |z|2)m+λ+n+1

(1− 〈u, z〉)n+m+λ+1(1− 〈z, w〉)n+m+λ+1

](0)

= −4(m+ n+ λ+ 1) + 4(m+ n+ λ+ 1)2〈u,w〉.Inserting this relation into the expression of Bm,λS given in Proposition 3.2 shows

Δ(Bm,λS)(0) = −4(m+ n+ λ+ 1)(Bm,λS)(0) (4.11)

+ 4(m+ n+ λ+ 1)2cλ+m

cλ

∫Bn

∫Bn

(1− 〈u,w〉)m〈u,w〉S∗Kλ

w(u) dvλ(u)dvλ(w).

On the other hand the same proposition implies that

cλcλ+m

(Bm,λS) (0)− cλcλ+m+1

(Bm+1,λS) (0)

=

∫Bn

∫Bn

(1− 〈u,w〉)m〈u,w〉S∗Kλ

w(u) dvλ(u)dvλ(w).(4.12)

Combining the equations (4.11) and (4.12) now gives

Δ(Bm,λS)(0) = 4(m+ n+ λ+ 1)(m+ n+ λ)(Bm,λS

)(0)

− 4cλ+m

cλ+m+1(m+ n+ λ+ 1)2

(Bm+1,λS)(0).

According to (2.1) we have

cλ+m

cλ+m+1(m+ n+ λ+ 1)2 = (n+m+ λ+ 1)(λ+m+ 1)

and we have shown the following relation:

Proposition 4.8. Let S ∈ L(A2λ(B

n)). For all m ∈ Z+ and λ > −1 it holds

Δ[Bm,λS

]= 4(m+n+λ+1)

[(m+n+λ)

(Bm,λS

)−(m+λ+1)(Bm+1,λS

)]. (4.13)

Moreover, for all k,m we have

ΔBm,λ

(Bk,λS

)= Bm,λ

(ΔBk,λS

). (4.14)


Proof. It suffices to prove (4.14). According to Proposition 3.8 and using (4.13)we have

ΔBm,λ

(Bk,λS

)= ΔBk,λ

(Bm,λS

)= 4(k + n+ λ+ 1)

[(k + n+ λ)Bk,λBm,λS − (k + λ+ 1)Bk+1,λBm,λS

]= 4(k + n+ λ+ 1)

[(k + n+ λ)Bm,λBk,λS − (k + λ+ 1)Bm,λBk+1,λS

]= Bm,λ

(ΔBk,λS

),

which shows the assertion. �

For the remaining part of the section we specialize to the case of dimensionn = 1. Proposition 4.8 then implies

Bm,λS −Bm+1,λS =Δ[Bm,λS]

4(m+ λ+ 2)(m+ λ+ 1)(4.15)

and we can prove an analogue of Lemma 4.1 in [10]. We write D := B1 ⊂ C forthe open unit disc.

Proposition 4.9. Let S ∈ L(A2λ(D)) where λ > −1. Assume that ‖T

˜Δ(Bm,λS)‖ ≤ C

where C does not depend on m. Then we have the norm convergence

limm→∞ TBm,λS = S, (4.16)

Proof. According to (4.15) we can write

TBm+1,λS = TB0,λS −m∑

k=0

{TBk,λS − TBk+1,λS

}= TB0,λS −

m∑k=0

T˜Δ(Bk,λS)

4(k + λ+ 2)(k + λ+ 1).

From the boundedness assumption on ‖T˜Δ(Bk,λS)‖ we conclude that the right-

hand side of the equation converges in norm to some operator R ∈ L(A2λ(D)). The

continuity property of the usual Berezin transform B0,λ, cf. (3.4) implies that

limm→∞B0,λTBm,λS = B0,λR.

On the other hand note that Proposition 3.8 and Corollary 3.11 imply the pointwiseconvergence

B0,λ(TBm,λS) = B0,λBm,λ(S) = Bm,λB0,λ(S) −→ B0,λS

and it follows that B0,λS = B0,λR. Finally the injectivity of B0,λ shows thatS = R. �


Appendix: Toeplitz operators with measure symbols

The (k, λ)-Berezin transform for complex-valued regular measures on the weightedp-Bergman space over the unit ball was defined and studied in detail in [6], where,in particular, the approximation results were obtained. As was mentioned, this(k, λ)-Berezin transform cannot be applied to any bounded liner operator actingon A2

λ(Bn), which forced us to introduce the modified definition (3.3). In this

appendix, following arguments of [6], we remark that the approximation results of[6] remain valid for our definition of the (m,λ)-Berezin transform.

Recall that a positive finite regular measure ν on Bn is called Carleson mea-sure for A2

λ(Bn) if there is a constant C > 0 independent of h ∈ A2

λ(Bn) such

that ∫Bn

|h(z)|2dν(z) ≤ C

∫Bn

|h(z)|2dvλ(z).This means that the inclusion of A2

λ(Bn) into L2(Bn, dν) is well defined and con-

tinuous. Consider the operator Tν defined for all h ∈ H∞(Bn) and z ∈ Bn by

Tνh(z) :=

∫Bn

h(w)

(1− 〈z, w〉)n+λ+1dν(w). (A.1)

Then Tν is densely defined on A2λ(B

n) and it admits a bounded extension to anelement in L(A2

λ(Bn)) if and only if ν is a Carlson measure. In the case where μ is

a complex measure such that ν = |μ| is Carleson we can define Tμ in an analogousway and it can be checked that one obtains a bounded operator on A2

λ(Bn), as

well. We call Tμ the Toeplitz operator with measure symbol μ.Now we calculate the (m,λ)-Berezin transform for Toeplitz operators of the

form (A.1). First note that by a direct application of Fubini’s theorem we have⟨Tνf, g

⟩λ=

∫Bn

f(w)g(w)dν(w) =:⟨f, g⟩ν.

In particular, it holds⟨(Tν)zw

k, wk⟩λ=⟨TνUzw

k, Uzwk⟩λ

=⟨Uzw

k, Uzwk⟩ν

=

∫Bn

(1− |z|2)n+λ+1

|1− 〈z, w〉|2(n+λ+1)|φk

z(w)|2dν(w).

By plugging the last relation into the definition of the (m,λ)-Berezin transformand using (3.2) one obtains the integral transform

Bm,λ(ν)(z) : = (Bm,λTν)(z)

=cλ+m

cλ

m∑|k|=0

Cm,k

∫Bn

∣∣φkz(w)

∣∣2 (1− |z|2)n+λ+1

|1− 〈z, w〉|2(n+λ+1)dν(w)

=cλ+m

cλ

∫Bn

(1 − |φz(w)|2)m (1 − |z|2)n+λ+1

|1− 〈z, w〉|2(n+λ+1)dν(w).


We insert the well-known relation

1− |φz(w)|2 =(1− |z|2)(1− |w|2)

|1− 〈z, w〉|2 (A.2)

and finally obtain

Bm,λ(ν)(z) =cλ+m

cλ

∫Bn

(1− |z|2)n+λ+m+1

|1− 〈z, w〉|2(n+λ+m+1)(1− |w|2)mdν(w). (A.3)

In [6], Section 4 the (m,λ)-Berezin transform Bm(ν) of ν was defined as

Bm(ν)(z) :=cmcλ

∫Bn

(1− |φz(w)|2)n+1+m

(1− |w|2)n+1+λdν(w).

A comparison with (A.3) shows that the restriction of Bm,λ to Toeplitz opera-

tors Tν with measure symbols ν is related to Bm(ν)(z) by a simple shift in theparameter m:

Bm,λ(ν) = Bm+λ(ν).

Hence an approximation result completely analogous to Theorem 4.7 in [6] holdsfor Bm,λ(ν). We only restate the result in the case p = q = 2 according to thenotation in [6].

Theorem A.1. Let ν be a complex measure such that |ν| is Carleson. Fix f, g ∈A2

λ(Bn) and put h := fg ∈ L1(Bn, dvλ). Then∣∣∣ ∫

Bn

Bm,λ(ν)hdvλ −∫Bn

hdν∣∣∣ ≤ C(m)‖B0,λ(|ν|)‖∞‖f‖λ‖g‖λ, (A.4)

where limm→∞ C(m) = 0. In particular, for all λ > −1 we have the followingnorm convergence

limm→∞TBm,λ(ν) = Tν .

References

[1] F.A. Berezin, Covariant and contravariant symbols of operators (Russian). Izv. Akad.Nauk. SSSR Ser. Mat. 36 (1972), 1134–1167.

[2] W. Bauer, C. Herrera Yanez, N. Vasilevski, Eigenvalue characterization of radialoperators on weighted Bergman spaces over the unit ball, Integr. Equ. Oper. Theory,78(2) (2014), 271–300.

[3] B.R. Choe, Y.J. Lee, Pluriharmonic symbols of commuting Toeplitz operators, IllinoisJ. Math. 37 (1993), 424–436.

[4] M. Englis, Density of algebras generated by Toeplitz operators on Bergman spaces,Ark. Mat. 30 no. 2 (1992), 227–243.

[5] I.S. Gradshteyn, I.M. Ryzhik, Tables of integrals, series, and products, AcademicPress, 1980.

[6] M. Mitkovski, D. Suarez, and B.D. Wick, The essential norm of operators on Apα(Bn),

Integr. Equ. Oper. Theory 75, no. 2 (2013), 197–233.


[7] K. Nam, D. Zheng, and C. Zhong, m-Berezin transform and compact operators, Rev.Mat. Iberoamericana, 22(3) (2006), 867–892.

[8] W. Rudin, Function theory in the unit ball of Cn, Fundamental principles of Math-ematical Science 241, Springer-Verlag, New York-Berlin, 1980.

[9] D. Suarez, Approximation and symbolic calculus for Toeplitz algebras on the Bergmanspace, Rev. Mat. Iberoamericana 20(2) (2004), 563–610.

[10] D. Suarez , The eigenvalues of limits of radial Toeplitz operators, Bull. London Math.Soc. v. 40 (2008), 631–641.

[11] K. Zhu, Operator theory in function spaces, Marcel Dekker, Inc., 1990.

[12] K. Zhu, Spaces of holomorphic functions in the unit ball, Springer-Verlag, 2005.

Wolfram BauerMathematisches InstitutGeorg-August-UniversitatBunsenstr. 3–5D-37073 Gottingen, Germany

e-mail: [email protected]

Crispin Herrera Yanez and Nikolai VasilevskiDepartamento de MatematicasCINVESTAV del I.P.N.Av. IPN 2508, Col. San Pedro ZacatencoMexico D.F. 07360, Mexico

e-mail: [email protected]@math.cinvestav.mx


Normal and CohyponormalWeighted Composition Operators on H2

Carl C. Cowen, Sungeun Jung and Eungil Ko

Abstract. In this paper we study normal and cohyponormal weighted compo-sition operators on the Hardy space H2. We show that if Wf,ϕ is cohyponor-mal, then f is outer and ϕ is univalent. Moreover, we prove that when thecomposition map ϕ has the Denjoy–Wolff point in the open unit disk, Wf,ϕ iscohyponormal if and only if it is normal; in this case, f and ϕ can be expressedas linear fractional maps. As a corollary, we find the polar decomposition ofthe cohyponormal operator Wf,ϕ. Finally, we examine the commutant of acohyponormal weighted composition operator.

Mathematics Subject Classification (2010). 47B20, 47B38, 47B33.

Keywords. Weighted composition operator, composition operator, normal op-erator, hyponormal operator, cohyponormal operator, polar decomposition,commutant, inner-outer factorization.

1. Introduction

Let L(H) be the algebra of all bounded linear operators on a separable complexHilbert space H. If T ∈ L(H), we write r(T ), σ(T ), and σp(T ) for the spectralradius, the spectrum, and the point spectrum of T , respectively.

Let D denote the open unit disk in the complex plane C. The Hardy spaceH2(D) = H2 consists of all the analytic functions on D having power series rep-resentation with square summable complex coefficients. The space H∞(D) = H∞

consists of all the functions that are analytic and bounded on D. If ϕ is an ana-lytic mapping from D into itself, the composition operator Cϕ on H2 is defined byCϕh = h ◦ ϕ for all h ∈ H2. The composition operator Cϕ is bounded on H2 byLittlewood subordination theorem (see [10] for more details). It is well known that

This research was supported by Basic Science Research Program through the National Re-

search Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology(2012R1A2A2A02008590).


70 C.C. Cowen, S. Jung and E. Ko

the composition operator Cϕ defined on H2 is normal if and only if ϕ(z) = γzwhere |γ| ≤ 1 (see [9] or [10]).

For an analytic function f on D and an analytic selfmap ϕ of D, the operatorWf,ϕ : H2 → H2 given by

Wf,ϕh = f · (h ◦ ϕ), h ∈ H2

is called a weighted composition operator. If f is bounded on D, then Wf,ϕ isclearly bounded on H2. For f ∈ H∞, the multiplication operator on H2 is givenby Mfh = fh for all h ∈ H2. Remark that Wf,ϕ can be written by Wf,ϕ = MfCϕ

if f ∈ H∞. We refer to [1], [7], and [8] for more details about weighted compositionoperators. In particular, C.C. Cowen and E. Ko characterized self-adjoint weightedcomposition operators on H2 (see [8]), and P. Bourdon and S.K. Narayan providedseveral properties of normal weighted composition operators on H2 (see [1]).

An operator T ∈ L(H) is said to be hyponormal if T ∗T ≥ TT ∗. We say thatT ∈ L(H) is cohyponormal if T ∗ is hyponormal, i.e.,

TT ∗ ≥ T ∗T, or equivalently, ‖T ∗x‖ ≥ ‖Tx‖ for all x ∈ H.

We refer the reader to [4] and [11] for hyponormal operators. In this paperwe study normal and cohyponormal weighted composition operators on the HardyspaceH2. We show that ifWf,ϕ is cohyponormal, then f is outer and ϕ is univalent.Moreover, we prove that when the composition map ϕ has the Denjoy–Wolff pointin the open unit disk, Wf,ϕ is cohyponormal if and only if it is normal; in thiscase, f and ϕ can be expressed as linear fractional maps. As a corollary, we findthe polar decomposition of the cohyponormal operator Wf,ϕ. Finally, we examinethe commutant of a cohyponormal weighted composition operator.

2. Preliminaries

In this section, we recall some definitions needed for our program. Let ∂D denotethe unit circle in the complex plane C. For each β ∈ D, the function Kβ(z) =

11−βz

∈ H2, called the reproducing kernel for H2 at β, has the property that

〈f,Kβ〉 = f(β) for every f ∈ H2 and β ∈ D. It is well known that the linear spanof the reproducing kernels {Kβ : β ∈ D} is dense in H2. In [6], C.C. Cowen gavean adjoint formula of a composition operator whose symbol is a linear fractionalselfmap of D. If ϕ(z) = az+b

cz+d is a linear fractional selfmap of D, then C∗ϕ =

MgCσM∗h where g(z) = 1

−bz+d, σ(z) = az−c

−bz+d, and h(z) = cz + d. It follows from

[6] that σ is a selfmap of D and g ∈ H∞. Notice that W ∗f,ϕKβ = f(β)Kϕ(β) when

Wf,ϕ is bounded on H2 and β ∈ D (see [8]); indeed, for any β ∈ D and f ∈ H2

〈f,W ∗f,ϕKβ〉 = 〈f · (h ◦ ϕ),Kβ〉 = f(β)h(ϕ(β)) = 〈h, f(β)Kϕ(β)〉.

In particular, C∗ϕKβ = Kϕ(β) because Cϕ = W1,ϕ.

For any selfmap ϕ of D and each positive integer n, we write ϕ1 := ϕ andϕn+1 := ϕ ◦ ϕn, which is called the iterate of ϕ for n. We also denote by ϕ0 the

Normal and Cohyponormal Weighted Composition Operators on H2 71

identical function on D. When ϕ is any analytic selfmap of D, we call w ∈ D a fixedpoint of ϕ if limr→1− ϕ(rw) = w. We say that a function f has a nontangentiallimit at ζ ∈ ∂D if limz→ζ f(z) exists in the nontangential region

Γ(ζ, α) = {z ∈ D : |z − ζ| < α(1− |z|)}for each α > 1. An analytic selfmap ϕ of D is said to have a finite angular derivative

at ζ ∈ ∂D if there exists η ∈ ∂D such that ϕ(z)−ηz−ζ has a finite nontangential limit

as z → ζ. This limit is written as ϕ′(ζ) if it exists (see [10, page 50] for moredetails).

It is well known that for an analytic function ϕ : D → D, if ϕ is neither theidentity map nor an elliptic automorphism of D, then there is a point w of D sothat the iterates of ϕ converges uniformly to w on compact subsets of D. Moreover,w is the unique fixed point of ϕ in D for which |ϕ′(w)| ≤ 1. We say that the uniquefixed point w is the Denjoy–Wolff point of ϕ.

Let ϕ be an automorphism of D. Then ϕ is of the form

ϕ(z) =az + b

bz + a

for all z ∈ D, where a and b in C with |a|2 − |b|2 = 1. When b �= 0, it is easy tocalculate that

iIm(a)±√|b|2 − (Im(a))2

b

are the fixed points of ϕ. If | Im(a)| = |b|, then ϕ is called parabolic, and we saythat ϕ is hyperbolic if | Im(a)| < |b|. Remark that ϕ is parabolic if and only if ithas one fixed point lying on ∂D, while ϕ is hyperbolic if and only if it has twofixed points lying on ∂D. If | Im(a)| > |b|, then ϕ is said to be elliptic. We notethat ϕ is elliptic if and only if one of its fixed points is inside D and another isoutside D. In this sense, this type also includes the case when b = 0, i.e., when 0and ∞ are the fixed points of ϕ.

3. Main results

Throughout this section, we examine several properties of the weight function fand the composition symbol ϕ of a cohyponormal weighted composition operatorWf,ϕ ∈ L(H2).

Lemma 3.1. If f ∈ H∞, then ker(M∗f ) = (ψH2)⊥ where f = ψF , ψ is an inner

function, and F is an outer function.

Proof. If g ∈ (ψH2)⊥, then for any h ∈ H2 we get that

〈M∗f g, h〉 = 〈g,Mfh〉 = 〈g, ψFh〉 = 0.

Hence M∗f g = 0 and so (ψH2)⊥ ⊆ ker(M∗

f ).


On the other hand, if there exists g ∈ ker(M∗f ) \ (ψH2)⊥, then there exists

u ∈ H2 such that 〈g, ψu〉 �= 0 but M∗f g = 0. Since F is outer, we have 1

F ∈ H∞

and so it follows that

0 =⟨M∗

f g,u

F

⟩= 〈g, ψu〉 �= 0,

which is a contradiction. Thus (ψH2)⊥ ⊇ ker(M∗f ). �

Theorem 3.2. Let ϕ be a nonconstant analytic selfmap of D and let f ∈ H∞ be notidentically zero. If Wf,ϕ is cohyponormal, then f is outer; hence never vanisheson D and ϕ is univalent.

Proof. Suppose that Wf,ϕ is cohyponormal. If Wf,ϕh = 0 for some h ∈ H2, thenf(z)h(ϕ(z)) ≡ 0. Since f is not identically zero, h(ϕ(z)) ≡ 0 on D. Since ϕ isnonconstant, h(z) ≡ 0 on D. Hence ker(Wf,ϕ) = {0}. Since Wf,ϕ is cohyponormal,ker(W ∗

f,ϕ) ⊆ ker(Wf,ϕ) = {0}. Since ker(M∗f ) ⊆ ker(W ∗

f,ϕ), we have ker(M∗f ) =

{0}. Since f ∈ H2 is not identically zero, f can be factorized as f = ψF where ψis inner and F is outer. Then Mf = MψMF = MFMψ and M∗

f = M∗FM

∗ψ. Since ψ

is inner, Mψ is an isometry of H2 onto ψH2. If ψ is not constant, then H2 �= ψH2.Hence Lemma 3.1 implies that ker(M∗

f ) = (ψH2)⊥ �= {0}, a contradiction. Thusψ must be constant, and so f is an outer function. In particular, f never vanisheson D.

Assume that ϕ is not univalent on D. Then we can choose distinct pointsα, β ∈ D such that ϕ(α) = ϕ(β). Since f never vanishes on D, we obtain thatf(α) �= 0 and f(β) �= 0. Set g = 1

f(α)Kα− 1

f(β)Kβ. If g is the zero function in H2,

then we have f(α)1−αz = f(β)

1−βzfor all z ∈ D, but it gives the contradiction that α = β.

Hence g is not the zero function in H2. Since W ∗f,ϕg = Kϕ(α) − Kϕ(β) = 0 and

W ∗f,ϕ is hyponormal, it holds that Wf,ϕg = 0. This means that f(z)g(ϕ(z)) = 0 for

all z ∈ D. Moreover, f has no zeros in D by the above argument. Thus g(ϕ(z)) = 0for all z ∈ D. Since ϕ(D) is a nonempty open subset of D, we get that g ≡ 0 on D,which is a contradiction. Therefore, ϕ should be univalent. �

From the proof of Theorem 3.2 we observe that a cohyponormal weightedcomposition operator Wf,ϕ is Fredholm if and only if it has closed range. For anoperator T ∈ L(H), a vector x ∈ H is said to be cyclic if the linear span of theorbit O(x, T ) := {T nx}∞n=0 is norm dense in H. If there is a cyclic vector x for T ,then we say that T is a cyclic operator. If the orbit O(x, T ) is norm dense in Hfor some x ∈ H, i.e., O(x, T ) = H, then T is called hypercyclic operator and x iscalled a hypercyclic vector. It is obvious that every hypercyclic operator is cyclic.

Corollary 3.3. Let ϕ be an analytic selfmap, not an elliptic automorphism, of Dwith a fixed point in D and let f ∈ H∞ \ {0}. If Wf,ϕ is cohyponormal, then W ∗

f,ϕ

is cyclic.

Proof. We note that there exists a point z0 ∈ D so that the sequence {ϕn(z0)}∞n=0

consists of pairwise distinct points in D which converges to the interior fixed point


of ϕ by [17, Lemma 1] and [16, Section 5.2, Proposition 1]. Let g ∈ H2 be suchthat g ⊥ ∨∞

n=0{(W ∗f,ϕ)

nKz0}. Since it holds that for any positive integer n

Wnf,ϕ = Wf ·(f◦ϕ)·(f◦ϕ2)···(f◦ϕn−1),ϕn

,

we obtain that

0 = 〈g, (W ∗f,ϕ)

nKz0〉 = 〈Wnf,ϕg,Kz0〉

= f(z0)f(ϕ(z0))f(ϕ2(z0)) · · · f(ϕn−1(z0))g(ϕn(z0))

for any positive integer n. Since f has no zeros in D by Theorem 3.2, it followsthat g(ϕn(z0)) = 0 for any positive integer n. Thus g ≡ 0 on D and so we have∨∞

n=0{(W ∗f,ϕ)

nKz0} = H2. �

Next we give a lower bound and an upper bound for the spectral radius of acohyponormal weighted composition operator.

Lemma 3.4. Let ϕ be an analytic selfmap of D and let f ∈ H∞. Then

|f(0)f(ϕ(0)) · · · f(ϕn−1(0))|√1− |ϕn(0)|2

≤ ‖Wnf,ϕ‖ ≤

2‖f‖n∞√1− |ϕn(0)|2

for each positive integer n.

Proof. It follows from [5, Theorem 2.1] that

‖Wnf,ϕ‖ = ‖Wf ·(f◦ϕ)···(f◦ϕn−1),ϕn

‖ ≤ ‖Mf ·(f◦ϕ)···(f◦ϕn−1)‖‖Cϕn‖

≤ ‖f‖n∞1 + |ϕn(0)|√1− |ϕn(0)|2

≤ 2‖f‖n∞√1− |ϕn(0)|2

.

On the other hand, it holds that

‖Wnf,ϕ‖ = ‖W ∗n

f,ϕ‖ ≥ ‖W ∗nf,ϕK0‖ = ‖W ∗

f ·(f◦ϕ)···(f◦ϕn−1),ϕnK0‖

= |f(0)f(ϕ(0)) · · · f(ϕn−1(0))| ‖Kϕn(0)‖

=|f(0)f(ϕ(0)) · · · f(ϕn−1(0))|√

1− |ϕn(0)|2.

Hence we complete the proof. �

Theorem 3.5. Let ϕ be an analytic selfmap of D with the Denjoy–Wolff point w,and let f ∈ H∞. If Wf,ϕ is cohyponormal, then the following statements hold:

(i) If w ∈ D, then

|f(w)| ≤ r(Wf,ϕ) ≤ ‖f‖∞.

(ii) If w ∈ ∂D and f is continuous at w, then

ϕ′(w)−12 |f(w)| ≤ r(Wf,ϕ) ≤ ϕ′(w)−

12 ‖f‖∞.


Proof. We may assume that f is not identically zero on D. By Lemma 3.4 we have

|f(0)f(ϕ(0)) · · · f(ϕn−1(0))| 1n(1− |ϕn(0)|2) 1

2n

≤ ‖Wnf,ϕ‖

1n ≤ 2

1n ‖f‖∞

(1− |ϕn(0)|2) 12n

for each positive integer n. Put xn = log |f(ϕn(0))| for each nonnegative integern; here, we note that f(ϕn(0)) �= 0 and f(w) �= 0 from Theorem 3.2. In addition,limn→∞ xn = log |f(w)|, which implies that

limn→∞ |f(0)f(ϕ(0)) · · · f(ϕn−1(0))| 1n = |f(w)|. (1)

On the other hand, since it holds for all n that

1

(1 − |ϕn(0)|2) 12n

≤ ‖Cnϕ‖

1n ≤ 2

1n

(1− |ϕn(0)|2) 12n

from [5, Theorem 2.1] or Lemma 3.4, we ensure that

r(Cϕ) = limn→∞

1

(1− |ϕn(0)|2) 12n

. (2)

Hence we obtain from (1) and (2) that r(Cϕ)|f(w)| ≤ r(Wf,ϕ) ≤ r(Cϕ)‖f‖∞, andso the proof follows from [10, Theorem 3.9]. �

Recall that for any α ∈ D and any positive integer j, the jth derivative

evaluation kernel at α, K[j]α , is the function in H2 so that

〈f,K [j]α 〉 = f (j)(α)

for f ∈ H2. In particular, K[0]α := Kα.

Lemma 3.6. Let ϕ be an analytic selfmap of D with a fixed point w ∈ D and letf ∈ H∞. If Wf,ϕ is a cohyponormal operator on H2, then

{f(w), f(w)ϕ′(w), f(w)(ϕ′(w))2, f(w)(ϕ′(w))3, . . . } ⊆ σp(Wf,ϕ).

Proof. Let g be any function in H2 and let n be any positive integer. Then

〈g,W ∗f,ϕK

[n]w 〉 = 〈Wf,ϕg,K

[n]w 〉 = 〈f · (g ◦ ϕ),K [n]

w 〉 = (f · (g ◦ ϕ))(n)(w)

=

n−1∑j=0

γj(w)g(j)(w) + f(w)g(n)(w)(ϕ′(w))n

= 〈g,n−1∑j=0

γj(w)K[j]w + f(w)(ϕ′(w))nK [n]

w 〉

where each γj is a function consisting of sums of products of derivatives of f andϕ at w. Since g is arbitrary in H2, we have

W ∗f,ϕK

[n]w =

n−1∑j=0

γj(w)K[j]w + f(w)(ϕ′(w))nK [n]

w (3)


for any positive integer n. Moreover, we get that

W ∗f,ϕKw = f(w)Kϕ(w) = f(w)Kw. (4)

For any nonnegative integer n, set Mn := span{Kw,K[1]w ,K

[2]w , . . . ,K

[n]w }. Then

W ∗f,ϕMn ⊆ Mn from (3) and (4). Since Mn is finite dimensional, it is closed and

so an invariant subspace for W ∗f,ϕ. Hence the operator W ∗

f,ϕ can be written as

W ∗f,ϕ =

(W ∗

f,ϕ|Mn B

0 C

)on H2 = Mn ⊕M⊥

n

where W ∗f,ϕ|Mn stands for the restriction of W ∗

f,ϕ to the invariant subspace Mn.

With respect to the basis {Kw,K[1]w ,K

[2]w , . . . ,K

[n]w } of Mn, the operator W ∗

f,ϕ|Mn

has the following matrix representation:

W ∗f,ϕ|Mn =

⎛⎜⎜⎜⎜⎝f(w) ∗ · · · ∗

f(w)ϕ′(w)...

. . ....

0 f(w)(ϕ′(w))n

⎞⎟⎟⎟⎟⎠ .

Since W ∗f,ϕ|Mn is a finite-dimensional upper triangular matrix, we obtain that

σp(W∗f,ϕ) ⊇ σp(W

∗f,ϕ|Mn) = {f(w), f(w)ϕ′(w), . . . , f(w)(ϕ′(w))n}.

Such an inclusion holds for every nonnegative integer n, and hence we have

{f(w), f(w)ϕ′(w), f(w)(ϕ′(w))2, f(w)(ϕ′(w))3, . . . } ⊆ σp(W∗f,ϕ).

Since Wf,ϕ is cohyponormal, we complete the proof. �

Theorem 3.7. Let ϕ be an analytic selfmap of D with the Denjoy–Wolff point w ∈ Dand let f ∈ H∞ \ {0}. Then the following statements are equivalent:

(i) Wf,ϕ is normal.

(ii) Wf,ϕ is cohyponormal.

(iii) The functions f and ϕ have the forms of

f(z) = f(w)1 − wϕ(z)

1− wzand ϕ(z) = a0 +

d1z

1− d0z(5)

where

a0 = ϕ(0) =w(1 − ϕ′(w))1− |w|2ϕ′(w)

, d0 =w(1 − ϕ′(w))

1− |w|2ϕ′(w),

and

d1 = ϕ′(0) =ϕ′(w)(1 − |w|2)2(1− |w|2ϕ′(w))2

.


Proof. Since every normal operator is cohyponormal, it suffices to show that (ii)⇒(iii) and (iii) ⇒ (i). Suppose that Wf,ϕ is cohyponormal. Since ϕ has the Denjoy–Wolff point w ∈ D, we have |ϕ′(w)| < 1. In addition, f(w) �= 0 by Theorem

3.2. Thus f(w) and f(w)ϕ′(w) are distinct eigenvalues of W ∗f,ϕ from the proof of

Lemma 3.6. Recall that if T is a hyponormal operator on a Hilbert space and uand v are eigenvectors corresponding to distinct eigenvalues of T , then u ⊥ v (see[4, Proposition 4.4]). Since Wf,ϕ is cohyponormal, we get that

ker(W ∗f,ϕ − f(w)) ⊥ ker(W ∗

f,ϕ − f(w)ϕ′(w)).

Since ϕ(w) = w, we obtain that W ∗f,ϕKw = f(w)Kw. Due to the cohyponor-

mality of Wf,ϕ, we have Wf,ϕKw = f(w)Kw, which ensures that

f(z) = f(w)1− wϕ(z)

1− wz. (6)

From the proof of Lemma 3.6, there exists an eigenvector g = β0Kw + β1K[1]w of

W ∗f,ϕ|M1 corresponding to the eigenvalue f(w)ϕ′(w). Then g is orthogonal to Kw,

and so it holds that

0 = 〈g,Kw〉 = β0Kw(w) + β1K[1]w (w).

Since Kw(z) = 11−wz and K

[1]w (z) = z

(1−wz)2 , we obtain that β0 = − w1−|w|2β1.

Therefore, we have g = −β1(w

1−|w|2Kw − K[1]w ). Here, it is easy to see that the

function w1−|w|2Kw −K

[1]w is not the zero map on D. Indeed, if w

1−|w|2Kw = K[1]w ,

then w(1 − wz)2 = (1 − |w|2)(1 − wz)z for all z ∈ D. Then we have w = 0 and−2|w|2 = 1−|w|2 by comparing the constant terms and the coefficients of z, which

is a contradiction. Thus w1−|w|2Kw−K

[1]w is an eigenvector for W ∗

f,ϕ corresponding

to the eigenvalue f(w)ϕ′(w). Since Wf,ϕ is cohyponormal, w1−|w|2Kw −K

[1]w is an

eigenvector for Wf,ϕ corresponding to the eigenvalue f(w)ϕ′(w). Hence we getthat

Wf,ϕ

(w

1− |w|2Kw −K [1]w

)= f(w)ϕ′(w)

(w

1− |w|2Kw −K [1]w

). (7)

Since K[1]w (z) = z

(1−wz)2 , it follows from (7) that

f(z)

{w

(1− |w|2)(1 − wϕ(z))− ϕ(z)

(1 − wϕ(z))2

}= f(w)ϕ′(w)

{(w

1− |w|2)(

1

1− wz

)− z

(1 − wz)2

}.

Since f(w) �= 0, we obtain that

ϕ(z) =(|w|2 − ϕ′(w))z − w(1 − ϕ′(w))w(1 − ϕ′(w))z − (1− |w|2ϕ′(w))

=− |w|2−ϕ′(w)

1−|w|2ϕ′(w)z +w(1−ϕ′(w))1−|w|2ϕ′(w)

1− w(1−ϕ′(w))1−|w|2ϕ′(w)z

. (8)


We note thatw(1 − ϕ′(w))1− |w|2ϕ′(w)

= ϕ(0) ∈ D.

Moreover, since

ϕ′(z) =ϕ′(w)(1 − |w|2)2

{(1− |w|2ϕ′(w)) − w(1− ϕ′(w))z}2 ,

we get that

ϕ′(0) =ϕ′(w)(1 − |w|2)2(1− |w|2ϕ′(w))2

.

Set a0 = ϕ(0), d1 = ϕ′(0), and d0 = w(1−ϕ′(w))

1−|w|2ϕ′(w). Then

a0d0 − d1 =|w|2(1 − ϕ′(w))2

(1 − |w|2ϕ′(w))2− ϕ′(w)(1 − |w|2)2

(1 − |w|2ϕ′(w))2=

|w|2 − ϕ′(w)1− |w|2ϕ′(w)

.

Thus ϕ(z) = a0−(a0d0−d1)z

1−d0z= a0+

d1z

1−d0z, which completes the proof of (ii) ⇒ (iii).

Assume that f and ϕ have the forms specified in (5). We want to show thatWf,ϕ is unitarily equivalent to Cϕ′(w)z that is a normal operator since |ϕ′(w)| < 1.From (8), we observe that

ϕ(z) =(|w|2 − ϕ′(w))z + w(ϕ′(w) − 1)

w(1− ϕ′(w))z + (|w|2ϕ′(w) − 1).

Hence, we have the following expression of ϕ:

ϕ = ηw ◦ (ϕ′(w)ηw)

where ηw(z) =w−z1−wz . Furthermore, it is obvious that

f = f(w)Kw

Kw ◦ ϕ.

Since the hyponormality is invariant under scalar multiples, we may assume thatf(w) = 1. This yields that

f(z) =Kw(z)

Kw(ϕ(z))=

1− |w|21− |w|2ϕ′(w) − w(1 − ϕ′(w))z

.

From [6, Theorem 2] we have

C∗ηw

= MKwCηwM∗h (9)

where h(z) = 1− wz. In addition, it holds that

M∗hM

∗ψw

= M∗hψw

=1

‖Kw‖I (10)

where ψw(z) =Kw

‖Kw‖ . It follows from (9) and (10) that

W ∗ψw ,ηw

= C∗ηw

M∗ψw

= MKwCηwM∗hM

∗ψw

=1

‖Kw‖MKwCηw .


Thus we get that

Wψw ,ηwCϕ′(w)zW∗ψw,ηw

=1

‖Kw‖MψwCηwCϕ′(w)zMKwCηw

=Kw

‖Kw‖2Kw ◦ (ϕ′(w)ηw)Cηw◦(ϕ′(w)ηw)

=Kw

‖Kw‖2Kw ◦ (ϕ′(w)ηw)Cϕ.

(11)

Since

Kw

‖Kw‖2Kw ◦ (ϕ′(w)ηw) =(1− |w|21− wz

)( 1− wz

1− |w|2ϕ′(w) − w(1− ϕ′(w))z)= f(z),

we obtain from (11) that

Wf,ϕ = Wψw,ηwCϕ′(w)zW∗ψw ,ηw

.

Since Wψw,ηw is unitary from [1, Theorem 6] and Cϕ′(w)z is normal, we concludethat Wf,ϕ is also normal. Hence (iii) ⇒ (i). �

Next we give an example which satisfies Theorem 3.7.

Example 3.8. Let ϕ be an analytic selfmap of D with the Denjoy–Wolff point 12 .

From the proof of Theorem 3.7 we obtain that

ϕ(z) =

(14 − ϕ′ ( 1

2

))z + 1

2

(ϕ′ ( 1

2

)− 1)

12

(1− ϕ′ ( 1

2

))z +

(14ϕ

′ ( 12

)− 1) .

and

f(z) =3/4(

1− 14ϕ

′ ( 12

))− 12

(1− ϕ′ ( 1

2

))z.

Then Wf,ϕ is unitarily equivalent to Cϕ′( 12 )z

. Hence Wf,ϕ is cohyponormal. In

particular, if ϕ′(12 ) is not real, then Wf,ϕ is not self-adjoint from [8, Theorem 5].

As an application of Theorem 3.7, we show that the symbol ϕ has an exteriorfixed point when b �= 0.

Corollary 3.9. Let ϕ be an analytic selfmap of D with the Denjoy–Wolff pointw ∈ D \ {0} and let f ∈ H∞ \ {0}. If Wf,ϕ is a cohyponormal operator on H2,then ϕ has another fixed point 1

w .

Proof. It follows from Theorem 3.7 that

ϕ(z) = a0 +d1z

1− d0z

where a0 = ϕ(0) = w(1−ϕ′(w))1−|w|2ϕ′(w) , d0 = w(1−ϕ′(w))

1−|w|2ϕ′(w), and d1 = ϕ′(0) = ϕ′(w)(1−|w|2)2

(1−|w|2ϕ′(w))2 .

In order to find the fixed points of ϕ, we must solve the equation ϕ(z) = z, that

is, d0z2 + (d1 − a0d0 − 1)z+ a0 = 0. Since w �= 0 and |ϕ′(w)| < 1, we have d0 �= 0.

Hence, if γ is another zero of the quadratic polynomial d0z2+(d1−a0d0−1)z+a0,

then wγ = a0

d0= w

w , i.e., γ = 1w . �


Corollary 3.10. Let ϕ be an analytic selfmap of D with the Denjoy–Wolff pointw ∈ D and let f ∈ H∞ \ {0}. If Wf,ϕ is a cohyponormal operator on H2, then itis compact and σ(Wf,ϕ) = {0, f(w)[ϕ′(w)]n : n = 0, 1, 2, . . .}.Proof. As in the proof of Theorem 3.7, we can show that Wf,ϕ is unitarily equiva-lent to Cϕ′(w)z. Since w ∈ D, we have |ϕ′(w)| < 1. Hence Cϕ′(w)z is compact by [12],and so Wf,ϕ is also compact. The result for the spectrum follows from [12]. �Corollary 3.11. Let ϕ be an analytic selfmap of D with the Denjoy–Wolff pointw ∈ D and ϕ′(w) �= 0, and let f ∈ H∞ \ {0}. If Wf,ϕ is a cohyponormal operatoron H2, then

Wf,ϕ =

∞∑n=0

f(w)[ϕ′(w)]nPn

where Pn is the orthogonal projection of H2 onto ker(Wf,ϕ − f(w)[ϕ′(w)]nI).

Proof. By Theorem 3.7 and Corollary 3.10, Wf,ϕ is compact and normal. Sincef(w) �= 0 and 0 < |ϕ′(w)| < 1, the set {f(w)(ϕ′(w))n : n = 0, 1, 2, . . .} consistsof distinct nonzero eigenvalues of Wf,ϕ by Corollary 3.10. Thus the proof followsfrom [3]. �Corollary 3.12. Let ϕ be an analytic selfmap of D with the Denjoy–Wolff pointw ∈ D and ϕ′(w) �= 0, and let f ∈ H∞ \ {0}. If Wf,ϕ is a cohyponormal operatoron H2, the following assertions hold:

(i) each Pn has finite rank,

(ii)( ∨ {Pn(H

2) : n ≥ 0})⊥ = {0}, and(iii) ‖Wf,ϕ‖ = |f(w)|where Pn is the orthogonal projection of H2 onto ker(Wf,ϕ − f(w)[ϕ′(w)]nI).

Proof. The proof follows from [3, Corollary II.7.8] and Theorem 3.2 (for (ii), usethe fact that ker(Wf,ϕ) = {0}). �

If T = U |T | is the polar decomposition of T ∈ L(H), then T := |T | 12U |T | 12called the Aluthge transform of T .

Corollary 3.13. Let ϕ be an analytic selfmap of D with the Denjoy–Wolff pointw ∈ D and ϕ′(w) �= 0, and let f ∈ H∞ \ {0}. If Wf,ϕ is a cohyponormal operator

on H2, then Wf,ϕ = Wf,ϕ.

Proof. Let f(w) = |f(w)|eiθ and ϕ′(w) = |ϕ′(w)|eiθ . If Wf,ϕ = U |Wf,ϕ| is thepolar decomposition of the weighted composition operator Wf,ϕ, then it is trivial

that U =∑∞

n=0 eiθeinθPn and |Wf,ϕ| 12 =

∑∞n=0 |f(w)|

12 |ϕ′(w)|n2 Pn where Pn de-

notes the orthogonal projection of H2 onto ker(Wf,ϕ − f(w)[ϕ′(w)]nI) for each n.Hence we get that

Wf,ϕ = |Wf,ϕ| 12U |Wf,ϕ| 12 =∞∑

n=0

f(w)(ϕ′(w))nPn = Wf,ϕ.

So we complete our proof. �


Corollary 3.14. Let ϕ be an analytic selfmap of D with the Denjoy–Wolff pointw ∈ D and ϕ′(w) �= 0, and let f ∈ H∞ \ {0}. If Wf,ϕ is a cohyponormal operatoron H2, then it is diagonalizable.

Proof. Since Wf,ϕ is compact and normal by Theorem 3.7 and Corollary 3.10, theproof follows from [3, Corollary II.7.9]. �

Corollary 3.15. Let ϕ(z) = az+bcz+1 be a linear fractional transformation mapping D

into itself such that ϕ(1) = 1 and ϕ(λ) = λ for some λ ∈ D, and let f ∈ H∞ \ {0}.Then Wf,ϕ is cohyponormal if and only if ϕ is the identity map and f is constanton D.

Proof. Since 1 = ϕ(1) = a+bc+1 and λ = ϕ(λ) = aλ+b

cλ+1 , we get that a+ b− c = 1 and

λa+ b− λ2c = λ, and so

a = 1+ (λ + 1)c and b = −λc. (12)

Thus it holds that ⎧⎪⎪⎨⎪⎪⎩ϕ(0) = −λc,ϕ′(0) = (c+ 1)(λc+ 1), and

ϕ′(λ) = (c+1)(λc+1)(λc+1)2 = c+1

λc+1 .

(13)

Here, λc+ 1 �= 0 since ϕ(0) �= 1. If Wf,ϕ is cohyponormal, then ϕ has the form in(5) of Theorem 3.7. Assume λ �= 0. Then we obtain from (13) that

−λc = ϕ(0) =λ(1− ϕ′(λ))1− |λ|2ϕ′(λ)

=λ(λ− 1)c

λ(1 − λ)c+ (1− |λ|2)and thus λ2(λ − 1)c(c+ 1) = 0. Since λ �= 0 and λ �= 1, we have c = 0 or c = −1.However, since there exist at most two fixed points of ϕ in C and λ �= 0, it followsthat ϕ(0) �= 0 and so c �= 0. Thus c = −1 and then ϕ′(1) = ϕ′(λ) = 0, butthis contradicts to the Denjoy–Wolff theorem. Hence λ = 0. This implies that

ϕ(z) = (c+1)zcz+1 by (12). We note that c �= −1 since ϕ is a linear fractional map. On

the other hand, with the same notations we used in (5), a0 = ϕ(0) = 0, d0 = 0,and d1 = ϕ′(0) = c+ 1 from (13). Then it follows that

(c+ 1)z

cz + 1= ϕ(z) = a0 +

d1z

1− d0z= (c+ 1)z for all z ∈ D,

which yields that c = 0. Thus ϕ is the identity map on D. Moreover, the weightfunction f is constant on D by (5).

Conversely, if ϕ(z) ≡ z and f(z) ≡ γ on D for some constant γ ∈ C, thenWf,ϕ = γI is trivially cohyponormal. �

Remark. Although replacing ϕ with an analytic selfmap of D in Corollary 3.15,we have the same result. Let Wf,ϕ be a cohyponormal operator on H2, where ϕis an analytic selfmap of D with the Denjoy–Wolff point in the open unit disk andϕ(1) = 1, and let f ∈ H∞ \ {0}. By Theorem 3.7, ϕ should be a linear fractional


transformation. Hence Wf,ϕ is a scalar multiple of the identity operator on H2

from Corollary 3.15.

Example 3.16. Set ϕ(z) = sz1−(1−s)z for some 0 < s < 1. Then ϕ is an analytic

selfmap of D with ϕ(0) = 0 and ϕ(1) = 1. Therefore it follows from Corollary 3.15that Wf,ϕ is not cohyponormal for any f ∈ H∞ \ {0}.

Provided that ϕ is an elliptic automorphism of D, we also get that cohyponor-mality and normality of a weighted composition operator Wf,ϕ are equivalent.

Proposition 3.17. Let ϕ be an elliptic automorphism of D and let f ∈ H∞. ThenWf,ϕ is cohyponormal on H2 if and only if it is normal on H2.

Proof. Suppose that Wf,ϕ is cohyponormal on H2. From [13, Lemma 3.0.6], Wf,ϕ

is similar to a weighted composition operator with composition map ϕ(z) = ϕ′(w)zwhere w is the fixed point of ϕ in D and |ϕ′(w)| = 1. In addition, if a hyponormaloperator is similar to a normal operator, then it is normal by [15]. Hence it suffices

to assume that ϕ(z) = γz for some |γ| = 1. Since W ∗f,ϕK0 = f(0)K0 and Wf,ϕ is

cohyponormal, we have Wf,ϕK0 = f(0)K0. This implies that f(z) ≡ f(0) on D,and so Wf,ϕ = f(0)Cγz is normal. The converse statement is trivial. �

Now we consider normal and cohyponormal weighted composition operatorswhose symbols are linear fractional transformations. If ϕ(z) = az+b

cz+d is a linear

fractional selfmap of D and f(z) = kcz+d , then we obtain from [1, Proposition 12]

that Wf,ϕ is normal if and only if

f(α)f(z)

1− ϕ(α)ϕ(z)=

kg(α)f(σ(α))

1 − ϕ(σ(α))z

for any α, z ∈ D, where g(z) = 1−bz+d

, σ(z) = az−c−bz+d

, and h(z) = cz + d. In

particular, if ϕ(z) = a0 +d1z

1−d0zis a selfmap of D and f(z) = 1

1−d0zwhere d0 ∈ D,

then it is easy to see that Wf,ϕ is normal if and only if

|a0| = |d0| and d0 + a0d1 − |a0|2d0 = a0 + d0d1 − a0|d0|2. (14)

Hence, if ϕ = a0 +d1z

1−d0zhas a fixed point w ∈ D and f(z) = 1

1−d0zwhere⎧⎪⎪⎪⎨⎪⎪⎪⎩

a0 = ϕ(0) = w(1−ϕ′(w))1−|w|2ϕ′(w)

d0 = w(1−ϕ′(w))

1−|w|2ϕ′(w)∈ D

d1 = ϕ′(0) = ϕ′(w)(1−|w|2)2(1−|w|2ϕ′(w))2 ,

then the equalities in (14) are true, which implies that Wf,ϕ is normal and socohyponormal.

Remark. Let ϕ(z) = a0+d1z

1−d0zbe a selfmap of D with a fixed point w ∈ D, where

a0 = ϕ(0) = w(1−ϕ′(w))1−|w|2ϕ′(w) , d0 = w(1−ϕ′(w))

1−|w|2ϕ′(w)∈ D, and d1 = ϕ′(0) = ϕ′(w)(1−|w|2)2

(1−|w|2ϕ′(w))2 ,

and let f ∈ H∞ \ {0}.


(i) Suppose that Wf,ϕ = γWg,ψ for some γ ∈ C \ {0} where ψ(z) = a0 +a1z

1−a0z

for some real a1 and g(z) = c1−a0z

for some real c. Since 1γWf,ϕ = Wg,ψ is

self-adjoint from [8, Theorem 5], we obtain that(1

γWf,ϕKα

)(0) =

(1

γW ∗

f,ϕKα

)(0) (15)

for any α ∈ D, which ensures that ϕ′(w) is real and f(z) = γc1−a0z

.

(ii) Assume that ϕ′(w) is real and f(z) = γc1−a0z

for some γ ∈ C. Since ϕ′(w) is

real, a0 = d0 and d1 is real. Hence it follows from [8, Theorem 5] that Wh,ϕ isself-adjoint where h(z) = c

1−a0z. Since γWh,ϕ = Wγh,ϕ = Wf,ϕ, we conclude

that Wf,ϕ is normal and hence cohyponormal.

Next we examine the commutant of a cohyponormal weighted compositionoperator. We denote the commutant of an operator T ∈ L(H) by {T }′.Theorem 3.18. Let ϕ be an analytic selfmap of D with the Denjoy–Wolff pointw ∈ D and let f ∈ H∞ \ {0}. Suppose that Wf,ϕ is cohyponormal. If ψ is ananalytic selfmap of D and g ∈ H∞ such that g(w) �= 0 and Wg,ψ ∈ {Wf,ϕ}′, thenw is a fixed point of ψ.

Proof. Since ϕ(w) = w ∈ D, we have{W ∗

g,ψW∗f,ϕKw = f(w)W ∗

g,ψKw = f(w)g(w)Kψ(w)

W ∗f,ϕW

∗g,ψKw = g(w)W ∗

f,ϕKψ(w) = g(w)f(ψ(w))Kϕ(ψ(w)).

Since Wg,ψ ∈ {Wf,ϕ}′, we have f(w)g(w)Kψ(w) = g(w)f(ψ(w))Kϕ(ψ(w)). From

the hypothesis that g(w) �= 0, we obtain that

f(w)Kψ(w) = f(ψ(w))Kϕ(ψ(w)), i.e.,

f(w)

1− ψ(w)z=

f(ψ(w))

1− ϕ(ψ(w))z

for all z ∈ D. Hence f(w) = f(ψ(w)) and f(w)ϕ(ψ(w)) = f(ψ(w))ψ(w). Theseidentities yield that f(w)ϕ(ψ(w)) = f(w)ψ(w). Since Wf,ϕ is cohyponormal, fnever vanishes on D from Theorem 3.2. In particular, f(w) �= 0. This implies thatϕ(ψ(w)) = ψ(w) ∈ D. By the Denjoy–Wolff theorem, ϕn → w and ϕn → ψ(w)uniformly on compact subsets of D. Thus we get that ψ(w) = w. �

Corollary 3.19. Let ϕ be an analytic selfmap of D with the Denjoy–Wolff point w ∈D and let f ∈ H∞. Suppose that Wf,ϕ is cohyponormal. If there exists an analyticselfmap ψ of D, not an elliptic automorphism of D, such that Cψ ∈ {Wf,ϕ}′, thenthe following statements hold:

(i) f is constant.

(ii) If f is not identically zero on D and w �= 0, then ϕ is the identity map on D.


Proof. (i) We may assume that f is not identically zero on D. It follows fromTheorem 3.18 that ψ(w) = w. From [2, Theorem 12] we obtain that (Wf,ϕh)(w) =h(w)Wf,ϕ(1) for any h ∈ H2. This implies that f(w)h(ϕ(w)) = h(w)f(z) for anyh ∈ H2 and z ∈ D. Taking h ≡ 1, we have that f(z) ≡ f(w) on D.

(ii) Assume that w �= 0. By (i) and Theorem 3.7, it holds that

f(w) = f = f(w)Kw

CϕKw.

Since f(w) �= 0, we get that CϕKw = Kw, which yields that ϕ(z) ≡ z on D. �

Corollary 3.20. Let ϕ be an analytic selfmap of D with the Denjoy–Wolff pointw ∈ D and let f ∈ H∞ \ {0}. Suppose that Wf,ϕ is cohyponormal. If ψ is ananalytic selfmap of D and g ∈ H∞ with g(w) �= 0 such that Wg,ψ ∈ {Wf,ϕ}′, thenthe space {h ∈ H2 : h(w) = 0} is a nontrivial invariant subspace for Wg,ψ.

Proof. It follows from Theorem 3.18 that

(Wg,ψh)(w) = g(w)h(ψ(w)) = g(w)h(w) (16)

for all h ∈ H2. Hence {h ∈ H2 : h(w) = 0} is invariant for Wg,ψ. Moreover,{h ∈ H2 : h(w) = 0} = {Kw}⊥ is clearly nontrivial. �

Corollary 3.21. Let ϕ be an analytic selfmap of D with the Denjoy–Wolff pointw ∈ D, and let f ∈ H∞ \ {0}. Suppose that Wf,ϕ is cohyponormal. If ψ is ananalytic selfmap of D and g ∈ H∞ with g(w) �= 0 such that Wg,ψ ∈ {Wf,ϕ}′, then

σp(Wg,ψ) ⊂ σp(Wg,ψ |M) ∪ {g(w)}and

σ(Wg,ψ) = σ(Wg,ψ |M) ∪ {g(w)}where M = {h ∈ H2 : h(w) = 0}.Proof. Corollary 3.20 implies that M is an invariant subspace for Wg,ψ . If λ ∈σp(Wg,ψ), then Wg,ψh = λh for some h ∈ H2 \ {0}. It follows from Theorem 3.18that

λh(w) = (Wg,ψh)(w) = g(w)h(w).

Hence either h(w) = 0 or λ = g(w), and thus σp(Wg,ψ) ⊂ σp(Wg,ψ |M) ∪ {g(w)}.In addition, we can write

Wg,ψ =

(Wg,ψ |M A

0 B

)on H2 =M⊕M⊥.

Since B is a linear operator on the one-dimensional spaceM⊥ = span{Kw},choose λ ∈ C such that σ(B) = σp(B) = {λ}. Obviously, σ(Wg,ψ |M) ∩ σ(B) hasno interior point. Thus the identity σ(Wg,ψ) = σ(Wg,ψ |M) ∪ σ(B) holds by [14,

Corollary 8]. Since σ(B∗) = σp(B∗) = {λ}, we get that

(W ∗g,ψ − λ)Kw =

((Wg,ψ |M)∗ − λ 0

A∗ B∗ − λ

)(0Kw

)=

(0

(B∗ − λ)Kw

)= 0,


which ensures that

0 = g(w)Kψ(w) − λKw = (g(w)− λ)Kw

by Theorem 3.18. This implies that λ = g(w) and so σ(Wg,ψ) = σ(Wg,ψ |M) ∪{g(w)}. �

References

[1] P. Bourdon and S.K. Narayan, Normal weighted composition operators on the Hardyspace H2(D), J. Math. Anal. Appl. 367(2010), 278–286.

[2] B. Cload, Commutants of composition operators, Ph.D. thesis, Univ. Toronto, 1997.

[3] J.B. Conway, A course in functional analysis, Springer-Verlag, New York, secondedition, 1990.

[4] J.B. Conway, The theory of subnormal operators, Mathematical Surveys and Mono-graphs, 36, Americal Mathematical Society, Providence, Rhode Island, 1991.

[5] C.C. Cowen, Composition operators on H2, J. Operator Theory 9(1983), 77–106.

[6] C.C. Cowen, Linear fractional composition operator on H2, Int. Eq. Op. Th.11(1988), 151–160.

[7] C.C. Cowen, G. Gunatillke, and E. Ko, Hermitian weighted composition operatorsand Bergman extremal functions, Complex Anal. Oper. Theory, to appear.

[8] C.C. Cowen and E. Ko, Hermitian weighted composition operator on H2, Trans.Amer. Math. Soc. 362(2010), 5771–5801.

[9] C.C. Cowen and T. Kriete, Subnormality and composition operators on H2, J. Funct.Analysis. 81(1988), 298–319.

[10] C.C. Cowen and B.D. MacCluer, Composition operators on spaces of analytic func-tions, CRC Press, 1995.

[11] M. Martin and M. Putinar, Lectures on hyponormal operators, Operator Theory:Advances and Applications, 39, Birkhauser Verlag, Basel, 1989.

[12] G. Gunatillake, Weighted composition operators, Ph.D. thesis, Purdue Univ., 1992.

[13] G. Gunatillake, Invertible weighted composition operators, J. Funct. Anal. 261(2011)831–860.

[14] J.K. Han, H.Y. Lee, and W.Y. Lee, Invertible completions of 2× 2 upper triangularoperator matrices, Proc. Amer. Math. Soc. 128(1999), 119–123.

[15] M.O. Otieno, On quasi-similarity and w-hyponormal operators, Opuscula Math.27(2007), 73–81.

[16] J.H. Shapiro, Composition operators and classical function theory, Springer-Verlag,New York, 1993.

[17] T. Worner, Commutants of certain composition operators, Acta Sci. Math. (Szeged)68(2002), 413–432.


Carl C. CowenDepartment of Mathematical SciencesIUPUIIndianapolis, Indiana 46202, USAe-mail: [email protected]

Sungeun JungInstitute of Mathematical SciencesEwha Womans University120-750 Seoul, Koreae-mail: [email protected]

Eungil KoDepartment of MathematicsEwha Womans University120-750 Seoul, Koreae-mail: [email protected]


A Subnormal Toeplitz Completion Problem

Raul E. Curto, In Sung Hwang and Woo Young Lee

Abstract. We give a brief survey of subnormality and hyponormality ofToeplitz operators on the vector-valued Hardy space of the unit circle. Wealso solve the following subnormal Toeplitz completion problem: Completethe unspecified rational Toeplitz operators (i.e., the unknown entries are ra-tional Toeplitz operators) of the partial block Toeplitz matrix

G :=

[Tω1 ?? Tω2

](ω1 and ω2 are finite Blaschke products)

to make G subnormal.

Mathematics Subject Classification (2010). Primary 47B20, 47B35.

Keywords. (Block) Toeplitz operators, bounded type functions, matrix-valuedrational functions, Halmos’ Problem 5, Abrahamse’s Theorem, hyponormal,subnormal, completion.

1. Hyponormality and subnormality of Toeplitz operators:A brief survey

1.1. Which operators are subnormal ?

Let H be a complex Hilbert space and let B(H) be the algebra of bounded linearoperators acting on H. An operator T ∈ B(H) is said to be hyponormal if itsself-commutator [T ∗, T ] := T ∗T − TT ∗ is positive (semi-definite), and subnormalif there exists a normal operator N on some Hilbert space K ⊇ H such that His invariant under N and N |H = T . The notion of subnormality was introducedby P.R. Halmos in 1950 and the study of subnormal operators has been highlysuccessful and fruitful: we refer to [Con] for details. Indeed, the theory of subnormal

The work of the first named author was partially supported by NSF Grant DMS-0801168. Thework of the second author was supported by Basic Science Research Program through the Na-tional Research Foundation of Korea (NRF) funded by the Ministry of Education, Science andTechnology (2011-0022577). The work of the third author was supported by Basic Science Re-

search Program through the National Research Foundation of Korea (NRF) grant funded by theKorea government(MEST) (2012-0000939).


88 R.E. Curto, I.S. Hwang and W.Y. Lee

operators has made significant contributions to a number of problems in functionalanalysis, operator theory, mathematical physics, and other fields. Oddly however,the question “Which operators are subnormal ?” is difficult to answer. In general, itis quite intricate to examine whether a normal extension exists for an operator. Ofcourse, there are a couple of constructive methods for determining subnormality;one of them is the Bram–Halmos criterion of subnormality ([Br]), which statesthat an operator T ∈ B(H) is subnormal if and only if

∑i,j(T

ixj , Tjxi) ≥ 0 for

all finite collections x0, x1, . . . , xk ∈ H. It is easy to see that this is equivalent tothe following positivity test:⎡⎢⎢⎢⎣

I T ∗ . . . T ∗k

T T ∗T . . . T ∗kT...

.... . .

...T k T ∗T k . . . T ∗kT k

⎤⎥⎥⎥⎦ ≥ 0 (1)

for all k ≥ 1. Condition (1) provides a measure of the gap between hyponormalityand subnormality. In fact, the positivity condition (1) for k = 1 is equivalent tothe hyponormality of T , while subnormality requires the validity of (1) for all k.If we denote by [A,B] := AB − BA the commutator of two operators A and B,and if we define T to be k-hyponormal whenever the k × k operator matrix

Mk(T ) := ([T ∗j, T i])ki,j=1

is positive, or equivalently, the (k+1)×(k+1) operator matrix in (1) is positive (viathe operator version of the Cholesky algorithm), then the Bram–Halmos criterioncan be rephrased as saying that T is subnormal if and only if T is k-hyponormalfor every k ≥ 1 ([CMX]). But it still may not be possible to test the positivitycondition (1) for every positive integer k, in general. Hence the following questionis interesting and challenging:

Are there feasible tests for the subnormality of an operator ? (2)

Recall ([At], [CMX], [CoS]) that T ∈ B(H) is said to be weakly k-hyponormal if

LS(T, T 2, . . . , T k) :=

⎧⎨⎩k∑

j=1

αjTj : α = (α1, . . . , αk) ∈ Ck

⎫⎬⎭consists entirely of hyponormal operators. If k = 2 then T is called quadraticallyhyponormal, and if k = 3 then T is said to be cubically hyponormal. Similarly,T ∈ B(H) is said to be polynomially hyponormal if p(T ) is hyponormal for everypolynomial p ∈ C[z]. It is known that k-hyponormal⇒ weakly k-hyponormal, butthe converse is not true in general. k-hyponormality and weak k-hyponormalityhave been considered by many authors with an aim at understanding the gapbetween hyponormality and subnormality ([Cu1], [Cu2], [CuF1], [CuF2], [CuF3],[CLL], [CL1], [CL2], [CL3], [CMX], [DPY], [McCP]). The study of this gap hasbeen only partially successful. For example, such a gap is not yet well describedfor Toeplitz operators on the Hardy space of the unit circle. For weighted shifts,

A Subnormal Toeplitz Completion Problem 89

positive results appear in [Cu1] and [CuF3], although no concrete example of aweighted shift which is polynomially hyponormal but not subnormal has yet beenfound (the existence of such weighted shifts was established in [CP1] and [CP2]).The Bram–Halmos criterion on subnormality indicates that 2-hyponormality isgenerally far from subnormality. There are special classes of operators, however,for which these two notions are equivalent. For example, in [CL1, Theorem 3.2],it was shown that 2-hyponormality and subnormality coincide for Toeplitz oper-ators Tϕ with trigonometric polynomial symbols ϕ ∈ L∞. On the other hand,2-hyponormality and subnormality enjoy some common properties. One of themis the following fact ([CL2]):

If T ∈ B(H) is 2-hyponormal then ker[T ∗, T ] is invariant for T . (3)

In fact, since the invariance of ker[T ∗, T ] for T is one of the most important proper-ties for subnormal operators, we may, in view of (3), expect that 2-hyponormalityand subnormality coincide for special classes of operators. Indeed, in Section 2, weshall see this phenomenon for a Toeplitz completion problem.

1.2. (Block) Toeplitz operators and bounded type functions

Toeplitz and Hankel operators arise in a variety of problems in several fields ofmathematics and physics, and nowadays the theory of Toeplitz and Hankel op-erators has become a very wide area. Let T ≡ ∂ D be the unit circle in thecomplex plane C. Let L2 ≡ L2(T) be the set of all square-integrable measur-able functions on T and let H2 ≡ H2(T) be the corresponding Hardy space. LetH∞ ≡ H∞(T) := L∞∩H2, that is, H∞ is the set of bounded analytic functions onthe unit disk D. Given ϕ ∈ L∞, the Toeplitz operator Tϕ and the Hankel operatorHϕ are defined by

Tϕg := P (ϕg) and Hϕg := JP⊥(ϕg) (g ∈ H2),

where P and P⊥ denote the orthogonal projections that map from L2 onto H2

and (H2)⊥, respectively, and where J denotes the unitary operator on L2 definedby J(f)(z) = zf(z).

We recall that a function ϕ ∈ L∞ is said to be of bounded type (or in theNevanlinna class) if there are analytic functions ψ1, ψ2 ∈ H∞ such that

ϕ(z) =ψ1(z)

ψ2(z)for almost all z ∈ T.

It is well known [Ab, Lemma 3] that if ϕ ∈ L∞ then

ϕ is of bounded type ⇐⇒ kerHϕ �= {0} .Assume that both ϕ and ϕ are of bounded type. Since TzHψ = HψTz for all

ψ ∈ L∞, it follows from Beurling’s Theorem that kerHϕ− = θ0H2 and kerHϕ+

=

θ+H2 for some inner functions θ0, θ+. We thus have b := ϕ−θ0 ∈ H2, and hence

we can write

ϕ− = θ0b, and similarly ϕ+ = θ+a for some a ∈ H2. (4)


In the factorization (4), we will always assume that θ0 and b are coprime and θ+and a are coprime. In (4), θ0b and θ+a are called coprime factorizations of ϕ−and ϕ+, respectively. By Kronecker’s Lemma [Ni, p. 183], if f ∈ H∞ then f is arational function if and only if rankHf <∞, which implies that

f is rational ⇐⇒ f = θb with a finite Blaschke product θ. (5)

For a Hilbert space X , let L2X ≡ L2

X (T) be the Hilbert space of X -valued normsquare-integrable measurable functions on T and let H2

X ≡ H2X (T) be the corre-

sponding Hardy space. We observe that L2Cn = L2⊗Cn and H2

Cn = H2⊗Cn. If Φis a matrix-valued function in L∞

Mn≡ L∞

Mn(T) (= L∞⊗Mn) then TΦ : H2

Cn → H2Cn

denotes the block Toeplitz operator with symbol Φ defined by

TΦF := Pn(ΦF ) for F ∈ H2Cn ,

where Pn is the orthogonal projection of L2Cn onto H2

Cn . A block Hankel operatorwith symbol Φ ∈ L∞

Mnis the operator HΦ : H2

Cn → H2Cn defined by

HΦF := JnP⊥n (ΦF ) for F ∈ H2

Cn ,

where P⊥n is the orthogonal projection of L2

Cn onto (H2Cn)⊥ and Jn denotes the

unitary operator on L2Cn given by Jn(F )(z) := zInF (z) for F ∈ L2

Cn (where In isthe n× n identity matrix). If we set H2

Cn = H2 ⊕ · · · ⊕H2 then we see that

TΦ =

⎡⎢⎣Tϕ11 . . . Tϕ1n

...Tϕn1 . . . Tϕnn

⎤⎥⎦ and HΦ =

⎡⎢⎣Hϕ11 . . . Hϕ1n

...Hϕn1 . . . Hϕnn

⎤⎥⎦ ,

where

Φ =

⎡⎢⎣ϕ11 . . . ϕ1n

...ϕn1 . . . ϕnn

⎤⎥⎦ ∈ L∞Mn

.

For Φ ∈ L∞Mn

, we write

Φ(z) := Φ∗(z). (6)

For Φ ∈ L∞Mn

, we also write

Φ+ := PnΦ ∈ H2Mn

and Φ− :=(P⊥n Φ)∗ ∈ H2

Mn.

Thus we can write Φ = Φ∗− + Φ+ . However, it will often be convenient toallow the constant term in Φ−. When this is the case, Φ−(0)∗ will not be zero;however, we will still ensure that Φ(0) = Φ+(0) + Φ−(0)∗.

A matrix-valued function Θ ∈ H∞Mn×m

(= H∞ ⊗Mn×m) is called inner if Θ

is isometric a.e. on T. The following basic relations can be easily derived:

T ∗Φ = TΦ∗ , H∗

Φ = H˜Φ (Φ ∈ L∞

Mn);

TΦΨ − TΦTΨ = H∗Φ∗HΨ (Φ,Ψ ∈ L∞

Mn); (7)

HΦTΨ = HΦΨ, HΨΦ = T ∗˜ΨHΦ (Φ ∈ L∞

Mn,Ψ ∈ H∞

Mn). (8)


For a matrix-valued function Φ = [φij ] ∈ L∞Mn

, we say that Φ is of bounded typeif each entry φij is of bounded type and that Φ is rational if each entry φij is arational function.

For a matrix-valued function Φ ∈ H2Mn×r

, we say that Δ ∈ H2Mn×m

is a left

inner divisor of Φ if Δ is an inner matrix function such that Φ = ΔA for someA ∈ H2

Mm×r(m ≤ n). We also say that two matrix functions Φ ∈ H2

Mn×rand

Ψ ∈ H2Mn×m

are left coprime if the only common left inner divisor of both Φ and

Ψ is a unitary constant and that Φ ∈ H2Mn×r

and Ψ ∈ H2Mm×r

are right coprime

if Φ and Ψ are left coprime. Two matrix functions Φ and Ψ in H2Mn

are said to

be coprime if they are both left and right coprime. We remark that if Φ ∈ H2Mn

issuch that detΦ is not identically zero then any left inner divisor Δ of Φ is square,i.e., Δ ∈ H2

Mn. If Φ ∈ H2

Mnis such that detΦ is not identically zero then we say

that Δ ∈ H2Mn

is a right inner divisor of Φ if Δ is a left inner divisor of Φ.

The shift operator S on H2Cn is defined by

S :=

n∑j=1

⊕Tz.

The following fundamental result known as the Beurling–Lax–Halmos Theo-rem is useful in the sequel.

The Beurling–Lax–Halmos theorem. A nonzero subspace M of H2Cn is invariant

for the shift operator S on H2Cn if and only if M = ΘH2

Cm , where Θ is an innermatrix function in H∞

Mn×m(m ≤ n). Furthermore, Θ is unique up to a unitary

constant right factor; that is, if M = ΔH2Cr (for Δ an inner function in H∞

Mn×r),

then m = r and Θ = ΔW , where W is a unitary matrix mapping Cm onto Cm.

As is customarily done, we say that two matrix-valued functions A and Bare equal if they are equal up to a unitary constant right factor. Observe by (8)that for Φ ∈ L∞

Mn, HΦS = HΦTzIn = HΦ·zIn = HzIn·Φ = T ∗

zInHΦ, which implies

that the kernel of a block Hankel operator HΦ is an invariant subspace of theshift operator on H2

Cn . Thus, if kerHΦ �= {0}, then by the Beurling–Lax–HalmosTheorem,

kerHΦ = ΘH2Cm

for some inner matrix function Θ. We note that Θ need not be a square matrix.For example, let θi (i = 0, 1, 2) be a scalar inner function such that θ1 and θ2 arecoprime and let q ∈ L∞ be such that kerHq = {0}. Define

Θ :=1√2

[θ0θ1θ0θ2

]and Φ :=

[θ0θ1 θ0θ2qθ2 −qθ1

].

Then a straightforward calculation shows that kerHΦ = ΘH2 (cf. [GHR,Example 2.9]).

The following result was shown in [GHR, Theorem 2.2].


Theorem 1.1 ([GHR]). For Φ ∈ L∞Mn

, the following statements are equivalent:

(i) Φ is of bounded type;(ii) kerHΦ = ΘH2

Cn for some square inner matrix function Θ;(iii) Φ = AΘ∗, where A ∈ H∞

Mnand A and Θ are right coprime.

For an inner matrix function Θ ∈ H2Mn

, we write

HΘ := H2Cn #ΘH2

Cn .

In view of Theorem 1.1, if Φ ∈ L∞Mn

is such that Φ and Φ∗ are of boundedtype then Φ+ and Φ− can be written in the form

Φ+ = Θ1A∗ and Φ− = Θ2B

∗, (9)

where Θ1 and Θ2 are inner, A,B ∈ H2Mn

, Θ1 and A are right coprime, and Θ2

and B are right coprime. In (9), Θ1A∗ and Θ2B

∗ will be called right coprimefactorizations of Φ+ and Φ−, respectively.

In general, it is not easy to check the condition “B and Θ are right coprime”.But if Θ ≡ θIn for a finite Blaschke product θ, then we have a tractable criterion(cf. [CHL2, Lemma 3.3]):

Θ and B are right coprime ⇐⇒ B(α) is invertible for each zero α of θ. (10)

1.3. Hyponormality of Toeplitz operators

An elegant and useful theorem of C. Cowen [Co4] characterizes the hyponormalityof a Toeplitz operator Tϕ by properties of the symbol ϕ ∈ L∞(T). This result makesit possible to answer an algebraic question coming from operator theory – namely,is Tϕ hyponormal ? – by studying the function ϕ itself. Normal Toeplitz operatorswere characterized by a property of their symbol in the early 1960’s by A. Brownand P.R. Halmos [BH], and so it is somewhat surprising that 25 years passed beforethe exact nature of the relationship between the symbol ϕ ∈ L∞ and the positivityof the self-commutator [T ∗

ϕ, Tϕ] was understood (via Cowen’s Theorem). As Cowennotes in his survey paper [Co3], the intensive study of subnormal Toeplitz operatorsin the 1970s and early 80s is one explanation for the relatively late appearance ofthe sequel to the Brown-Halmos work. The characterization of hyponormality viaCowen’s Theorem requires one to solve a certain functional equation in the unitball of H∞.

Cowen’s theorem ([Co4], [NT]). For each ϕ ∈ L∞, let

E(ϕ) ≡ {k ∈ H∞ : ||k||∞ ≤ 1 and ϕ− kϕ ∈ H∞}.Then Tϕ is hyponormal if and only if E(ϕ) is nonempty.

Cowen’s Theorem has been used in [CHL1], [CL1], [CL2], [FL], [Gu1], [Gu2], [GS],[HKL1], [HKL2], [HL1], [HL2], [HL3], [Le], [NT] and [Zhu], which have been de-voted to the study of hyponormality for Toeplitz operators on H2. Particularattention has been paid to Toeplitz operators with polynomial symbols, rationalsymbols, and bounded type symbols [HL2], [HL3], [CHL1]. However, the case of


arbitrary symbol ϕ, though solved in principle by Cowen’s theorem, is in practicevery complicated. Indeed, it may not even be possible to find tractable necessaryand sufficient condition for the hyponormality of Tϕ in terms of the Fourier coef-ficients of the symbol ϕ unless certain assumptions are made about ϕ. To date,tractable criteria for the cases of trigonometric polynomial symbols (resp. rationalsymbols) were derived from a Caratheodory–Schur interpolation problem ([Zhu])(resp. a tangential Hermite–Fejer interpolation problem ([Gu1]) or the classicalHermite–Fejer interpolation problem ([HL3])). Very recently, a tractable and ex-plicit criterion on the hyponormality of Toeplitz operators having bounded typesymbols was established via the triangularization theorem for compressions of theshift operator ([CHL1]).

When one studies the hyponormality (also, normality and subnormality) ofthe Toeplitz operator Tϕ one may, without loss of generality, assume that ϕ(0) = 0;this is because hyponormality is invariant under translation by scalars.

In 2006, Gu, Hendricks and Rutherford [GHR] characterized the hyponormal-ity of block Toeplitz operators in terms of their symbols. Their characterizationfor hyponormality of block Toeplitz operators TΦ resembles Cowen’s Theorem ex-cept for an additional condition which is trivially satisfied in the scalar case – thenormality of the symbol, i.e., Φ∗Φ = ΦΦ∗.

Theorem 1.2 (Hyponormality of block Toeplitz operators, Gu–Hendricks–Ruther-ford [GHR]). For each Φ ∈ L∞

Mn, let

E(Φ) :={K ∈ H∞

Mn: ||K||∞ ≤ 1 and Φ−KΦ∗ ∈ H∞

Mn

}.

Then TΦ is hyponormal if and only if Φ is normal and E(Φ) is nonempty.

In [GHR], the normality of block Toeplitz operator TΦ was also characterizedin terms of the symbol Φ, under a “determinant” assumption on the symbol Φ.

Theorem 1.3 (Normality of block Toeplitz operators, Gu–Hendricks–Rutherford[GHR]). Let Φ ≡ Φ+ +Φ∗− be normal. If detΦ+ is not identically zero then

TΦ is normal⇐⇒ Φ+ − Φ+(0)

=(Φ− − Φ−(0)

)U for some constant unitary matrix U.

(11)

Until now, tractable criteria for the hyponormality of block Toeplitz operatorsTΦ with matrix-valued trigonometric polynomials, rational functions or boundedtype functions Φ have been established via interpolation problems or the so-calledtriangularization theorem for compressions of the shift operator ([GHR], [HL4],[HL5], [CHL1]).

1.4. Halmos’ Problem 5

In view of the preceding argument, it is natural and significant to elucidate thesubnormality of Toeplitz operators. In 1970, P.R. Halmos addressed a problem onsubnormality of Toeplitz operators acting on H2, the so-called Halmos’ Problem5 in his lectures “Ten problems in Hilbert space” [Hal1]:


Halmos’ Problem 5. Is every subnormal Toeplitz operator either normal or ana-lytic ?

A Toeplitz operator Tϕ is called analytic if ϕ ∈ H∞. Any analytic Toeplitz operatoris easily seen to be subnormal: indeed, Mϕ is a normal extension of Tϕ, whereMϕ is the normal operator of multiplication by ϕ on L2. Thus the question isnatural because the two classes, the normal and analytic Toeplitz operators, arewell understood and are subnormal. In the 1970’s, interesting partial (affirmative)answers appeared. Thus, when in 1979 Halmos wrote a report on progress on histen problems (cf. [Hal2]), he stated that “some very good mathematics had goneinto that answer” on Problem 5. He then conjectured that the future of Problem 5was hopeful in the affirmative direction. However, in 1984, Halmos’ Problem 5 wasanswered in the negative by C. Cowen and J. Long [CoL]: they found an analyticfunction ψ for which Tψ+αψ (0 < α < 1) is subnormal – in fact, this Toeplitz

operator is unitarily equivalent to a subnormal weighted shift Wβ with weight

sequence β ≡ {βn}, where βn = (1 − α2n+2)12 for n = 0, 1, 2, . . . . A similar result

was independently obtained by S. Sun ([Sun1], [Sun2], [Sun3]). Unfortunately,these constructions do not provide an intrinsic connection between subnormalityand the theory of Toeplitz operators.

Until now researchers have been unable to characterize subnormal Toeplitzoperators in terms of their symbols. In fact it may not even be possible to findtractable necessary and sufficient condition for the subnormality of Tϕ in terms oftheir symbols unless certain assumptions are made about ϕ. On the other hand,surprisingly, as C. Cowen notes in [Co2], some analytic Toeplitz operators are uni-tarily equivalent to non-analytic Toeplitz operators; i.e., the analyticity of Toeplitzoperators is not invariant under unitary equivalence. In this sense, we might askwhether Cowen and Long’s non-analytic subnormal Toeplitz operator is unitarilyequivalent to an analytic Toeplitz operator. It was shown in [CHL2] that Cowenand Long’s non-analytic subnormal Toeplitz operator Tϕ is not unitarily equiv-alent to any analytic Toeplitz operator. Consequently, even if we interpret “is”in Halmos’ Problem 5 as “is up to unitary equivalence,” the answer to Halmos’Problem 5 is still negative. Thus we would like to reformulate Halmos’ Problem 5as follows:

Halmos’ Problem 5 reformulated. Which Toeplitz operators are subnormal ?

Directly connected with Halmos’ Problem 5 is the following question:

Which subnormal Toeplitz operators are normal or analytic ? (12)

In 1976, M.B. Abrahamse proved that the answer to Halmos’ question is affirmativefor Toeplitz operators with bounded type symbols ([Ab]):

Abrahamse’s theorem ([Ab, theorem]). Let ϕ ∈ L∞ be such that ϕ or ϕ is ofbounded type. If Tϕ is hyponormal and ker[T ∗

ϕ, Tϕ] is invariant under Tϕ then Tϕ

is normal or analytic.


Consequently, if ϕ ∈ L∞ is such that ϕ or ϕ is of bounded type, then everysubnormal Toeplitz operator must be either normal or analytic. Partial answersto question (12) have been obtained by many authors (cf. [AIW], [Co2], [CoL],[CHL1], [CHL2], [CL1], [CL2], [CL3], [ItW], [NT]). More generally, we are inter-ested in the following question:

Which subnormal block Toeplitz operators are normal or analytic ? (13)

Question (13) is more difficult to answer, in comparison with the scalar-valuedcase. Indeed, Abrahamse’s Theorem does not hold for block Toeplitz operators(even with matrix-valued trigonometric polynomial symbol): For instance, if

Φ :=

[z + z 00 z

],

then

TΦ =

[U+ + U∗

+ 00 U+

](U+ := the unilateral shift on H2)

is neither normal nor analytic although TΦ is evidently subnormal.

Recall that an operator T ∈ B(H) is said to be quasinormal if T commuteswith T ∗T and is said to be pure if it has no nonzero reducing subspace on which itis normal. It is well known that quasinormal⇒ subnormal. On the other hand, in[ItW], it was shown that every quasinormal Toeplitz operator is either normal oranalytic, i.e., the answer to the Halmos’ Problem 5 is affirmative for quasinormalToeplitz operators. However, this is not true for the cases of matrix-valued symbols:indeed, if

Φ ≡[

z z + 2zz + 2z z

]. (14)

then TΦ is quasinormal, but it is neither normal nor analytic. Since

TΦ =[

U∗+ U∗

++2U+

U∗++2U+ U∗

+

],

it follows that if W := 1√2

[1 −11 1

], then W is unitary and

W ∗TΦW = 2

[U∗+ + U+ 0

0 −U+

],

which says that TΦ is unitarily equivalent to a direct sum of the normal operator2(U∗

++U+) and the analytic Toeplitz operator −2U+. This phenomenon is not anaccident. Indeed, very recently, in [CHKL], it was shown that every pure quasinor-mal operator with finite rank self-commutator is unitarily equivalent to a Toeplitzoperator with a matrix-valued analytic rational symbol and (as a corollary) thatevery pure quasinormal Toeplitz operator with a matrix-valued rational symbol isunitarily equivalent to an analytic Toeplitz operator.

Also, in [CHKL], the following theorem was obtained:


Theorem 1.4 (Abrahamse’s theorem for matrix-valued rational symbols, [CHKL]).Let Φ ≡ Φ∗

− + Φ+ ∈ L∞Mn

be a matrix-valued rational function. Thus in view of(9), we may write

Φ− = ΘB∗ (right coprime factorization).

Assume that Θ has an inner divisor of the form θIn, where θ is a nonconstantinner function. If

(i) TΦ is hyponormal;(ii) ker[T ∗

Φ, TΦ] is invariant for TΦ,

then TΦ is normal. Hence in particular, if TΦ is subnormal then TΦ is normal.

Theorem 1.4 may fail if we drop the assumption “Θ has a nonconstantdiagonal-constant inner divisor.” To see this, consider the matrix-valued func-tion in (14):

Φ ≡[

z z + 2zz + 2z z

].

We thus have

Φ− =

[z zz z

]=

(1√2

[1 z−1 z

])(1√2

[0 20 2

])∗,

where

Θ ≡ 1√2

[1 z−1 z

]and B ≡ 1√

2

[0 20 2

]are right coprime.

As we saw in the preceding, TΦ is quasinormal, and hence subnormal. Butclearly, TΦ is neither normal nor analytic. Here we note that Θ does not haveany nonconstant diagonal inner divisor of the form θIn with a nonconstant innerfunction θ.

1.5. A special subnormal Toeplitz completion

Given a partially specified operator matrix with some known entries, the problemof finding suitable operators to complete the given partial operator matrix so thatthe resulting matrix satisfies certain given properties is called a completion prob-lem. Dilation problems are special cases of completion problems: in other words,the dilation of T is a completion of the partial operator matrix

[T ?? ?

]. A partial

block Toeplitz matrix is simply an n×n matrix some of whose entries are specifiedToeplitz operators and whose remaining entries are unspecified. A subnormal com-pletion of a partial operator matrix is a particular specification of the unspecifiedentries resulting in a subnormal operator. In particular, to avoid the triviality, weare interested in the cases whose diagonal entries are specified. For example, if ωis a finite Blaschke product, then Tω is evidently subnormal, so that

[Tω 00 Tω

]is

itself subnormal. On the other hand,[Tω 1− TωTω

0 Tω

](ω is a finite Blaschke product) (15)


is a subnormal (even unitary) completion of the 2× 2 partial operator matrix[Tω ?? Tω

].

A subnormal Toeplitz completion of a partial block Toeplitz matrix is a sub-normal completion whose unspecified entries are Toeplitz operators. Then the fol-lowing question comes up at once: Does there exist a subnormal Toeplitz com-pletion of

[Tω ?? Tω

]? Evidently, (15) is not such a completion. To answer this

question, let

Φ ≡[ω ϕψ ω

](ϕ, ψ ∈ L∞).

If TΦ is hyponormal then by Theorem 1.2, Φ should be normal. Thus astraightforward calculation shows that

|ϕ| = |ψ| and ω(ϕ+ ψ) = ω(ϕ+ ψ),

which implies that ϕ = −ψ. Thus a direct calculation shows that

[T ∗Φ, TΦ] =

[∗ ∗∗ TωTω − 1

],

which is not positive semi-definite because TωTω − 1 is not. Therefore, there are

no hyponormal Toeplitz completions of

[Tω ?? Tω

]. The following question seems

to be more difficult: Does there exist a subnormal Toeplitz completion of[Tω ?? Tω

](ω is a finite Blaschke product) ?

Special cases of this question were successfully considered in [CHL1] and[CHL3]. In the next section, we consider a subnormal Toeplitz completion problem.

2. Subnormal Toeplitz completions

In this section we consider the following:

Problem A. Complete the unspecified rational Toeplitz operators of the partialblock Toeplitz matrix

G :=

[Tω1 ?? Tω2

](ω1 and ω2 are finite Blaschke products) (16)

to make G subnormal.

To answer Problem A, we need several auxiliary lemmas. We write

bα(z) :=z − α

1− αz(α ∈ D) .


We begin with:

Lemma 2.1. Suppose ϕ, ψ ∈ L∞. Then

[T ∗ϕ◦bα , Tψ◦bα ] ∼= [T ∗

ϕ, Tψ] (∼= denotes unitary equivalence).

In particular, Tϕ◦bα is hyponormal if and only if Tϕ is hyponormal.

Proof. By a well-known fact due to C. Cowen [Co1, Theorem 1], there exists aunitary operator V such that

Tϕ◦bα = V ∗TϕV and Tψ◦bα = V ∗TψV .

We thus have [T ∗ϕ◦bα , Tψ◦bα ] = V ∗[T ∗

ϕ, Tψ]V , which gives the result. �Lemma 2.2. Let ϕ, ψ ∈ L∞ be rational functions and let ω1 and ω2 be finiteBlaschke products. If

Φ :=

[ω1 ϕψ ω2

]is such that TΦ is hyponormal then ω1 = ω2.

Proof. We first observe (bα ◦ b−α)(z) = z. Thus, in view of Lemma 2.1 we mayassume that ω1(0) = 0. Then this lemma follows from a slight variation of theproof of [CHL3, Theorem 4.2], in which ω1 = bα and ω2 = bβ . �

In view of Lemma 2.2, for the problem (16), it suffices to consider the case

Φ :=

[ω ϕψ ω

](ϕ, ψ ∈ L∞ are rational; ω is a finite Blaschke product))

Lemma 2.3. Suppose Φ := Φ∗− + Φ+ ∈ L∞

Mnis a matrix-valued rational function.

Then we may write (cf. [CHL3, Lemma 3.1])

Φ+ = A∗Δ0Δ and Φ− = B∗Δ,

where Δ0Δ ≡ θIn with an inner function θ, B and Δ are left coprime and A,B ∈H2

Mn. If ker[T ∗

Φ, TΦ] is invariant under TΦ and K ∈ E(Φ), thencl ranHAΔ∗ ⊆ ker(I − T

˜KT ∗˜K).

(For the definition of K, see (6).)

Proof. This follows from formula (16) in [CHL2], together with a careful analysisthat the proof of (16) in [CHL2] does not employ the diagonal-constant-ness of Δ.

�Lemma 2.4. Let Φ ≡ Φ∗

− + Φ+ ∈ L∞Mn

be a matrix-valued rational function suchthat

Φ− :=

[ω ψ−ϕ− ω

],

where ω is a finite Blaschke product of the form

ω =

p∏i=1

bqii

(bi(z) :=

z − αi

1− αiz(αi �= αj if i �= j) and qi ≥ 1

).


If

Φ− = ΘB∗ (right coprime factorization),

then Θ has an inner divisor of the form biI2 for some i = 1, 2, . . . , p, except in thefollowing two cases:

(i) mi + ni = 2qi for all i = 1, 2, . . . , p;(ii) mi0 + ni0 > 2qi0 and mi0ni0 = 0 for some i0 ,

in the representation

ϕ− ≡ θ0a =( p∏i=1

bmi

i

)θ′0a and ψ− ≡ θ1b =

( p∏i=1

bni

i

)θ′1b

(coprimefactorizations

)(mi, ni = 0, 1, . . . and (θ′0θ

′1)(αi) �= 0 for all i = 1, 2, . . . , p).

Proof. By Theorem 1.1, kerHΦ∗− = ΘH2

C2 . We observe that for f, g ∈ H2,

Φ∗−

[fg

]∈ H2

C2 ⇐⇒[ω θ0a

θ1b ω

] [fg

]∈ H2

C2 ,

which implies that if

[fg

]∈ kerHΦ∗

− , then( p∏i=1

biqi)f+( p∏i=1

bimi)θ′0ag ∈ H2 and

( p∏i=1

bini)θ′1bf+

( p∏i=1

biqi)g ∈ H2. (17)

We split the proof into two cases.

Case 1 (0 ≤ mi0 +ni0 < 2qi0 for some i0 = 1, 2, . . . , d). In this case, ni0 < qi0or mi0 < qi0 . Suppose that mi0 < qi0 . Then by the first statement of (17) we have(∏

i�=i0

biqi−mi

)bi0

qi0−mi0 θ′0f ∈ H2,

which implies that f = bqi0−mi0

i0f1 for some f1 ∈ H2. In turn, by the second

statement of (17) we have(∏i�=i0

bini)bi0

mi0+ni0−qi0 θ′1bf1 +( p∏i=1

biqi)g ∈ H2.

Thus if mi0 + ni0 − qi0 ≤ 0, then g = bqi0i0

g1 for some g1 ∈ H2 and if insteadmi0 + ni0 − qi0 > 0, then

bi02qi0−mi0−ni0

(∏i�=i0

biqi−ni

)θ′1g ∈ H2 ,

which implies that g = b2qi0−mi0−ni0

i0g2 for some g2 ∈ H2. Therefore bi0I2 is an

inner divisor of Θ.If instead ni0 < qi0 then the same argument as the above gives that bi0I2 is

an inner divisor of Θ.


Case 2 (mi0 + ni0 > 2qi0 and mi0ni0 �= 0 for some i0).

(a) Suppose mi0 ≥ qi0 + 1. If

[fg

]∈ kerHΦ∗

− , then by the first statement of

(17) we have (∏i�=i0

bimi−qi

)bi0

mi0−qi0 θ′0ag ∈ H2,

which implies that g = bmi0−qi0i0

g1 for some g1 ∈ H2. In turn, by the secondstatement of (17) we have(∏

i�=i0

bini)bi0

ni0 θ′1bf +(∏i�=i0

biqi)bi0

2qi0−mi0 g1 ∈ H2.

Thus if 2qi0 ≤ mi0 , then f = bni0

i0f1 for some f1 ∈ H2 and if instead 2qi0 >

mi0 , then (∏i�=i0

bini−qi

)bi0

mi0+ni0−2qi0 θ′1bf ∈ H2 ,

which implies that f = bmi0+ni0−2qi0i0

f2 for some f2 ∈ H2. Therefore bi0I2 is aninner divisor of Θ.

(b) Suppose mi0 < qi0 +1. Then ni0 ≥ qi0 +1 and the same argument as theCase 2(a) gives that bi0I2 is an inner divisor of Θ.

From Case 1 and Case 2, we can conclude that Θ has an inner divisor of theform biI2 for some i = 1, 2, . . . , p except the casesmi+ni = 2qi for all i = 1, 2, . . . , pand mi0 + ni0 > 2qi0 with mi0ni0 = 0 for some i0. This completes the proof. �Lemma 2.5. Let Φ ≡ Φ∗− + Φ+ ∈ L∞

Mnbe a matrix-valued rational function such

that

Φ− :=

[ω ϕ−ψ− ω

],

where ω is a finite Blaschke product of the form

ω =

p∏i=1

bqii

(bi(z) :=

z − αi

1− αiz, qi ≥ 1

),

ϕ− ≡ θ0a =( p∏i=1

bmi

i


( p∏i=1

bni

i

)θ′1b

(coprime

factorizations

)(mi, ni = 0, 1, . . . and (θ′0θ

′1)(αi) �= 0 for all i = 1, 2, . . . , p). If αi0 = 0, mi0 > 2qi0

and ni0 = 0 for some i0, then

kerHΦ∗− ⊆ 1√|α|2 + 1

[zmi0−qi0 θ′0 −αzmi0−qi0+1θ′0

αθ′1 zθ′1

]H2

C2

(α := − a′(0)

θ′′1 (0)

),

where

a′ :=(∏i�=i0

biMi−mi

)a and θ′′1 :=

(∏i�=i0

bMi−qii

)θ′1

(Mi := max(mi, qi) for i �= i0).


Proof. Observe that for f, g ∈ H2,

Φ∗−

[fg

]∈ H2

C2 ⇐⇒[ω θ1b

θ0a ω

] [fg

]∈ H2

C2 ,

which implies that if

[fg

]∈ kerHΦ∗

− , then

( p∏i=1

biqi)f+( p∏i=1

bini)θ′1bg ∈ H2 and

( p∏i=1

bimi)θ′0af+

( p∏i=1

biqi)g ∈ H2. (18)

It follows from the first statement of (18) that g = θ′1g1 for some g1 ∈ H2. In turn( p∏i=1

biqi)f +

( p∏i=1

bini)bg1 ∈ H2.

Since ni0 = 0, we have f = zqi0f1 for some f1 ∈ H2. Thus, by the secondstatement of (18) we have(∏

i�=i0

bimi)zmi0−qi0 θ′0af1 +

( p∏i=1

biqi)g ∈ H2 , (19)

so that (∏i�=i0

bimi−qi

)zmi0−2qi0 θ′0af1 ∈ H2.

Since mi0 > 2qi0 , it follows that f1 = θ′0zmi0−2qi0 f2 for some f2 ∈ H2. Thus,

by (19) we have (∏i�=i0

bimi)zqi0af2 +

( p∏i=1

biqi)θ′1g1 ∈ H2. (20)

Then it follows from (20) that(∏i�=i0

biMi−mi

)af2 +

(∏i�=i0

bMi−qii

)θ′1g1 ∈ zqi0H2. (21)

Write

a′ :=(∏i�=i0

biMi−mi

)a and θ′′1 :=

(∏i�=i0

bMi−qii

)θ′1.

Then we have a′(0) �= 0 and θ′′1 (0) �= 0, and by (21) we have

g1(0) = αf2(0)(α := − a′(0)

θ′′1 (0)

).

Therefore, we have[fg

]∈ kerHΦ∗

− =⇒ f = zmi0−qi0 θ′0f2, g = θ′1g1, and g1(0) = αf2(0). (22)


Put

Ω :=1√|α|2 + 1

[zmi0−qi0 θ′0 −αzmi0−qi0+1θ′0

αθ′1 zθ′1

].

Then Ω is inner, and for h1, h2 ∈ H2,

Ω

[h1

h2

]=

1√|α|2 + 1

[zmi0−qi0 θ′0h1 − αzmi0−qi0+1θ′0h2

αθ′1h1 + zθ′1h2

]=

1√|α|2 + 1

[zmi0−qi0 θ′0

(h1 − αzh2

)θ′1(αh1 + zh2

) ].

Since (αh1 + zh2)(0) = αh1(0) = α(h1 − αzh2)(0), it follows from (22) that

kerHΦ∗− ⊆ ΩH2

C2 ,

which gives the result. �

To answer Problem A, we recall ([CHL2, Lemma 3.2]) that if Φ ≡ Φ∗−+Φ+ ∈

L∞Mn

is such that Φ and Φ∗ are of bounded type, we may write, as in (9),

Φ+ = Θ1A∗ and Φ− = ΘB∗ (right coprime factorizations).

If TΦ is hyponormal, then

Θ1 = ΘΘ0 for some inner matrix function Θ0; (23)

in other words, Θ is a left inner divisor of Θ1.

We are ready for:

Theorem 2.6. Let ϕ, ψ ∈ L∞ be rational functions and consider

G :=

[Tω1

Tϕ

Tψ Tω2

](ωi is a finite Blaschke product for i = 1, 2) . (24)

Then the following statements are equivalent:

1. G is normal; 2. G is subnormal; 3. G is 2-hyponormal;4. G is hyponormal and ker[G∗, G] is invariant for G;5. ω1 = ω2 =: ω and the following condition holds:

ϕ = eiδ1ω + ζ and ψ = eiδ2ϕ (ζ ∈ C; δ1, δ2 ∈ [0, 2π)) , (25)

except in the following case:

mi + ni = 2qi for some i = 1, 2, . . . , p , (26)

in the representation

ω :=

p∏i=1

bqii

(bi(z) :=

z − αi

1− αiz, qi ≥ 1

),

ϕ− ≡ θ0a =( p∏i=1

bmi

i


( p∏i=1

bni

i

)θ′1b

(coprime

factorizations

)(mi, ni = 0, 1, . . . and (θ′0θ′1)(αi) �= 0 for all i = 1, 2, . . . , p).


Proof. Clearly, (1) ⇒ (2) and (2) ⇒ (3). Also (3) ⇒ (4) is evident becauseker[T ∗, T ] is invariant under T for every 2-hyponormal operator T ∈ B(H) (cf.[CL2]). Moreover, (5) ⇒ (1) follows from a straightforward calculation.

(4) ⇒ (5): By Lemma 2.2, ω1 = ω2 =: ω. Thus we may write

Φ ≡[ω ϕψ ω

]≡ Φ∗

− +Φ+ =

[ω ψ−ϕ− ω

]∗+

[0 ϕ+

ψ+ 0

]and assume that TΦ is hyponormal and ker[TΦ, T

∗Φ] is invariant for TΦ. Since, by

Theorem 1.2, Φ is normal, we have

|ϕ| = |ψ|, (27)

and also there exists a function K ≡ [ k1 k2

k3 k4

] ∈ H∞M2

such that Φ∗−−KΦ∗+ ∈ H2

M2,

i.e., [ω ϕ−ψ− ω

]−[k1 k2k3 k4

] [0 ψ+

ϕ+ 0

]∈ H2

M2,

which implies that ϕ+ and ψ+ are not identically zero and hence detΦ+ is notidentically zero.

We now split the proof into three cases.

Case 1 (mi0 = ni0 = 0 for some i0). In this case, by Lemma 2.4 and Theorem1.4, we can conclude that TΦ is normal. Since detΦ+ is not identically zero, itfollows from Theorem 1.3 that Φ+ − Φ−U ∈ Mn(C) for some constant unitarymatrix U ≡ [ c1 c2

c3 c4 ]. We observe

Φ+ − Φ−U ∈Mn(C)⇐⇒[0 ϕ+

ψ+ 0

]−[ω θ1bθ0a ω

] [c1 c2c3 c4

]∈Mn(C)

(28)

=⇒

⎧⎪⎪⎪⎨⎪⎪⎪⎩c1ω + c3θ1b = ξ1

c4ω + c2θ0a = ξ2

ϕ+ = c2ω + c4θ1b+ ξ3

ψ+ = c3ω + c1θ0a+ ξ4

(ξi ∈ C for i = 1, . . . , 4) ,

which givesc1Hω = −c3Hθ1b

and c4Hω = −c2Hθ0a. (29)

Thus if c1 �= 0 then c3 �= 0 and hence ω = θ1, which is a contradiction becauseω(αi0) = 0, but θ1(αi0 ) �= 0. Thus c1 = 0 and similarly, c4 = 0. Since U is unitary,it follows that |c2| = |c3| = 1, and hence θ1b and θ0a are constants. Thus, againby (28), we have

ϕ = ϕ+ = eiδ1ω + β1 and ψ = ψ+ = eiδ2ω + β2 (δ1, δ2 ∈ [0, 2π); β1, β2 ∈ C).

Since |ϕ| = |ψ|, it follows thatϕ = eiδ1ω + ζ and ψ = eiδ2ϕ (δ1, δ2 ∈ [0, 2π); ζ ∈ C)).

Case 2((i) 0 < mi0 + ni0 < 2qi0 ; or (ii) mi0 + ni0 > 2qi0 (mi0ni0 �= 0)

for some i0). In this case, by Lemma 2.4 and Theorem 1.4, we can conclude that


TΦ is normal. By case assumption, we have mi0 �= qi0 or ni0 �= qi0 . Suppose thatmi0 �= qi0 . Then ω �= θ0, and hence by (29) we have c2 = c4 = 0. Therefore U isnot unitary, a contradiction.

If instead ni0 �= qi0 then the same argument as above gives that U is notunitary, a contradiction. Thus this case cannot occur.

Case 3 (mi + ni > 2qi (mini = 0) for all i = 1, . . . , p). Fix i0 (1 ≤ io ≤ p);we may, without loss of generality, assume that ni0 = 0 (and hence, mi0 > 2qi0).By Lemma 2.1, we may also assume that bi0 = z. It follows from Theorem 1.2 that

there exists a matrix function K ≡ [ k1 k2

k3 k4

] ∈ E(Φ), so that[ω ϕ−ψ− ω

]−[k1 k2k3 k4

] [0 ψ+

ϕ+ 0

]∈ H2

M2,

which implies that {ω − k2ϕ+ ∈ H2, θ1b− k4ϕ+ ∈ H2

ω − k3ψ+ ∈ H2, θ0a− k1ψ+ ∈ H2.(30)

Since ||K||∞ ≤ 1 and hence ||ki||∞ ≤ 1 for each i = 1, . . . , 4, the following Toeplitzoperators are all hyponormal (by Cowen’s Theorem):

Tω+ϕ+ , Tθ1b+ϕ+, Tω+ψ+ , Tθ0a+ψ+

. (31)

Put Mi := max(mi, qi) and Ni := max(ni, qi). Then by (31) and a scalar-valuedversion of (23), we can see that

ϕ+ = zqi0∏i�=i0

bNi

i θ′1θ3d and ψ+ = zmi0

∏i�=i0

bMi

i θ′0θ2c (coprime factorizations),

where θ2 and θ3 are finite Blaschke products. Thus, in particular, c(0) �= 0 andd(0) �= 0. Thus, by (30), we can see that

k3(0) = 0 and k4(0) = 0 : (32)

indeed, in (30),

ω − k3ψ+ ∈ H2 =⇒ zqi0∏i�=i0

biqi − k3z

mi0

∏i�=i0

biMi

θ′0θ2c ∈ H2

=⇒ zmi0−qi0∏i�=i0

biMi−qiθ′0θ2 − k3c ∈ zmi0H2

=⇒ k3(0) = 0 (since mi0 > 2qi0)

and

θ1b− k4ϕ+ ∈ H2 =⇒( d∏i=1

bini)θ′1b− k4z

qi0∏i�=i0

biNiθ′1θ3d ∈ H2

=⇒ zqi0(∏i�=i0

bNi−ni

i

)θ3b− k4d ∈ zqi0H2

=⇒ k4(0) = 0 ,


which proves (32). Write

θ2 = zl2θ′2 and θ3 = zl3θ′3 (θ′2(0) �= 0, θ′3(0) �= 0).

Then we can write

Φ+ =

[0 zqi0+l3

∏i�=i0

bNi

i θ′1θ′3d

zmi0+l2∏

i�=i0bMi

i θ′0θ′2c 0

].

On the other hand, write

a′ :=(∏i�=i0

biMi−mi

)a, θ′′1 :=

(∏i�=i0

bMi−qii

)θ′1

and

α := − a′(0)θ′′1 (0)

and ν :=1√|α|2 + 1

.

Note that

Φ− =

[ω θ0a

θ1b ω

].

Since Φ∗− is of bounded type, it follows from Theorem 1.1 that there exists a

square inner matrix function Δ such that kerH˜Φ∗

−= ΔH2

C2 and

Φ∗− = BΔ∗ (right coprime factorization).

Thus, by Lemma 2.5 we have

kerH˜Φ∗

−= ΔH2

C2⊆ ΩH2

C2 and Φ− = B∗Δ (left coprime factorization) ,

(33)where

Ω = ν

[zmi0−qi0 θ′0 αθ′1

−αzmi0−qi0+1θ′0 zθ′1

].

Since ΔH2C2⊆ ΩH2

C2 , it follows that Ω is a left inner divisor of Δ. Thus, wecan write

Δ = ΩΩ1 for some Ω1, so that Δ = Ω1Ω.

We suppose that qi0 + l3 ≤ mi0 + l2 and write r := (mi0 + l2)− (qi0 + l3) ≥ 0.Then there exist finite Blaschke products θ4 and θ5 with θi(0) �= 0 (i = 4, 5) suchthat

Φ+ =∏i�=i0

bmax(Mi,Ni)i (zmi0+l2θ′1θ

′3θ

′0θ

′2)I2

[0 θ5θ

′1θ

′3c

zrθ4θ′0θ

′2d 0

]∗≡ (θI2)A

∗,

where θ :=∏

i�=i0bmax(Mi,Ni)i (zmi0+l2θ′1θ′3θ′0θ′2). Since HAΔ∗ = HAΩ∗Ω∗

1, it follows

that

ranHAΔ∗ ⊇ ranHAΩ∗ . (34)


Observe that

AΩ∗ = ν

[0 θ5θ

′1θ

′3c

zrθ4θ′0θ

′2d 0

] [zmi0−qi0 θ′0 αθ′1

−αzmi0−qi0+1θ′0 zθ′1

]∗= ν

[αθ5θ

′3c zθ5θ

′3c

zr−mi0+qi0 θ4θ′2d −αzr−mi0+qi0−1θ4θ

′2d

].

If r ≤ mi0 − qi0 , then we have

HAΩ∗

[0

zmi0−qi0−r

]= ν

[Hz(z

mi0−qi0−rθ5θ′3c)

−αHz(θ4θ′2d)

].

Since (θ4θ′2d)(0) �= 0, it follows from Lemma 2.3, (33) and (34) that[

β1

]∈ cl ranHAΩ∗ ⊆ cl ranHAΔ∗ ⊆ ker(I − T

˜KT ∗˜K) for some β ∈ C. (35)

It thus follows from (32) and (35) that[β1

]= T

˜KT ∗˜K

[β1

]=

[T˜k1

T˜k3

T˜k2

T˜k4

][T˜k1

T˜k2

T˜k3

T˜k4

] [β1

]

=

[T˜k1

T˜k3

T˜k2

T˜k4

] [(βk1(0) + k2(0))

0

]=

[k1(βk1(0) + k2(0))

k2(βk1(0) + k2(0))

],

which implies that k1 is a constant and k2 is a nonzero constant. Again by (30),

ω − k2ϕ+ ∈ H2 =⇒ ωzqi0∏i�=i0

biNiθ′1θ3d ∈ H2

=⇒ qi ≥ ni (i �= i0) and θ′1θ3d ∈ H2

=⇒ ni = 0 (i �= i0) and θ′1θ3 = 1 ,

(36)

where the last implication follows from the observation that if ni �= 0 then by thecase assumption, mi = 0 and hence, 2qi < ni ≤ qi, a contradiction. We thus haveni = 0 for all i = 1, . . . , p. Since θ′1 = 1, it follows that

θ1 = 1 and hence, ψ− = 0.

In turn, mi > 2qi for all i = 1, . . . , p, so that θ0 is nonconstant, and henceϕ− = θ0a �= 0. Since by (30), θ0a − k1ψ+ ∈ H2, it follows that k1 �= 0. We thushave

θ0a− k1ψ+ ∈ H2 =⇒ θ0a− k1zmi0

∏i�=i0

biMi

θ′0θ2c ∈ H2

=⇒p∏

i=1

bimi

a− k1zmi0

∏i�=i0

biMi

θ2c ∈ H2

=⇒ θ2c ∈ H2 (because mi > 2qi and hence, Mi = mi)

=⇒ θ2 = 1 .

(37)


Therefore, we have

ϕ+ = zq∏i�=i0

bNi

i θ′1θ3d = ωd (q := qi0) and ψ+ = zmi0

∏i�=i0

bMi

i θ′0θ2c = θ0c.

Since by (27), |ϕ| = |ψ|, we have

|ωd+ θ0a| = |ϕ+ + ϕ−| = |ψ+| = |θ0c| (where a ∈ Hθ0 , d ∈ Hzω, c ∈ Hzθ0) ,

which implies

ωθ0(ωd+ θ0a)(ωd+ θ0a) = ωθ0cc ,

so that

ad = z((θ0c)(zω)c− (θ0d)(zω)d− (θ0a)(θ0d)(zω

2)− (θ0a)(zω)a). (38)

Since a ∈ Hθ0 , c ∈ Hzθ0 , d ∈ Hzω and mi ≥ 2qi for all i = 1, . . . , p, it follows

that θ0a ∈ H2, θ0c ∈ H2 and θ0d = (∏p

i=1 bmi

i ) θ′0d =(∏p

i=1 bmi−qii θ′0

)(ωd) ∈ H2.

Thus, (38) implies that ad = zh for some h ∈ H2, and hence (ad)(0) = 0, acontradiction. Therefore this case cannot occur.

If instead r > mi0 − qi0 , then the same argument as before leads to a contra-diction. Moreover, by the same argument as in the case qi0 + l3 ≤ mi0 + l2, thecase qi0 + l3 > mi0 + l2 cannot occur either.

Therefore, Case 3 cannot occur. This proves the implication (4) ⇒ (5).

This completes the proof. �

Remark 2.7. From the proof of Theorem 2.6 we can see that if G is given by (24)then G is subnormal if and only if G is normal, except in the case (26). Howeverwe need not expect that the exceptional case (26) implies normality of G. Forexample, if

Φ :=

[ω ω + 2ω

ω + 2ω ω

](ω is a finite Blaschke product)

then TΦ satisfies the case (26) (where mi = ni = qi and a = b = θ′0 = θ′1 = 1). Astraightforward calculation shows that TΦ is not normal. Since

TΦ =

[Tω Tω + 2Tω

Tω + 2Tω Tω

],

it follows that if W := 1√2

[1 −11 1

], then W is unitary and

W ∗TΦW = 2

[Tω + Tω 0

0 −Tω

],

which says that TΦ is unitarily equivalent to a direct sum of the normal operator2(Tω+Tω) and the analytic Toeplitz operator−2Tω. From this viewpoint, we mightconjecture that every subnormal rational Toeplitz operator is unitarily equivalentto a direct sum of a normal operator and an analytic Toeplitz operator. Howeverwe have been unable to settle this conjecture.


Acknowledgment

The authors are deeply indebted to the referee for many helpful comments thathelped improved the presentation and mathematical content of the paper.

References

[Ab] M.B. Abrahamse, Subnormal Toeplitz operators and functions of bounded type,Duke Math. J. 43 (1976), 597–604.

[AIW] I. Amemiya, T. Ito, and T.K. Wong, On quasinormal Toeplitz operators,Proc. Amer. Math. Soc. 50(1975), 254–258.

[At] A. Athavale On joint hyponormality of operators, Proc. Amer. Math. Soc.103(1988), 417–423.

[Br] J. Bram, Subnormal operators, Duke Math. J. 22(1955), 75–94.

[BH] A. Brown and P.R. Halmos, Algebraic properties of Toeplitz operators, J. ReineAngew. Math. 213(1963/1964), 89–102.

[Con] J.B. Conway, The Theory of Subnormal Operators, Math. Surveys and Mono-graphs, 36(1991), Amer. Math. Soc. Providence, Rhode Island.

[CoS] J.B. Conway and W. Szymanski, Linear combination of hyponormal operators,Rocky Mountain J. Math. 18(1988), 695–705.

[Co1] C.C. Cowen, On equivalence of Toeplitz operators, J. Operator Theory 7(1982),167–172.

[Co2] C.C. Cowen, More subnormal Toeplitz operators, J. Reine Angew. Math.367(1986), 215–219.

[Co3] C.C. Cowen, Hyponormal and subnormal Toeplitz operators, Surveys of SomeRecent Results in Operator Theory, I (J.B. Conway and B.B. Morrel, eds.),Pitman Research Notes in Mathematics, Volume 171, Longman, 1988, pp. (155–167).

[Co4] C.C. Cowen, Hyponormality of Toeplitz operators, Proc. Amer. Math. Soc.103(1988), 809–812.

[CoL] C.C. Cowen and J. Long, Some subnormal Toeplitz operators, J. Reine Angew.Math. 351(1984), 216–220.

[Cu1] R.E. Curto, Fredholm and invertible n-tuples of operators. The deformation prob-lem, Trans. Amer. Math. Soc. 266(1981), 129–159

[Cu2] R.E. Curto, Quadratically hyponormal weighted shifts, Integral Equations Oper-ator Theory, 13(1990), 49–66.

[CuF1] R.E. Curto and L.A. Fialkow, Recursiveness, positivity, and truncated momentproblems, Houston J. Math. 17(1991), 603–635.

[CuF2] R.E. Curto and L.A. Fialkow, Recursively generated weighted shifts and the sub-normal completion problem, Integral Equations Operator Theory, 17(1993), 202–246.

[CuF3] R.E. Curto and L.A. Fialkow, Recursively generated weighted shifts and the sub-normal completion problem II, Integral Equations Operator Theory, 18(1994),369–426.


[CHKL] R.E. Curto, I.S. Hwang, D. Kang and W.Y. Lee, Subnormal and quasinormalToeplitz operator with matrix-valued rational symbols, Adv. Math. 255(2014),562–585.

[CHL1] R.E. Curto, I.S. Hwang and W.Y. Lee, Hyponormality and subnormality of blockToeplitz operators, Adv. Math. 230(2012), 2094–2151.

[CHL2] R.E. Curto, I.S. Hwang and W.Y. Lee, Which subnormal Toeplitz operators areeither normal or analytic ?, J. Funct. Anal. 263(8)(2012), 2333–2354.

[CHL3] R.E. Curto, I.S. Hwang and W.Y. Lee, Abrahamse’s Theorem for matrix-valued symbols and subnormal Toeplitz completions, (preprint 2012) (arXiv:1301.6901,2013).

[CLL] R.E. Curto, S.H. Lee and W.Y. Lee, Subnormality and 2-hyponormality forToeplitz operators, Integral Equations Operator Theory, 44(2002), 138–148.

[CL1] R.E. Curto and W.Y. Lee, Joint hyponormality of Toeplitz pairs, MemoirsAmer. Math. Soc. 712, Amer. Math. Soc., Providence, 2001.

[CL2] R.E. Curto and W.Y. Lee, Towards a model theory for 2–hyponormal operators,Integral Equations Operator Theory 44(2002), 290–315.

[CL3] R.E. Curto and W.Y. Lee, Subnormality and k-hyponormality of Toeplitz opera-tors: A brief survey and open questions, Operator Theory and Banach Algebras(Rabat, 1999), 73–81, Theta, Bucharest, 2003.

[CMX] R.E. Curto, P.S. Muhly and J. Xia, Hyponormal pairs of commuting opera-tors, Contributions to Operator Theory and Its Applications (Mesa, AZ, 1987)(I. Gohberg, J.W. Helton and L. Rodman, eds.), Operator Theory: Advancesand Applications, vol. 35, Birkhauser, Basel–Boston, 1988, 1–22.

[CP1] R.E. Curto and M. Putinar, Existence of non-subnormal polynomially hyponor-mal operators, Bull. Amer. Math. Soc. (N.S.), 25(1991), 373–378.

[CP2] R.E. Curto and M. Putinar, Nearly subnormal operators and moment problems,J. Funct. Anal. 115(1993), 480–497.

[DPY] R.G. Douglas, V.I. Paulsen, and K. Yan, Operator theory and algebraic geometryBull. Amer. Math. Soc. (N.S.) 20(1989), 67–71.

[FL] D.R. Farenick and W.Y. Lee, Hyponormality and spectra of Toeplitz operators,Trans. Amer. Math. Soc. 348(1996), 4153–4174.

[Gu1] C. Gu, A generalization of Cowen’s characterization of hyponormal Toeplitz op-erators, J. Funct. Anal. 124(1994), 135–148.

[Gu2] C. Gu, On a class of jointly hyponormal Toeplitz operators, Trans. Amer. Math.Soc. 354(2002), 3275–3298.

[GHR] C. Gu, J. Hendricks and D. Rutherford, Hyponormality of block Toeplitz opera-tors, Pacific J. Math. 223(2006), 95–111.

[GS] C. Gu and J.E. Shapiro, Kernels of Hankel operators and hyponormality ofToeplitz operators, Math. Ann. 319(2001), 553–572.

[Hal1] P.R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76(1970),887–933.

[Hal2] P.R. Halmos, Ten years in Hilbert space, Integral Equations Operator Theory2(1979), 529–564.

[HKL1] I.S. Hwang, I.H. Kim and W.Y. Lee, Hyponormality of Toeplitz operators withpolynomial symbols, Math. Ann. 313(2)(1999), 247–261.


[HKL2] I.S. Hwang, I.H. Kim and W.Y. Lee, Hyponormality of Toeplitz operators withpolynomial symbols: An extremal case, Math. Nach. 231(2001), 25–38.

[HL1] I.S. Hwang and W.Y. Lee, Hyponormality of trigonometric Toeplitz operators,Trans. Amer. Math. Soc. 354(2002), 2461–2474.

[HL2] I.S. Hwang and W.Y. Lee, Hyponormality of Toeplitz operators with rationalsymbols, Math. Ann. 335 (2006), 405–414.

[HL3] I.S. Hwang and W.Y. Lee, Hyponormal Toeplitz operators with rational symbols,J. Operator Theory 56(2006), 47–58.

[HL4] I.S. Hwang and W.Y. Lee, Block Toeplitz Operators with rational symbols, J.Phys. A: Math. Theor. 41(18)(2008), 185207.

[HL5] I.S. Hwang and W.Y. Lee, Block Toeplitz Operators with rational symbols (II),J. Phys. A: Math. Theor. 41(38)(2008), 385206.

[Le] W.Y. Lee, Cowen sets for Toeplitz operators with finite rank selfcommutators, J.Operator Theory 54(2)(2005), 301–307.

[ItW] T. Ito and T.K. Wong, Subnormality and quasinormality of Toeplitz operators,Proc. Amer. Math. Soc. 34(1972), 157–164.

[McCP] S. McCullough and V. Paulsen, A note on joint hyponormality, Proc. Amer.Math. Soc. 107(1989), 187–195.

[NT] T. Nakazi and K. Takahashi, Hyponormal Toeplitz operators and extremal prob-lems of Hardy spaces, Trans. Amer. Math. Soc. 338(1993), 753–769.

[Ni] N.K. Nikolskii, Treatise on the shift operator, Springer, New York, 1986.

[Sun1] S. Sun, On hyponormal weighted shift, Chinese Ann. Math. Ser. B 5 (1984), no.1, 101–108. (A Chinese summary appears in Chinese Ann. Math. Ser. A 5 (1984),no. 1, 124.)

[Sun2] S. Sun, On Toeplitz operators in the θ-class, Sci. Sinica Ser. A 28 (1985), no. 3,235–241.

[Sun3] S. Sun, On hyponormal weighted shift, II, Chinese Ann. Math. Ser. B 6 (1985),no. 3, 359–361. (A Chinese summary appears in Chinese Ann. Math. Ser. A 6(1985), no. 4, 516.)

[Zhu] K. Zhu, Hyponormal Toeplitz operators with polynomial symbols, Integral Equa-tions Operator Theory 21(1996), 376–381.

Raul E. CurtoDepartment of Mathematics, University of IowaIowa City, IA 52242, USAe-mail: [email protected]

In Sung HwangDepartment of Mathematics, Sungkyunkwan UniversitySuwon 440-746, Koreae-mail: [email protected]

Woo Young LeeDepartment of Mathematics, Seoul National UniversitySeoul 151-742, Koreae-mail: [email protected]


Generalized Repeated Interaction Modeland Transfer Functions

Santanu Dey and Kalpesh J. Haria

Abstract. Using a scheme involving a lifting of a row contraction we introducea toy model of repeated interactions between quantum systems. In this modelthere is an outgoing Cuntz scattering system involving two wandering sub-spaces. We associate to this model an input/output linear system which leadsto a transfer function. This transfer function is a multi-analytic operator, andwe show that it is inner if we assume that the system is observable. Finallyit is established that transfer functions coincide with characteristic functionsof associated liftings.

Mathematics Subject Classification (2010). Primary 47A13; Secondary 47A20,46L53, 47A48, 47A40, 81R15.

Keywords. Repeated interaction, quantum system, multivariate operator the-ory, row contraction, contractive lifting, outgoing Cuntz scattering system,transfer function, multi-analytic operator, input-output formalism, linear sys-tem, observability, scattering theory, characteristic function.

1. Introduction

In page 287 of the article [9] the author has commented the following while com-paring [9] with [4, 5]: In [4] a row contraction A on a Hilbert space H with aone-dimensional eigenspace is considered and the theory of minimal isometric di-lations is used. The characteristic function introduced in [5] is a multi-analyticoperator associated to a lifting and the ergodic case is studied in detail in [4]. In[9] minimality is not considered but one starts with an interaction U (which is aunitary operator) in a scheme similar to [4] and obtains a multi-analytic operatorwhich represents the transfer function of an input-output system associated withthe interaction. It is expected that the scheme developed [9] is more directly appli-cable to physical models. In the setting of [5] the assumption of a one-dimensionaleigenspace is dropped and the theory is much more general in another direction.A further integration of these schemes in the future may help to remove unnec-


112 S. Dey and K.J. Haria

essarily restrictive assumptions of the toy model considered in [9] and lead to thestudy of other and of more realistic models.

This paper achieves some of these objectives. In the model of repeated inter-actions between quantum systems, also called a noncommutative Markov chain,studied in [9] (cf. [8]) for given three Hilbert spaces H,K and P with unit vectorsΩH,ΩK and ΩP an interaction is defined to be a unitary operator U : H ⊗ K →H⊗P such that

U(ΩH ⊗ ΩK) = ΩH ⊗ ΩP . (1.1)

Define K∞ :=⊗∞

i=1K and P∞ :=⊗∞

i=1 P as infinite tensor products of Hilbertspaces with distinguished unit vectors. We denote mth copy of K in K∞ by Km

and set K[m,n] := Km ⊗ · · · ⊗ Kn for m ≤ n. Similar notations are also used withrespect to P . The repeated interaction is defined as

U(n) := Un . . . U1 : H⊗K∞ → H⊗P[1,n] ⊗K[n+1,∞)

where Ui’s are copies of U on the factors H ⊗ Ki of the infinite tensor productsand Ui’s leaves other factors fixed. Equation (1.1) tells us that the tensor productof the vacuum vectors ΩH,ΩK (along with ΩP) represents a state of the coupledsystem which is not affected by the interaction U. This entire setting representsinteractions of an atom with light beams or fields. In particular ΩH in [9] is thoughtof as the vacuum state of an atom, and ΩK and ΩP as a state indicating the absenceof photons.

In the generalized repeated interaction model that we introduce in this articlewe use a pair of unitaries to encode the interactions instead of one unitary asfollows:Let H be a (closed) subspace ofH, and U : H⊗K → H⊗P and U : H⊗K → H⊗Pbe two unitaries such that

U(h⊗ ΩK) = U(h⊗ ΩK) for all h ∈ H. (1.2)

We fix {ε1, . . . , εd} to be an orthonormal basis of P . The equation (1.2) is the analogof the equation (1.1) for our model and thus our model can be used for the settingwhere a quantum system interacts with a stream of copies of another quantumsystem in such a way that there is no backaction (so we get a Markovian type ofdynamics) and such that there is a certain kind of subprocess. In the model of [9]the vacuum state ΩH of an atom plays an important role. For a model describinginteraction of a quantum system with a stream of copies of another quantum sytemwe need that the computations do not involve any fixed unit vector ΩH and weare able to achieve this in our model by using a pair of unitaries. Instead of ΩH

we now have a kind of subprocess, described by U , which can be treated on thesame level as the full process, described by U.

The main condition imposed on the unitary U : H⊗K → H⊗P in order toget a generalized interaction model is that U(H ⊗ ΩK) ⊂ H ⊗ P (cf. Proposition3.1 of [10] for an interesting consequence of this assumption). We can then define

U restricted to H ⊗ΩK as U restricted to H ⊗ΩK, and assume that H⊗P is bigenough to allow a unitary extension U : H ⊗ K → H ⊗ P . The focus of the study

Generalized Repeated Interaction Model and Transfer Functions 113

done here, as also in [9], is to bring out that certain multi-analytic operators ofthe multivariate operator theory are associated to noncommutative Markov chainsand related models, and that these operators can be exploited as powerful tools.These operators occur as central objects in various context such as in the systemstheory related works (cf. [3]) and noncommutative multivariable operator theoryrelated works (cf. [14], [15]).

A tuple T = (T1, . . . , Td) of operators Ti’s on a common Hilbert space L is

called a row contraction if∑d

i=1 TiT∗i ≤ I. In particular if

∑di=1 TiT

∗i = I, then

the tuple T = (T1, . . . , Td) is called coisometric. We introduce the notation Λfor the free semigroup with generators 1, . . . , d. Suppose T1, . . . , Td ∈ B(L) for a

Hilbert space L. If α ∈ Λ is the word α1 . . . αn with length |α| = n, where eachαj ∈ {1, . . . , d}, then Tα denote Tα1 . . . Tαn . For the empty word ∅ we define |∅| = 0and T∅ = I.

The unitary U : H⊗K → H⊗P from our model can be decomposed as

U(h⊗ ΩK) =d∑

j=1

E∗j h⊗ εj for h ∈ H, (1.3)

where Ej ’s are some operators in B(H), for j = 1, . . . , d. Likewise there exist some

operators Cj ’s in B(H) such that

U(h⊗ ΩK) =d∑

j=1

C∗j h⊗ εj for h ∈ H. (1.4)

Observe that∑d

j=1 EjE∗j = I and

∑dj=1 CjC

∗j = I, i.e., E and C are coisometric

tuples. By equation (1.2)

E∗j h = C∗

j h for all h ∈ H, j = 1, . . . , d.

We recall from [5] that such tuple E = (E1, . . . , Ed) is called a lifting ofC = (C1, . . . , Cd).

From a physicist perspective our model is a Markovian approximation of therepeated interaction between a quantum system and a stream of copies of anotherquantum system in such a way that there is no backaction. The change of anobservable X ∈ B(H) until time n, compressed to H, is written as

Zn(X) := PHU(n)∗(X ⊗ I)U(n)|H.

From equation (1.3) it follows that Zn(X) = Zn(X) where Z(X) =∑d

i=1 EiXE∗i :

B(H) → B(H) and Z is called the transition operator of the noncommutativeMarkov chain.

In Section 2 we develop our generalized repeated interaction model and obtaina coisometric operator which intertwines between the minimal isometric dilationsof E and C, and which will be crucial for the further investigation in this article.Using this an outgoing Cuntz scattering system in the sense of [3] is constructedfor our model in Section 3. Popescu introduced the minimal isometric dilation in


[13] and the characteristic function in [14] of a row contraction, and systematicallydeveloped an extensive theory of row contractions (cf. [16], [17]). We use some ofthe concepts from Popescu’s theory in this work.

For the outgoing Cuntz scattering system in Section 4 we give a Λ-linearsystem with an input-output formalism. A multi-analytic operator appears hereas the transfer function and in the next section we show that this transfer functioncan be derived from the intertwining coisometry of Section 2. In the scatteringinterpretation of the transfer function this now mediates between two processes.This together with a nice product formula obtained in Proposition 2.1 tells us thatthis identification of transfer function is a reminiscent of the scattering operatorconstruction using wave operators in Lax–Phillips scattering theory [12], equation(1.5) (cf. [18]), with one of the processes moving forward combined with the othermoving backward. In [20] and [7] there are other approaches to transfer functions.Several works on transfer functions and on quantum systems using linear systemtheory can be found in recent theoretical physics and control theory surveys.

In Section 5 we investigate in regard to our model what the notion of observ-ability implies for the scattering theory and the theory of liftings. Some techniquesused here are similar to those of scattering theory of noncommutative Markovchains introduced in [11]. Characteristic functions for liftings, introduced in [5],are multi-analytic operators which classify certain class of liftings. Our model gen-eralizes the setting of [9], and a comparison is done in Section 6 between thetransfer function of our model and the characteristic function for the associatedlifting using the series expansion of the transfer function obtained in Section 4.As a consequence mathematically generalized interaction models get firmly linkedinto the theory of functional models.

2. A generalised repeated interaction model

We begin with three Hilbert spaces H,K and P with unit vectors ΩK ∈ K andΩP ∈ P , and unitaries U and U as in equation (1.2). In K∞ =

⊗∞i=1K and

P∞ =⊗∞

i=1 P define ΩK∞ :=

⊗∞i=1 Ω

K and ΩP∞ :=

⊗∞i=1 Ω

P respectively. Wedenotemth copy of ΩK in ΩK

∞ by ΩKm and in terms of this we introduce the notation

ΩK[m,n] := ΩK

m⊗· · ·⊗ΩKn . Identify K[m,n] with ΩK

[1,m−1]⊗K[m,n]⊗ΩK[n+1,∞), H with

H⊗ΩK∞ as a subspace of H⊗K∞ and H with H ⊗ΩK

∞ as a subspace of H ⊗K∞.Similar notations with respect to P are also used. For simplicity we assume thatd is finite but all the results here can be derived also for d =∞.

Associate a row contraction E to the unitary U as in equation (1.3) anddefine isometries

V Ej (h⊗ η) := U∗(h⊗ εj)⊗ η for j = 1, . . . , d,

on the elementary tensors h ⊗ η ∈ H ⊗ K∞ and extend it linearly to obtain

V Ej ∈ B(H⊗K∞) for j = 1, . . . , d. We recall that a lifting T = (T1, . . . , Td) of any

row contraction S = (S1, . . . , Sd) is called its isometric dilation if Ti’s are isometries


with orthogonal ranges. It can be easily verified that VE= (V E

1 , . . . , V Ed ) on the

space H⊗K∞ is an isometric dilation of E = (E1, . . . , Ed). If h ∈ H and k1 ∈ K,then there exist hi ∈ H for i = 1, . . . , d such that U∗(

∑di=1 hi ⊗ εi) = h ⊗ k1

because U is a unitary. This implies

d∑i=1

V Ei (hi ⊗ ΩK

∞) = h⊗ k1 ⊗ ΩK[2,∞).

In addition if k2 ∈ K, thend∑

i=1

V Ei (hi ⊗ k2 ⊗ ΩK

[2,∞)) = U∗(d∑

i=1

hi ⊗ εi)⊗ k2 ⊗ ΩK[3,∞) = h⊗ k1 ⊗ k2 ⊗ ΩK

[3,∞).

By induction we conclude that

H⊗K∞ = span{V Eα (h⊗ ΩK

∞) : h ∈ H, α ∈ Λ},i.e., V

Eis the minimal isometric dilation of E. Note that the minimal isometric

dilation is unique up to unitary equivalence (cf. [13]).

Similarly, associate a row contraction C to the unitary U as in equation (1.4)and define isometries

V Cj (h⊗ η) := U∗(h⊗ εj)⊗ η for j = 1, . . . , d (2.1)

on the elementary tensors h ⊗ η ∈ H ⊗ K∞ and extend it linearly to obtain

V Cj ∈ B(H ⊗ K∞) for j = 1, . . . , d. The tuple V

C= (V C

1 , . . . , V Cd ) on the space

H ⊗ K∞ is the minimal isometric dilation of C = (C1, . . . , Cd). Recall that

Um : H⊗K∞ → H⊗K[1,m−1] ⊗ Pm ⊗K[m+1,∞)

is nothing but the operator which acts as U on H⊗Km and fixes other factors ofthe infinite tensor products. Similarly, we define Um using U .

Proposition 2.1. Let Pn := PH⊗ IP[1,n]⊗ IK[n+1,∞)

∈ B(H⊗P[1,n]⊗K[n+1,∞)) forn ∈ N. Then

sot− limn→∞ U∗

1 . . . U∗nPnUn . . . U1

exists and this limit defines a coisometry W : H ⊗ K∞ → H ⊗ K∞. Its adjoint

W ∗ : H ⊗ K∞ → H⊗K∞ is given by

W ∗ = sot− limn→∞U∗

1 . . . U∗nUn . . . U1.

Here sot stands for the strong operator topology.

Proof. At first we construct the adjoint W ∗. For that consider the dense subset⋃m≥1 H ⊗ K[1,m] of H ⊗ K∞ and let an arbitrary simple tensor element of this

dense subset be h⊗k1⊗· · ·⊗k�⊗ΩK[�+1,∞) for some � ∈ N, h ∈ H and ki ∈ Ki. Set

ap = U∗1 . . . U∗

p Up . . . U1(h⊗ k1⊗ · · ·⊗ k�⊗ΩK[�+1,∞)) for p ∈ N. Since U(h⊗ΩK) =


U(h ⊗ ΩK) for all h ∈ H, we have a� = a�+n for all n ∈ N. Therefore we deducethat

limn→∞U∗

1 . . . U∗nUn . . . U1(h⊗ k1 ⊗ · · · ⊗ k� ⊗ ΩK

[�+1,∞))

exists. Because U and U are unitaries, we obtain an isometric extension W ∗ to the

whole of H ⊗ K∞. Thus its adjoint is a coisometry W : H⊗K∞ → H ⊗K∞.

Now we will derive the limit form for W as claimed in the statement of theproposition. If h⊗ η ∈ H⊗K[1,k], h⊗ η ∈ H ⊗ K[1,n] and k ≤ n, then

〈W (h⊗ η), h⊗ η〉 = 〈h⊗ η, W ∗(h⊗ η)〉= 〈h⊗ η, U∗

1 . . . U∗nUn . . . U1(h⊗ η)〉

= 〈U∗1 . . . U∗

nPnUn . . . U1(h⊗ η), h⊗ η〉.Consequently W = sot− lim

n→∞ U∗1 . . . U∗

nPnUn . . . U1 on a dense subset and therefore

it can be extended to the whole of H⊗K∞. �Observe that

W ∗(h⊗ ΩK∞) = h⊗ ΩK

∞ for all h ∈ H. (2.2)

Next we show that this coisometry W intertwines between V Ej and V C

j for allj = 1, . . . , d. For j = 1, . . . , d, define

Sj : H⊗K∞ → H⊗P1 ⊗K[2,∞),

h⊗ η �→ h⊗ εj ⊗ η.

The following are immediate:

(1) S∗j (h⊗ p1 ⊗ η) = 〈εj , p1〉(h⊗ η) for (h⊗ p1 ⊗ η) ∈ H⊗ P1 ⊗K[2,∞).

(2) V Ej (h⊗ η) = U∗

1Sj(h⊗ η) for h⊗ η ∈ H ⊗K∞.

(3) V Cj (h⊗ η) = U∗

1Sj(h⊗ η) for h⊗ η ∈ H ⊗ K∞.

Proposition 2.2. If W is as in Proposition 2.1, then

W V Ej = V C

j W , V Ej W ∗ = W ∗V C

j for all j = 1, . . . , d.

Proof. If h ∈ H, η ∈ K∞, h ∈ H and ki ∈ Ki, then by the three observations thatwere noted preceding this proposition we obtain for j = 1, . . . , d

〈 W V Ej (h⊗ η), h⊗ k1 ⊗ · · · ⊗ k� ⊗ ΩK

[�+1,∞) 〉= 〈 U∗(h⊗ εj)⊗ η, U∗

1 . . . U∗� U� . . . U1(h⊗ k1 ⊗ · · · ⊗ k� ⊗ ΩK

[�+1,∞)) 〉.Substituting U(h⊗ k1) =

∑i h

(i) ⊗ k(i)1 where h(i) ∈ H and k

(i)1 ∈ K we obtain

〈 W V Ej (h⊗ η), h⊗ k1 ⊗ · · · ⊗ k� ⊗ ΩK

[�+1,∞) 〉=⟨h⊗ εj ⊗ η, U∗

2 . . . U∗� U� . . . U2

(∑i(h(i) ⊗ k

(i)1 )

⊗ k2 ⊗ · · · ⊗ k� ⊗ ΩK[�+1,∞)

) ⟩


=∑

i〈εj , k

(i)1 〉〈 h⊗ η, W ∗(h(i) ⊗ k2 ⊗ · · · ⊗ k� ⊗ ΩK

[�+1,∞)) 〉= 〈 W (h⊗ η), S∗

j U1(h⊗ k1 ⊗ · · · ⊗ k� ⊗ ΩK[�+1,∞)) 〉

= 〈 U∗1SjW ((h⊗ η), h⊗ k1 ⊗ · · · ⊗ k� ⊗ ΩK

[�+1,∞) 〉= 〈 V C

j W (h⊗ η), h⊗ k1 ⊗ · · · ⊗ k� ⊗ ΩK[�+1,∞) 〉.

Hence W V Ej = V C

j W for all j = 1, . . . , d. To obtain the other equation of theproposition we again use the last two of the three observations as follows: Forj = 1, . . . , d

W ∗V Cj (h⊗ k1 ⊗ · · · ⊗ k� ⊗ ΩK

[�+1,∞))

= W ∗U∗1 (h⊗ εj ⊗ k1 ⊗ · · · ⊗ k� ⊗ ΩK

[�+2,∞))

= U∗1U

∗2 . . . U∗

�+1U�+1 . . . U2U1U∗1 (h⊗ εj ⊗ k1 ⊗ · · · ⊗ k� ⊗ ΩK

[�+2,∞))

= U∗1U

∗2 . . . U∗

�+1U�+1 . . . U2Sj(h⊗ k1 ⊗ · · · ⊗ k� ⊗ ΩK[�+1,∞))

= U∗1SjU

∗1 . . . U∗

� U� . . . U1(h⊗ k1 ⊗ · · · ⊗ k� ⊗ ΩK[�+1,∞))

= V Ej W ∗(h⊗ k1 ⊗ · · · ⊗ k� ⊗ ΩK

[�+1,∞)). �

Further define

(H⊗K∞)◦ := (H⊗K∞)# (H ⊗ ΩK∞),

(H ⊗ K∞)◦ := (H ⊗ K∞)# (H ⊗ ΩK∞) and H◦ := H# H.

(2.3)

Let∑k

i=1 ξi ⊗ ηi ∈ (H⊗K∞)◦ and h ∈ H. Then for j = 1, . . . , d⟨V Ej

(∑iξi ⊗ ηi

), h⊗ ΩK

∞⟩=⟨∑

iU∗(ξi ⊗ εj)⊗ ηi, h⊗ ΩK

∞⟩

=⟨∑

iξi ⊗ εj ⊗ ηi, U(h⊗ ΩK

1 )⊗ ΩK[2,∞)

⟩= 0

because U maps into H⊗P and∑k

i=1 ξi⊗ηi⊥H⊗ΩK. Therefore V Ej (H⊗K∞)◦ ⊂

(H⊗K∞)◦ for j = 1, . . . , d. Similarly V Cj (H⊗K∞)◦ ⊂ (H⊗K∞)◦ for j = 1, . . . , d.

Set V Ej := V E

j |(H⊗K∞)◦ and V Cj := V C

j |(H⊗K∞)◦ for j = 1, . . . , d. If we define

W ∗ := W ∗|(H⊗K∞)◦ ,

then by equation (2.2) it follows that W ∗ ∈ B((H ⊗ K∞)◦, (H ⊗ K∞)◦). Theoperator W ∗ is an isometry because it is a restriction of an isometry and W , the

adjoint of W ∗, is the restriction of W to (H⊗K∞)◦, i.e., W = W |(H⊗K∞)◦ .

Remark 2.3. It follows that

WV Ej = V C

j W

for j = 1, . . . , d.


3. Outgoing Cuntz scattering systems

In this section we aim to construct an outgoing Cuntz scattering system (cf. [3])for our model. This will assist us in the next section to work with an input-outputformalism and to associate a transfer function to the model.

Following are some notions from the multivariable operator theory:

Definition 3.1. Suppose T = (T1, . . . , Td) is a row contraction where Ti ∈ B(L).(1) If Ti’s are isometries with orthogonal ranges, then the tuple T = (T1, . . . ,

Td) is called a row isometry.(2) If spanj=1,...,dTjL = L and T = (T1, . . . , Td) is a row isometry, then T is

called a row unitary.(3) If there exist a subspace E of L such that L =

⊕α∈Λ TαE and T = (T1, . . . , Td)

is a row isometry, then T is called a row shift and E is called a wanderingsubspace of L w.r.t. T .

Definition 3.2. A collection (L, V = (V1, . . . , Vd),G+∗ ,G) is called an outgoing

Cuntz scattering system (cf. [3]), if V is a row isometry on the Hilbert spaceL, and G+∗ and G are subspaces of L such that

(1) for E∗ := L # spanj=1,...,dVjL, the tuple V |G+∗ is a row shift where G+

∗ =⊕α∈Λ VαE∗.

(2) there exist E := G # spanj=1,...,dVjG with G =⊕

α∈Λ VαE , i.e., V |G is a rowshift.

In the above definition the part (1) is the Wold decomposition (cf. [13]) ofthe row isometry V and therefore G+

∗ can be derived from V . But G+∗ is included

in the data because it helps in describing the scattering phenomenon. We continue

using the notations from the previous section. V Ej ’s are isometries with orthogonal

ranges and because (εj)dj=1 is an orthonormal basis of P , we have

spanj=1,...,dVEj (H⊗K∞) = H⊗K∞.

Thus VEis a row unitary on H⊗K∞. Now using the fact that V E

j = V Ej |(H⊗K∞)◦

we infer that V Ej ’s are isometries with orthogonal ranges. Therefore V E is a row

isometry on (H⊗K∞)◦.

Proposition 3.3. If Y := H ⊗ (ΩK1 )

⊥ ⊗ ΩK[2,∞) ⊂ H ⊗ K∞, then

W ∗Y ⊥ spanj=1,...,d V Ej (H⊗K∞)◦.

Proof. By Proposition 2.1 it is easy to see that

W ∗Y = U∗1 U1Y ⊂ H⊗K1 ⊗ ΩK

[2,∞). (3.1)


Let hi ∈ H and ki ⊥ ΩK1 for i = 1, . . . , n, i.e.,

∑i hi ⊗ ki ⊗ ΩK

[2,∞) ∈ Y. For∑k hk ⊗ ηk ∈ (H⊗K∞)◦ with hk ∈ H and ηk ∈ K∞

〈 W ∗(∑

i hi ⊗ ki ⊗ ΩK[2,∞)), V

Ej (∑

k hk ⊗ ηk) 〉= 〈 U∗U(

∑i hi ⊗ ki)⊗ ΩK

[2,∞),∑

k U∗(hk ⊗ εj)⊗ ηk 〉

= 〈 U(∑

i hi ⊗ ki)⊗ ΩK[2,∞),

∑k hk ⊗ εj ⊗ ηk 〉 = 0.

The last equality holds because∑

k hk ⊗ ηk ⊥ H ⊗ ΩK∞.

Thus W ∗Y ⊥ spanj=1,...,dVEj (H⊗K∞)◦. �

The following Proposition gives an explicit description of the Wold decom-position of V E :

Proposition 3.4. If Y is defined as in the previous proposition, then W ∗Y is awandering subspace of V E, i.e., V E

α (W ∗Y) ⊥ V Eβ (W ∗Y) whenever α, β ∈ Λ, α �=

β, and

W ∗Y = (H⊗K∞)◦ # spanj=1,...,dVEj (H⊗K∞)◦.

Proof. By Proposition 3.3 it is immediate that V Eα (W ∗Y) ⊥ V E

β (W ∗Y) wheneverα, β ∈ Λ, α �= β and W ∗Y ⊂ (H ⊗ K∞)◦ # spanj=1,...,dV

Ej (H ⊗ K∞)◦. The only

thing that remains to be shown is that

(H⊗K∞)◦ # spanj=1,...,dVEj (H⊗K∞)◦ ⊂W ∗Y.

Let x ∈ (H ⊗K∞)◦ # spanj=1,...,dVEj (H ⊗K∞)◦. Write down the decompo-

sition of x as x1 ⊕ x2 w.r.t. W ∗Y ⊕ (W ∗Y)⊥. So x − x1 = x2 is orthogonal toboth spanj=1,...,dV

Ej (H ⊗ K∞)◦ and W ∗Y. Now we show that if any element in

(H ⊗ K∞)◦ is orthogonal to spanj=1,...,dVEj (H ⊗ K∞)◦ and W ∗Y, then it is the

zero vector. Let x0 be such an element. Because x0 ∈ (H⊗K∞)◦ and x0 ⊥W ∗Y,x0 ⊥ U∗(H ⊗ εj)⊗ ΩK

[2,∞)

for j = 1, . . . , d. This implies x0 ⊥ spanj=1,...,dVEj (H ⊗ ΩK

∞). We also know that

x0 ⊥ spanj=1,...,dVEj (H⊗K∞)◦(= spanj=1,...,dV

Ej (H⊗K∞)◦).

Therefore

x0 ⊥ spanj=1,...,dVEj (H⊗K∞).

Since VE

is a row unitrary, x0 ⊥ H ⊗ K∞. So x0 = 0 and hence x = x1 ∈ W ∗Y.We conclude that (H⊗K∞)◦ # spanj=1,...,dV

Ej (H⊗K∞)◦ ⊂W ∗Y. �

Proposition 3.5. If E := H ⊗ (ΩK1 )

⊥ ⊗ ΩK[2,∞) ⊂ (H ⊗ K∞)◦, then V E

α E ⊥ V Eβ E

whenever α, β ∈ Λ, α �= β and (H⊗K∞)◦ = H◦ ⊕⊕α∈ΛVEα E.


Proof. If |α| = |β| and α �= β, then it is easy to see that V Eα E ⊥ V E

β E because

ranges of V Ej ’s are mutually orthogonal. If |α| �= |β| (without loss of generality we

can assume that |α| > |β|), then by taking the inner product at the tensor factorK|α|+1 we obtain V E

α E ⊥ V Eβ E .

To prove the second part of the proposition, observe that for n ∈ N,

H⊗K[1,n] ⊗ ΩK[n+1,∞]

= (H⊗ ΩK∞)⊕ (H⊗ (ΩK

1 )⊥ ⊗ ΩK

[2,∞))⊕ (H⊗K1⊗(ΩK

2 )⊥ ⊗ ΩK

[3,∞))⊕ · · · ⊕ (H⊗K[1,n−1] ⊗ (ΩKn )

⊥ ⊗ ΩK[n+1,∞))

= (H ⊗ ΩK∞)⊕ (H◦ ⊗ ΩK

∞)⊕ E ⊕d⊕

j=1

V Ej E ⊕ · · · ⊕

d⊕|α|=n−1

V Eα E .

Taking n→∞ we have the following:

H⊗K∞ = (H ⊗ ΩK∞)⊕ (H◦ ⊗ ΩK

∞)⊕⊕α∈Λ

V Eα E .

Since (H⊗K∞)◦ = (H⊗K∞)# (H ⊗ ΩK∞), it follows that

(H⊗K∞)◦ = H◦ ⊕⊕α∈Λ

V Eα E . �

We sum up Propositions 3.3, 3.4 and 3.5 in the following theorem:

Theorem 3.6. For a generalized repeated interaction model involving unitaries Uand U as before set Y := H ⊗ (ΩK

1 )⊥ ⊗ ΩK

[2,∞) and E := H ⊗ (ΩK1 )

⊥ ⊗ ΩK[2,∞). If

E∗ := W ∗Y, G+∗ :=⊕

α∈Λ V Eα E∗ and G :=

⊕α∈Λ V E

α E, then the collection

((H⊗K∞)◦, V E = (V E1 , . . . , V E

d ),G+∗ ,G)

is an outgoing Cuntz scattering system such that (H⊗K∞)◦ = H◦ ⊕ G.Remark 3.7. Applying arguments similar to those used for proving the second partof the Proposition 3.5 one can prove the following:

(H ⊗ K∞)◦ =⊕α∈Λ

V Cα Y.

We refer the reader to Proposition 3.1 of [10] for a result in a similar direction.

4. Λ-linear systems and transfer functions

We would demonstrate that the outgoing Cuntz scattering system ((H ⊗ K∞)◦,V E = (V E

1 , . . . , V Ed ), G+∗ ,G) from Theorem 3.6 has interesting relations with a

generalization of the linear systems theory that is associated to our interaction


model. For a given model involving unitaries U and U as before, let us define theinput space as

U := E = H⊗ (ΩK1 )

⊥ ⊗ ΩK[2,∞) ⊂ (H⊗K∞)◦

and the output space as

Y = H ⊗ (ΩK1 )

⊥ ⊗ ΩK[2,∞) ⊂ (H ⊗ K∞)◦.

Here we assume that a quantum system A interacts with a stream of copiesof another quantum system B and we assume H is the (quantum mechanical)Hilbert space of A. Let Ki be the Hilbert space of a part of a stream of copiesof B at time i immediately before the interaction with A. Let the Hilbert spacePi be that the part of a stream of copies of B at time i immediately after theinteraction with A. ΩK and ΩP denote states indicating that no copy of quantumsystem B is present and so no interaction is taking place at time i. Then η ∈ U =H⊗ (ΩK

1 )⊥ ⊗ ΩK

[2,∞) ⊂ H⊗K∞ represents a vector state with copies of quantum

system B arriving at time 1 and stimulating an interaction between the streamof copies of A and B, but no further copy of B arriving at later times. But someactivity is induced which goes on for a longer period.

Note that H ⊗ K = H ⊕ U and H ⊗ K = H ⊕ Y. So U maps H ⊕ U ontoH ⊗ P and U maps H ⊕ Y onto H ⊗ P . Using unitaries U and U we defineFj : H → U and Dj : H → Y for j = 1, . . . , d by

d∑j=1

F ∗j η ⊗ εj := U(0⊕ η),

d∑j=1

D∗j y ⊗ εj := U(0⊕ y) for η ∈ U and y ∈ Y. (4.1)

Combining equation (4.1) with equations (1.3) and (1.4) we have for h ∈ H, η ∈U , h ∈ H and y ∈ Y

U(h⊕ η) =d∑

j=1

(E∗j h+ F ∗

j η)⊗ εj , (4.2)

U(h⊕ y) =

d∑j=1

(C∗j h+D∗

j y)⊗ εj (4.3)

respectively. Using equation (4.3) it can be checked that

U∗(h⊗ εj) = Cj h⊕Dj h for h ∈ H; j = 1, . . . , d. (4.4)

Let us define

C :=

d∑j=1

DjPHE∗j : H → Y, D :=

d∑j=1

DjPHF ∗j : U → Y

where PH is the orthogonal projection onto H. It follows that

PY U∗P1U(h⊕ η) = Ch+ Dη (4.5)


where h ∈ H, η ∈ U , P1 is as in Proposition 2.1 and PY is the orthogonal projectiononto Y.

Define a colligation of operators (cf. [3]) using the operators E∗j ’s, F ∗

j ’s,

C and D by

CU,U :=

⎛⎜⎜⎜⎝E∗

1 F ∗1

......

E∗d F ∗

d

C D

⎞⎟⎟⎟⎠ : H⊕ U →d⊕

j=1

H⊕ Y.

From the colligation CU,U we get the following Λ-linear system∑

U,U :

x(jα) = E∗j x(α) + F ∗

j u(α), (4.6)

y(α) = Cx(α) + Du(α) (4.7)

where j = 1, . . . , d and α, jα are words in Λ, and

x : Λ→ H, u : Λ→ U , y : Λ→ Y.If x(∅) and u are known, then using

∑U,U we can compute x and y recursively. Such

a Λ-linear system is also called a noncommutative Fornasini–Marchesini system in[1] in reference to [6].

Let z = (z1, . . . , zd) be a d-tuple of formal noncommuting indeterminates.Define the Fourier transforms of x, u and y as

x(z) =∑α∈Λ

x(α)zα, u(z) =∑α∈Λ

u(α)zα, y(z) =∑α∈Λ

y(α)zα

respectively where zα = zαn . . . zα1 for α = αn . . . α1 ∈ Λ. Assuming that z-variables commute with the coefficients the input-output relation

y(z) = ΘU,U (z)u(z)

can be obtained on setting x(∅) := 0 where

ΘU,U (z) :=∑α∈Λ

Θ(α)

U,Uzα := D + C

∑β∈Λ,j=1,...,d

(Eβ)∗F ∗

j zβj. (4.8)

Here β = β1 . . . βn is the reverse of β = βn . . . β1 ∈ Λ and Θ(α)

U,Umaps U to Y. The

formal noncommutative power series ΘU,U is called the transfer function associated

to the unitaries U and U . The transfer function is a mathematical tool for encodingthe evolution of a Λ-linear system. For y(α) ∈ Y with

∑α∈Λ ‖y(α)‖2 < ∞, any

series∑

α∈Λ y(α)zα stands for a series converging to an element of �2(Λ,Y).Theorem 4.1. The map MΘU,U

: �2(Λ,U)→ �2(Λ,Y) defined by

MΘU,Uu(z) := ΘU,U (z)u(z)

is a contraction.


Proof. Observe that PY U∗P1U(h ⊗ ΩK∞) = 0 for all h ∈ H. Consider another

colligation which is defined as follows:

C◦U,U

:=

⎛⎜⎜⎜⎝E∗◦

1 F ∗◦1

......

E∗◦d F ∗◦

d

C◦ D

⎞⎟⎟⎟⎠ : H◦ ⊕ U →d⊕

j=1

H◦ ⊕ Y

where E∗◦j := PH◦E∗

j |H◦ : H◦ → H◦, F ∗◦j := PH◦F ∗

j : U → H◦ and C◦ := C|H◦ :H◦ → Y for j = 1, . . . , d. Recall that H◦ and (H⊗K∞)◦ were defined in equation

array (2.3). Consider the outgoing Cuntz scattering system ((H ⊗ K∞)◦, V E =(V E

1 , . . . , V Ed ),G+∗ ,G), with (H ⊗K∞)◦ = H◦ ⊕ G, constructed by us in Theorem

3.6. In Chapter 5.2 of [3] it is shown that there is an associated unitary colligation⎛⎜⎜⎜⎝E1 F1

......

Ed Fd

M N

⎞⎟⎟⎟⎠ : H◦ ⊕ E →d⊕

j=1

H◦ ⊕ E∗ (4.9)

such that (Ej , Fj) = PH◦(V Ej )∗|H◦⊕E and (M, N) = PE∗ |H◦⊕E . Recall that E

and E∗ were introduced in Proposition 3.5 and Theorem 3.6 respectively. Fromequations (4.2) and (4.5) we observe that (E∗◦

j , F ∗◦j ) = PH◦⊗εjU |H◦⊕E (identifying

H◦ with H◦ ⊗ εj) and (C◦, D) = PY U∗P1U |H◦⊕E . Using these observations weobtain the following relations:

U∗(E∗◦j , F ∗◦

j ) = U∗PH◦⊗εjU |H◦⊕E = PU∗(H◦⊗εj)|H◦⊕E = PV Ej H◦ |H◦⊕E

= V Ej PH◦(V E

j )∗|H◦⊕E = V Ej (Ej , Fj) (4.10)

for j = 1, . . . , d and

U∗U(C◦, D) = U∗UPY U∗P1U |H◦⊕E = U∗PUYP1U |H◦⊕E = U∗PUYU |H◦⊕E= PU∗UY |H◦⊕E = PW∗Y |H◦⊕E (by equation (3.1))

= PE∗ |H◦⊕E = (M, N). (4.11)

Let u(z)=∑

α∈Λu(α)zα∈ �2(Λ,U) with u(α)∈U such that

∑α∈Λ‖u(α)‖2<

∞. We would prove that

‖MΘU,Uu(z)‖2 ≤ ‖u(z)‖2.

Define x : Λ → H by equation (4.6) such that x(∅) = 0. Further, define x◦(α) :=PH◦x(α) for all α ∈ Λ. Now applying the projection PH◦ to relation (4.6) on both

sides and using the fact H is invariant under E∗j for j = 1, . . . , d we obtain the

following relation:

x◦(jα) = E∗◦j x◦(α) + F ∗◦

j u(α) for all α ∈ Λ, j = 1, . . . , d. (4.12)


Because PY U∗P1U1(h⊗ΩK∞) = 0 for all h ∈ H we conclude by equation (4.5) that

Ch = 0 for h ∈ H. (4.13)

This implies

Cx(α) = C◦x◦(α) for all α ∈ Λ. (4.14)

Define y : Λ→ Y by

y(α) := Cx(α) + Du(α) (4.15)

for all α ∈ Λ. Recall that the input-output relation stated just before the theorem is

y(z) =∑α∈Λ

y(α)zα = ΘU,U (z)u(z)(= MΘU,Uu(z)).

Using the unitary colligation given in equation (4.9) we have

‖x◦(α)‖2 + ‖u(α)‖2 =

d∑j=1

‖Ejx◦(α) + Fju(α)‖2 + ‖Mx◦(α) + Nu(α)‖2

=d∑

j=1

‖E∗◦j x◦(α) + F ∗◦

j u(α)‖2 + ‖C◦x◦(α) + Du(α)‖2

=

d∑j=1

‖x◦(jα)‖2 + ‖Cx(α) + Du(α)‖2

=

d∑j=1

‖x◦(jα)‖2 + ‖y(α)‖2

for all α ∈ Λ. In the above calculation equations (4.10), (4.11), (4.12), (4.14) and(4.15) respectively have been used. This gives us

‖u(α)‖2 − ‖y(α)‖2 =

d∑j=1

‖x◦(jα)‖2 − ‖x◦(α)‖2

for all α ∈ Λ. Summing over all α ∈ Λ with |α| ≤ n and using the fact thatx◦(∅) = 0 we obtain∑

|α|≤n

‖u(α)‖2 −∑|α|≤n

‖y(α)‖2 =∑

|α|=n+1

‖x◦(α)‖2 ≥ 0 for all n ∈ N.

Therefore ∑|α|≤n

‖y(α)‖2 ≤∑|α|≤n

‖u(α)‖2 for all n ∈ N.

Finally taking limit n→∞ both the sides we get that MΘU,Uis a contraction. �


MΘU,Uis a multi-analytic operator ([15]) (also called analytic intertwining

operator in [3]) because

MΘU,U

(∑α∈Λ

u(α)zαzj)= MΘU,U

(∑α∈Λ

u(α)zα)zj for j = 1, . . . , d,

i.e., MΘU,Uintertwines with right translation. The noncommutative power series

ΘU,U is called the symbol of MΘU,U.

5. Transfer functions, observability and scattering

We would now establish that the transfer function can be derived from the coisom-etry W of Section 2. In the last section d-tuple z = (z1, . . . , zd) of formal noncom-muting indeterminates were employed. Treat (zα)α∈Λ as an orthonormal basis of

�2(Λ,C). Assume Y and U to be the spaces associated with our model with uni-

taries U and U as in the last section. It follows from Remark 3.7 that there exista unitary operator Γ : (H ⊗ K∞)◦ → �2(Λ,Y) defined by

Γ(V Cα y) := yzα for all α ∈ Λ, y ∈ Y.

We observe the following intertwining relation:

Γ(V Cα y) = (Γy)zα. (5.1)

Similarly, using Theorem 3.6, we can define a unitary operator Γ : (H ⊗K∞)◦(=(H◦ ⊕ G))→ H◦ ⊕ �2(Λ,U) by

Γ(h⊕ V Eα η) := h⊕ ηzα for all α ∈ Λ

where h ∈ H◦, η ∈ U . In this case the intertwining relation is

Γ(V Eα η) = (Γη)zα. (5.2)

Using the coisometric operator W , which appears in Remark 2.3, we define ΓW

by the following commutative diagram:

(H⊗K∞)◦ W ��

Γ

��

(H ⊗ K∞)◦

Γ��

H◦ ⊕ �2(Λ,U) ΓW �� 2(Λ,Y),

(5.3)

i.e., ΓW = ΓWΓ−1.

Theorem 5.1. ΓW defined by the above commutative diagram satisfies

ΓW |�2(Λ, U) = MΘU,U.


Proof. Using the intertwining relation V Cj W = WV E

j from Remark 2.3, and equa-

tions (5.1) and (5.2) we obtain

ΓW (ηzβzj) = ΓWΓ−1(ηzβzj) = ΓWV Ej V E

β η

= ΓV Cj V C

β Wη = (ΓWη)zβzj = ΓW (ηzβ)zj

for η ∈ U , β ∈ Λ, j = 1, . . . , d. Hence, ΓW |�2(Λ, U) is a multi-analytic operator.

For computing its symbol we determine ΓW η for η ∈ U , where η is identified withηzφ ∈ �2(Λ,U). For α = αn−1 . . . α1 ∈ Λ let Pα be the orthogonal projection onto

Γ−1{f ∈ �2(Λ,Y) : f = yzα for some y ∈ Y}= V C

α Y = U∗1 . . . U∗

n−1(H ⊗ εα1 ⊗ · · · ⊗ εαn−1 ⊗ (ΩKn )

⊥ ⊗ ΩK[n+1,∞))

with Ui’s as in Proposition 2.1.Recall that the tuple E associated with the unitary U is a lifting of the tuple

C (associated with the unitary U) and so E can be written as a block matrix in

terms of C as follows: Ej =

(Cj 0Bj Aj

)for j = 1, . . . , d w.r.t. to the decomposition

H = H⊕H◦ where B and A are some row contractions. Because E is a coisometriclifting of C we have

d∑j=1

CjC∗j = I and

d∑j=1

CjB∗j = 0

(cf. [5]). Now using these relations and equations (4.2), (4.3) and (4.4) it can beeasily verified that

PαU∗1 . . . U∗

nPnUn . . . U1η = PαU∗1 . . . U∗

mPmUm . . . U1η for all m ≥ n, η ∈ U .Using the formula of W from Proposition 2.1 we obtain

PαWη = PαU∗1 . . . U∗

nPnUn . . . U1η for η ∈ U .Finally for η ∈ U

PαU∗1 . . . U∗

nPnUn . . . U1η =

{Dη if n = 1, α = ∅,V Cα (CE∗

αn−1. . . E∗

α2F ∗α1η) if n = |α|+ 1 ≥ 2.

This implies for η ∈ UΓWΓ−1η = ΓWη = Dη ⊕

∑|α|≥1

(CE∗αn−1

. . . E∗α2F ∗α1η)zα.

Comparing this with equation (4.8) we conclude that ΓW |�2(Λ, U) = MΘU,U. �

Note that the Theorem 4.1 and its proof concern the transfer function of theΛ-linear system and has nothing to do with the scattering theory. Theorem 5.1,on the other hand, is the scattering theory part in the sense of Lax–Phillips [12].The same function MΘU,U

relates the outgoing Fourier representation for a vector

in the ambient scattering Hilbert space to the incoming Fourier representation for


the same vector. This makes MΘU,Uthe scattering function for the outgoing Cuntz

scattering system. We introduce a notion from the linear systems theory for ourmodel:

Definition 5.2. The observability operator W0 : H◦ → �2(Λ,Y) is defined as therestriction of the operator ΓW to H◦, i.e., W0 = ΓW |H◦ .

It follows that W0h = (C(Eα)∗h)α∈Λ. Popescu has studied the similar types of

operators called Poisson kernels in [16].

Definition 5.3. If there exist k,K > 0 such that for all h ∈ H◦

k‖h‖2 ≤∑α∈Λ

‖C(Eα)∗h‖2 = ‖W0h‖2 ≤ K ‖h‖2,

then the Λ-linear system is called (uniformly) observable.

We illustrate below that the notion of observability is closely related to the scat-tering theory notions of noncommutative Markov chains. Observability of a systemfor dimH < ∞ is interpreted as the property of the system that in the absenceof U-inputs we can determine the original state h ∈ H◦ of the system from allY-outputs at all times. Uniform observability is an analog of this for dimH =∞.

We extend W0 to

W0 : (H ⊕ H◦)(= H) −→ H ⊕ �2(Λ,Y)by defining W0h := h for all h ∈ H. If W0 is uniformly observable, then using

k = k and K = max{1,K} the above inequalities can be extended to W0 on H as

k‖h‖2 ≤ ‖W0h‖2 ≤ K‖h‖2for all h ∈ H.

Before stating the main theorem of this section regarding observability werecall from [5] the following: Let C be a row contraction on a Hilbert space HC .

The lifting E of C is called subisometric [5] if the minimal isometric dilations VE

and VC

of E and C respectively are unitarily equivalent and the corresponding

unitary, which intertwines between V Ei and V C

i for all i = 1, 2, . . . , d, acts asidentity on HC . Some of the techniques used here are from the scattering theoryof noncommutative Markov chains (cf. [11], [8]).

Theorem 5.4. For any Λ-linear system associated to a generalized repeated inter-action model with unitaries U, U the following statements are equivalent:

(a) The system is (uniformly) observable.(b) The observability operator W0 is isometric.(c) The tuple E associated with the unitary U is a subisometric lifting of the

tuple C (associated with the unitary U).

(d) W : (H⊗K∞)◦ → (H ⊗ K∞)◦ is unitary.

If one of the above holds, then


(e) The transfer function ΘU,U is inner, i.e., MΘU,U: �2(Λ,U) → �2(Λ,Y) is

isometric.

If we have additional assumptions, viz. dimH < ∞ and dimP ≥ 2, then theconverse holds, i.e., (e) implies all of (a), (b), (c) and (d).

Proof. Clearly (d) ⇒ (b) ⇒ (a). We now prove (a) ⇒ (d). Because the system is

(uniformly) observable there exist k > 0 such that for all h ∈ H◦

k‖h‖2 ≤ ‖W0h‖2.Since

⋃m≥1H⊗K[1,m] is a dense subspace of H ⊗K∞, for any 0 �= η ∈ H ⊗ K∞

there exist n ∈ N and η′ ∈ H ⊗K[1,n] such that

‖η − η′‖ <√k√

k + 1‖η‖.

Let η0 ∈ H ⊗ K[1,n]. Suppose Un . . . U1η0 = h0 ⊗ p0 ⊗ ΩK[n+1,∞), where h0 ∈ H,

p0 ∈ P[1,n]. Then clearly

limN→∞

‖U∗1 . . . U∗

nU∗n+1 . . . U

∗NPNUN . . . Un+1Un . . . U1η0‖ = ‖W0h0‖‖p0‖

and thus by Proposition 2.1 it is equal to ‖Wη0‖. Because the system is (uniformly)observable,

‖W0h0‖‖p0‖ ≥√k‖h0‖‖p0‖.

Therefore ‖Wη0‖2 ≥ k‖η0‖2. However, in general Un . . . U1η0 =∑

j h(j)0 ⊗

p(j)0 ⊗ΩK

[n+1,∞) with h(j)0 ∈ H and some mutually orthogonal vectors p

(j)0 ∈ P[1,n].

By using the above inequality for each term of the summation and then addingthem we find that in general for all η0 ∈ H⊗K[1,n]

‖Wη0‖2 ≥ k‖η0‖2.In particular, for η′ ∈ H ⊗K[1,n] we have the above inequality. Therefore

‖Wη‖ ≥ ‖Wη′‖ − ‖W (η′ − η)‖≥√k‖η′‖ − ‖η − η′‖

≥√k‖η‖ − (

√k + 1)‖η − η′‖ > 0.

This implies Wη �= 0 for all 0 �= η ∈ H ⊗ K∞ and hence W is injective. Recall

that W is a coisometry and an injective coisometry is unitary. Further, because

W (h⊗ΩK∞) = h⊗ΩK

∞ for all h ∈ H it follows that W is unitary. This establishes(a)⇒ (d) and we have proved (a)⇔ (b)⇔ (d).

Next we prove (d)⇔ (c). Assume that (d) holds. Since W is unitary, clearly

W is unitary. We know that W intertwines between the minimal isometric dilations

VE

and VCof E and C respectively. Hence E is a subisometric lifting of C.


Conversely, if we assume (c), then by the definition of subisometric liftingthere exist a unitary operator

W1 : H⊗K∞ −→ H ⊗ K∞

which intertwines between VE

and VC, and W1 acts as an identity on H ⊗ ΩK∞.

To prove W is unitary it is enough to prove W is unitary. We show that W = W1.By the definition of the minimal isometric dilation we know that H ⊗ K∞ =

span{V Cα (h⊗ΩK

∞) : h ∈ H, α ∈ Λ}. For j = 1, . . . , d and h ∈ H, by equation (2.2)and Proposition 2.2,

W ∗V Cj (h⊗ ΩK

∞) = V Ej W ∗(h⊗ ΩK

∞) = V Ej (h⊗ ΩK

∞)

= W ∗1 V

Cj W1(h⊗ ΩK

∞) = W ∗1 V

Cj (h⊗ ΩK

∞).

Thus W ∗ = W ∗1 and hence W = W1.

To prove (d) ⇒ (e) we at first note that since W is unitary, ΓW is alsounitary. By Theorem 4.2, we have MΘU,U

= ΓW |�2(Λ,U). Since a restriction of a

unitary operator is an isometry, MΘU,Uis isometric.

Finally with the additional assumptions dimH <∞ and dimP ≥ 2, we show(e)⇒ (b). Define

Hscat := H ∩ W ∗(H ⊗ K∞) = H ⊕ {h ∈ H◦ : ‖W0h‖ = ‖h‖}.Since ‖W0h‖ = lim

n→∞ ‖U1 . . . UnPnUn . . . U1h‖ by Proposition 2.1, the following can

be easily verified:

U(Hscat ⊗ ΩK) ⊂ Hscat ⊗ P . (5.4)

Because MΘU,U= ΓW |�2(Λ,U) is isometric by (e), it can be checked that

U(H⊗ (ΩK)⊥) ⊂ Hscat ⊗ P . (5.5)

Combining equations (5.4) and (5.5) we have

U∗((H#Hscat)⊗ P) ⊂ (H#Hscat)⊗ ΩK.

Since dimH < ∞ and dimP ≥ 2, we obtain H # Hscat = {0}, i.e., H = Hscat.This implies W0 is isometric and hence (e)⇒ (b). �

6. Transfer functions and characteristic functions of liftings

Continuing with the study of our generalized repeated interaction model, fromequations (2.1) and (4.4) we obtain

V Cj (h⊗ ΩK

∞) = (Cj h⊕Djh)⊗ ΩK[2,∞) for h ∈ H and j = 1, . . . , d. (6.1)

Let DC := (I − C∗C)12 :

⊕di=1 H → ⊕d

i=1 H denote the defect operator and

DC := Range DC . The full Fock space over Cd (d ≥ 2) denoted by F is

F = C⊕ Cd ⊕ (Cd)⊗2 ⊕ · · · ⊕ (Cd)⊗

m ⊕ · · · .


The vector e∅ := 1⊕ 0⊕ · · · is called the vacuum vector. Let {e1, . . . , ed} be

the standard orthonormal basis of Cd. For α ∈ Λ and |α| = n, eα denote the vectoreα1⊗eα2⊗· · ·⊗eαn in the full Fock space F . We recall that Popescu’s construction

[13] of the minimal isometric dilation VC= (V C

1 , . . . , V Cd ) on H⊕ (F ⊗DC) of the

tuple C is

V Cj

(h⊕

∑α∈Λ

eα ⊗ dα

)= Cj h⊕

[e∅ ⊗ (DC)j h+ ej ⊗

∑α∈Λ

eα ⊗ dα

]for h ∈ H and dα ∈ DC where (DC)j h = DC(0, . . . , h, . . . , 0) (h is embedded atthe jth component). So

V Cj h = Cj h⊕ (e∅ ⊗ (DC)j h) for h ∈ H and j = 1, . . . , d. (6.2)

From equations (6.1) and (6.2) it follows that∥∥∥∥∥d∑

j=1

Dj hj

∥∥∥∥2 =

∥∥∥∥ d∑j=1

(DC)j hj

∥∥∥∥2 (6.3)

where hj ∈ H for j = 1, . . . , d. Let ΦC : span{Dj h : h ∈ H, j = 1, . . . , d} → DC bethe unitary given by

ΦC

( d∑j=1

Djhj

)=

d∑j=1

(DC)j hj for hj ∈ H and j = 1, . . . , d.

Similarly for Ei’s and Fi’s obtained from interaction U in equation (4.2) we set

DE := (I−E∗E)12 :⊕d

i=1H →⊕di=1H and DE := Range DE, and define another

unitary operator ΦE : span{Fjh : h ∈ H, j = 1, . . . , d} → DE by

ΦE

( d∑j=1

Fjhj

)=

d∑j=1

(DE)jhj for hj ∈ H and j = 1, . . . , d.

The second equation of (4.1) yields

d∑j=1

DjD∗j y = y for y ∈ Y.

This implies

span{Djh : h ∈ H, j = 1, . . . , d} = Y.Similarly, we can show that span{Fjh : h ∈ H, j = 1, . . . , d} = U . Thus ΦC is aunitary from Y onto DC and ΦE is a unitary from U onto DE . As a consequencewe have for i, j = 1, . . . , d

D∗jDi = (DC)

∗j (DC)i = δijI − C∗

jCi, (6.4)

F ∗j Fi = (DE)

∗j (DE)i = δijI − E∗

jEi. (6.5)


Define unitaries MΦC : �2(Λ,Y)→ F ⊗DC and ΦE : Uz∅ → e∅ ⊗DE by

MΦC

(∑α∈Λ

yαzα

):=∑α∈Λ

eα ⊗ ΦC(yα),

ΦE(uz∅) := e∅ ⊗ ΦEu

which would be useful in comparing transfer functions with characteristic func-tions.

Define D∗,A := (I − AA∗)12 : H◦ → H◦ and D∗,A := Range D∗,A. Because

E is a coisometric lifting of C, using Theorem 2.1 of [5] we conclude that thereexist an isometry γ : D∗,A → DC with γD∗,Ah = B∗h for all h ∈ H◦. Further, forh ∈ H◦

ΦCCh = ΦC

d∑j=1

DjPHE∗j h = ΦC

d∑j=1

DjPH(B∗j h⊕A∗

jh)

= ΦC

d∑j=1

DjB∗j h =

d∑j=1

(DC)jB∗j h

= DCB∗h = B∗h.

The last equality holds because for the coisometric tuple C the operator DC is theprojection onto DC and Range B∗ ⊂ DC . This implies

ΦCCh = γD∗,Ah. (6.6)

The characteristic function MC,E : F ⊗ DE → F ⊗ DC of lifting E of C,which was introduced in [5], and its symbol ΘC,E has the following expansion: For

i = 1, . . . , d and h ∈ H

ΘC,E(DE)ih = e∅ ⊗ [(DC)ih− γD∗,ABih]−∑|α|≥1

eα ⊗ γD∗,A(Aα)∗Bih, (6.7)

and for h ∈ H◦

ΘC,E (DE)ih = − e∅ ⊗ γD∗,AAih

+

d∑j=1

ej ⊗∑α

eα ⊗ γD∗,A(Aα)∗(δjiI −A∗

jAi)h.(6.8)

Theorem 6.1. Let U and U be unitaries associated with a generalized repeatedinteraction model, and the lifting E of C be the corresponding lifting. Then thecharacteristic function MC,E coincides with the transfer function ΘU,U , i.e.,

MΦCΘU,U (z) = ΘC,EΦE .


Proof. If h ∈ H and i = 1, . . . , d, then by equation (4.8)

MΦCΘU,U (z)(Fihz∅) = MΦC

[D z∅ +

∑β∈Λ,j=1,...,d

C(Eβ)∗F ∗

j zβj

](Fihz

∅)

= MΦC

[DFih z∅ +

∑β∈Λ,j=1,...,d

C(Eβ)∗F ∗

j Fih zβj].

(6.9)

Case 1. h ∈ H :

DFih =

d∑j=1

DjPHF ∗j Fih =

d∑j=1

DjPH(δijI − E∗jEi)h

= Dih−( d∑

j=1

DjPHE∗j

)Eih = Dih− CEih

= Dih− C(Cih⊕Bih) = Dih− CBih.

Second and last equalities follow from equations (6.5) and (4.13) respectively. Byequation (6.5) again we obtain∑

β∈Λ,j=1,...,d

C(Eβ)∗F ∗

j Fih zβj

=∑

β∈Λ,j=1,...,d

C(Eβ)∗(δijI − E∗

jEi)h zβj

=∑β∈Λ

C(Eβ)∗h zβi −

∑β∈Λ,j=1,...,d

C(Eβ)∗E∗

jEih zβj

= −∑

β∈Λ,j=1,...,d

C(Eβ)∗E∗

jEih zβj

(because C(Eβ)∗h = C(Cβ)

∗h = 0 by equation (4.13))

= −∑

β∈Λ,j=1,...,d

C(Eβ)∗((C∗

jCi +B∗jBi)h⊕A∗

jBih)zβj

= −∑

β∈Λ,j=1,...,d

C(Aβ)∗A∗

jBih zβj (by equation (4.13))

= −∑|α|≥1

C(Aα)∗Bih zα.

So by equation (6.9) we have for all i = 1, . . . , d and h ∈ H

MΦCΘU,U (z)(Fihz∅) = MΦC

[(Dih− CBih) z

∅ −∑|α|≥1

C(Aα)∗Bih zα

]= e∅ ⊗ ΦC(Dih− CBih)−

∑|α|≥1

eα ⊗ ΦC(C(Aα)∗Bih)


= e∅ ⊗ [(DC)ih− γD∗,ABih]−∑|α|≥1

eα ⊗ γD∗,A(Aα)∗Bih.

By equation (6.7) it follows that

MΦCΘU,U (z)(Fihz∅) = ΘC,E(e∅ ⊗ (DE)ih)

= ΘC,EΦE(Fihz∅).

Case 2. h ∈ H◦ :

DFih =

d∑j=1

DjPHF ∗j Fih =

d∑j=1

DjPH(δijI − E∗jEi)h

= DiPHh−( d∑

j=1

DjPHE∗j

)Eih = −CAih

The second equality follows from equation (6.5). By equations (6.5) and (4.13)again we obtain∑

β∈Λ,j=1,...,d

C(Eβ)∗F ∗

j Fih zβj =∑

β∈Λ,j=1,...,d

C(Eβ)∗(δijI − E∗

jEi)h zβj

=∑

β∈Λ,j=1,...,d

C(Aβ)∗(δijI −A∗

jAi)h zβj.

So by equation (6.9) we have for all i = 1, . . . , d and h ∈ H◦

MΦCΘU,U (z)(Fihz∅)

= MΦC

[−CAih z∅ +

∑β∈Λ,j=1,...,d

C(Aβ)∗(δijI −A∗

jAi)h zβj]

= −e∅ ⊗ ΦC(CAih) +∑

β∈Λ,j=1,...,d

ej ⊗ eβ ⊗ ΦC(C(Aβ)∗(δijI −A∗

jAi)h)

= −e∅ ⊗ γD∗,AAih+∑

β∈Λ,j=1,...,d

ej ⊗ eβ ⊗ γD∗,A(Aβ)∗(δijI −A∗

jAi)h.

By equation (6.8) it follows that

MΦCΘU,U (z)(Fihz∅) = ΘC,E(e∅ ⊗ (DE)ih)

= ΘC,EΦE(Fihz∅).

Hence we conclude that

MΦCΘU,U (z) = ΘC,EΦE . �

The transfer function is a notion affiliated with the input/state/output linearsystem, while the scattering function is a notion affiliated with the scattering the-ory in the sense of Lax–Phillips. For our repeated interaction model Theorem 6.1elucidates that the transfer function is identifiable with the characteristic function


of the associated lifting. This establishes a strong connection between a model forquantum systems and the multivariate operator theory. Connections between themwere also endorsed in other works like [2], [8], [4] and [10], and this indicates thatsuch approaches to quantum systems using multi-analytic operators are promising.

Acknowledgment

The first author received a support from UKIERI to visit Aberystwyth University,UK in July 2011 which was helpful for this project.

References

[1] J.A. Ball, G. Groenewald, T. Malakorn, Conservative structured noncommutativemultidimensional linear systems. The state space method generalizations and appli-cations, 179–223, Oper. Theory Adv. Appl., 161, Birkhauser, Basel (2006).

[2] B.V.R. Bhat, An index theory for quantum dynamical semigroups, Trans. Amer.Math. Soc., 348 (1996) 561–583.

[3] J.A. Ball, V. Vinnikov, Lax–Phillips scattering and conservative linear systems: aCuntz-algebra multidimensional setting, Mem. Amer. Math. Soc., 178 (2005).

[4] S. Dey, R. Gohm, Characteristic functions for ergodic tuples, Integral Equations andOperator Theory, 58 (2007), 43–63.

[5] S. Dey,; R. Gohm, Characteristic functions of liftings, J. Operator Theory, 65 (2011),17–45.

[6] E. Fornasini,; G. Marchesini, Doubly-indexed Dynamical Systems: State Space Modelsand Structural Properties, Math. Systems Theory, 12 (1978), 59–72.

[7] J. Gough, R. Gohm, Yanagisawa: Linear Quantum feedback Networks, Phys. Rev. A,78 (2008).

[8] R. Gohm, Noncommutative stationary processes, Lecture Notes in Mathematics,1839, Springer-Verlag, Berlin (2004).

[9] R. Gohm, Non-commutative Markov chains and multi-analytic operators, J. Math.Anal. Appl., 364 (2010), 275–288.

[10] R. Gohm, Transfer function for pairs of wandering subspaces, Spectral theory, mathe-matical system theory, evolution equations, differential and difference equations, 385–398, Oper. Theory Adv. Appl., 221, Birkhauser/Springer Basel AG, Basel, (2012).

[11] B. Kummerer, H. Maassen, A scattering theory for Markov chains, Infin. Dimens.Anal. Quantum Probab. Relat. Top. 3 (2000), 161–176.

[12] P.D. Lax, R.S. Phillips, Scattering theory, Pure and Applied Mathematics 26 Aca-demic press, New York-London, (1967).

[13] G. Popescu, Isometric dilations for infinite sequences of noncommuting operators,Trans. Amer. Math. Soc., 316 (1989), 523–536.

[14] G. Popescu, Characteristic functions for infinite sequences of noncommuting opera-tors, J. Operator Theory, 22 (1989), 51–71.

[15] G. Popescu, Multi-analytic operators on Fock spaces, Math. Ann., 303 (1995), 31–46.

[16] G. Popescu, Poisson transforms on some C∗-algebras generated by isometries, J.Funct. Anal., 161 (1999), 27–61.


[17] G. Popescu, Free holomorphic functions on the unit ball of B(H)n, J. Funct. Anal.,241 (2006), 268–333.

[18] M. Reed, B. Simon, Methods of modern mathematical physics. III. Scattering theory.Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, (1979).

[19] B. Sz.-Nagy, C. Foias, Harmonic analysis of operators on Hilbert space, North–Holland Publ., Amsterdam-Budapest (1970).

[20] M. Yanagisawa, H. Kimura, Transfer function approach to quantum control, part I:Dynamics of Quantum feedback systems, IEEE Transactions on Automatic control,48 (2003), no. 12, 2107–2120.

Santanu Dey and Kalpesh J. HariaDepartment of MathematicsIndian Institute of Technology BombayPowai, Mumbai-400076, Indiae-mail: [email protected]

[email protected]


Some Remarks on the Spectral ProblemUnderlying the Camassa–Holm Hierarchy

Fritz Gesztesy and Rudi Weikard

Dedicated with great pleasure to Ludwig Streit on the occasion of his 75th birthday.

Abstract. We study particular cases of left-definite eigenvalue problems

Aψ = λBψ, with A ≥ εI for some ε > 0 and B self-adjoint,

but B not necessarily positive or negative definite, applicable, in particular,to the eigenvalue problem underlying the Camassa–Holm hierarchy. In fact,we will treat a more general version where A represents a positive definiteSchrodinger or Sturm–Liouville operator T in L2(R; dx) associated with adifferential expression of the form τ = −(d/dx)p(x)(d/dx)+ q(x), x ∈ R, andB represents an operator of multiplication by r(x) in L2(R; dx), which, ingeneral, is not a weight, that is, it is not nonnegative (or nonpositive) a.e. onR. In fact, our methods naturally permit us to treat certain classes of distri-butions (resp., measures) for the coefficients q and r and hence considerablyextend the scope of this (generalized) eigenvalue problem, without having tochange the underlying Hilbert space L2(R; dx). Our approach relies on rewrit-

ing the eigenvalue problem Aψ = λBψ in the form A−1/2BA−1/2χ = λ−1χ,χ = A1/2ψ, and a careful study of (appropriate realizations of) the operator

A−1/2BA−1/2 in L2(R; dx).In the course of our treatment, we review and employ various necessary

and sufficient conditions for q to be relatively bounded (resp., compact) andrelatively form bounded (resp., form compact) with respect to T0 = −d2/dx2

defined on H2(R). In addition, we employ a supersymmetric formalism whichpermits us to factor the second-order operator T into a product of two first-order operators familiar from (and inspired by) Miura’s transformation linkingthe KdV and mKdV hierarchy of nonlinear evolution equations. We also treatthe case of periodic coefficients q and r, where q may be a distribution and rgenerates a measure and hence no smoothness is assumed for q and r.

Mathematics Subject Classification (2010). Primary 34B24, 34C25, 34K13,34L05, 34L40, 35Q58, 47A10, 47A75; Secondary 34B20, 34C10, 34L25, 37K10,47A63, 47E05.

Keywords. Camassa–Holm hierarchy, left-definite spectral problems, distribu-tional coefficients, Floquet theory, supersymmetric formalism.


138 F. Gesztesy and R. Weikard

1. Introduction

In this paper we are interested in a particular realization of a generalized left-definite spectral problem originally derived from the Camassa–Holm hierarchy ofintegrable nonlinear evolution equations.

Before specializing to the one-dimensional context at hand, we briefly addressthe notion of generalized spectral problems associated with operator pencils of thetype A− zB, z ∈ C, for appropriate densely defined and closed linear operators Aand B in a complex, separable Hilbert space H. As discussed in [90, Sect. VII.6],there are several (and in general, inequivalent) ways to reformulate such generalizedspectral problems. For instance, if B is boundedly invertible, one may consider thespectral problem for the operators B−1A or AB−1, and in some cases (e.g., ifB ≥ εIH for some ε > 0, a case also called a right-definite spectral problem) alsothat of B−1/2AB−1/2. Similarly, if A is boundedly invertible, the spectral problemfor the linear pencil A− zB can be reformulated in terms of the spectral problemsfor A−1B or BA−1, and sometimes (e.g., if A ≥ εIH for some ε > 0, a case alsocalled a left-definite spectral problem)) in terms of that of A−1/2BA−1/2.

There exists an enormous body of literature for these kinds of generalizedspectral problems and without any possibility of achieving completeness, we refer,for instance, to [2], [12], [55], [69], [73], [74], [75], [76], [100], [118], [133], and theextensive literature cited therein in the context of general boundary value prob-lems. In the context of indefinite Sturm–Liouville-type boundary value problemswe mention, for instance, [6], [8], [11], [13], [14], [15], [16], [18], [19], [20], [21], [23],[30], [31], [32], [37], [47], [52], [56], [85], [86], [88], [91], [92], [93], [101], [102], [119],[134], [137, Chs. 5, 11, 12], and again no attempt at a comprehensive account ofthe existing literature is possible due to the enormous volume of the latter.

The prime motivation behind our attempt to study certain left-definite eigen-value problems is due to their natural occurrence in connection with the Camassa–Holm (CH) hierarchy. For a detailed treatment and an extensive list of referenceswe refer to [60], [61, Ch. 5] and [62]. The first few equations of the CH hierarchy(cf., e.g., [61, Sect. 5.2] for a recursive approach to the CH hierarchy) explicitlyread (with u = u(x, t), (x, t) ∈ R2)

CH0(u) = 4ut0 − uxxt0 + uxxx − 4ux = 0,

CH1(u) = 4ut1 − uxxt1 − 2uuxxx − 4uxuxx + 24uux + c1(uxxx − 4ux) = 0,

CH2(u) = 4ut2 − uxxt2 + 2u2uxxx − 8uuxuxx − 40u2ux (1.1)

+ 2(uxxx − 4ux)G(u2x + 8u2

)− 8(4u− uxx)G(uxuxx + 8uux

)+ c1(−2uuxxx − 4uxuxx + 24uux) + c2(uxxx − 4ux) = 0, etc.,

for appropriate constants c�, � ∈ N. Here G is given by

G :{L∞(R; dx) → L∞(R; dx),

v �→ (Gv)(x) = 14

∫Rdy e−2|x−y|v(y), x ∈ R,

(1.2)

The Spectral Problem for the Camassa–Holm Hierarchy 139

and one observes that G is the resolvent of minus the one-dimensional Laplacianat energy parameter equal to −4, that is,

G =

(− d2

dx2+ 4

)−1

. (1.3)

The spectral problem underlying the CH hierarchy can then be cast in theform (with “prime” denoting d/dx),

Φ′(z, x) = U(z, x)Φ(z, x), (z, x) ∈ C× R, (1.4)

where

Φ(z, x) =

(φ1(z, x)φ2(z, x)

), U(z, x) =

( −1 1z[uxx(x)− 4u(x)] 1

),

(z, x) ∈ C× R.(1.5)

Eliminating φ2 in (1.4) then results in the scalar (weighted) spectral problem

−φ′′(z, x) + φ(z, x) = z[uxx(x) − 4u(x)]φ(z, x), (z, x) ∈ (C\{0})× R. (1.6)

In the specific context of the left-definite Camassa–Holm spectral problemwe refer to [9], [10], [17], [19], [30], [31], [32], [33], [34] [35], [37], [51], [62], [87], [96],[98], [108], [109], [110], [111], and the literature cited therein.

Rather than directly studying (1.6) in this note, we will study some of itsgeneralizations and hence focus on several spectral problems originating with thegeneral Sturm–Liouville equation

−(p(x)ψ′(z, x))′ + q(x)ψ(z, x) = zr(x)ψ(z, x), (z, x) ∈ (C\{0})× R, (1.7)

under various hypotheses on the coefficients p, q, r to be described in more detaillater on and with emphasis on the fact that r may change its sign. At this pointwe assume the following basic requirements on p, q, r (but we emphasize that lateron we will consider vastly more general situations where q and r are permitted tolie in certain classes of distributions):

Hypothesis 1.1.

(i) Suppose that p > 0 a.e. on R, p−1 ∈ L1loc(R; dx), and that q, r ∈ L1

loc(R; dx)are real-valued a.e. on R. In addition, assume that r �= 0 on a set of positiveLebesgue measure and that

± limx→±∞

∫ x

dx′ p(x′)−1/2 =∞. (1.8)

(ii) Introducing the differential expression

τ = − d

dxp(x)

d

dx+ q(x), x ∈ R, (1.9)


and the associated minimal operator Tmin in L2(R; dx) by

Tminf = τf,

f ∈ dom(Tmin) ={g ∈ L2(R; dx)

∣∣ g, (pg′) ∈ ACloc(R); supp (g) compact; (1.10)

τg ∈ L2(R; dx)},

we assume that for some ε > 0,

Tmin ≥ εIL2(R;dx). (1.11)

We note that our assumptions (1.8) and (1.11) imply that τ is in the limitpoint case at +∞ and −∞ (cf., e.g., [29], [59], [72], [122]). This permits one tointroduce the maximally defined self-adjoint operator T in L2(R; dx) associatedwith τ by

Tf = τf,

f ∈ dom(T ) ={g ∈ L2(R; dx)

∣∣ g, (pg′) ∈ ACloc(R); τg ∈ L2(R; dx)} (1.12)

(where ACloc(R) denotes the set of locally absolutely continuous functions on R).In particular, T is the closure of Tmin,

T = Tmin, (1.13)

and hence also

T ≥ εIL2(R;dx). (1.14)

Remark 1.2. By a result proven in Yafaev [138] (see also [68, pp. 110–115]), ifp = 1 and q ≥ 0 a.e. on R, (1.14) holds for some ε > 0 if and only if there existsc0 > 0 such that for some a > 0,∫ x+a

x

dx′ q(x′) ≥ c0, x ∈ R. (1.15)

If p is bounded below by some ε0 > 0 (which we may choose smaller than one),one has∫

R

dx[p(x)|u′(x)|2 + q(x)|u(x)|2] ≥ ε0

∫R

dx[|u′(x)|2 + ε−1

0 q(x)|u(x)|2]. (1.16)

Hence (1.15) is then still sufficient for (1.14) to hold.

We also note that Theorem 3 in [138] shows that q ≥ 0 is not necessary for

(1.14) to hold. In fact, if q2 ≥ 0, but∫ a+1

a dx q2(x) ≤ c for all a ∈ R, one finds

−(c+ 4c2)

∫R

dx |u(x)|2 ≤∫R

dx[|u′(x)|2 − q2(x)|u(x)|2

]. (1.17)

Hence if p = 1 and q = ε + c + 4c2 − q2 one obtains (1.14) even though q mayassume negative values.


Given these preparations, we now associate the weighted eigenvalue equation(1.7) with a standard self-adjoint spectral problem of the form

T−1/2rT−1/2 χ = ζχ,

χ(ζ, x) =(T 1/2ψ(z, ·))(x), ζ = 1/z ∈ C\{0}, x ∈ R,

(1.18)

for the integral operator T−1/2rT−1/2 in L2(R; dx), subject to certain additionalconditions on p, q, r. We use the particular notation T−1/2rT−1/2 to underscorethe particular care that needs to be taken with interpreting this expression asa bounded, self-adjoint operator in L2(R; dx) (pertinent details can be found in(2.37) and, especially, in (3.98)). It is important to note that in contrast to anumber of papers that find it necessary to use different Hilbert spaces in connectionwith a left-definite spectral problem (in some cases the weight r is replaced by |r|,in other situations the new Hilbert space is coefficient-dependent), our treatmentworks with one and the same underlying Hilbert space L2(R; dx).

We emphasize that rewriting (1.7) in the form (1.18) is not new. In particular,in the context of the CH spectral problem (1.6) this has briefly been used, forinstance, in [36] (in the periodic case), in [33] (in the context of the CH scatteringproblem), in [62] (in connection with real-valued algebro-geometric CH solutions),and in [109] (in connection with CH flows and Fredholm determinants). However,apart from the approach discussed in [18], [19], [20], most investigations associatedwith the CH spectral problem (1.6) appear to focus primarily on certain Liouville–Green transformations which transform (1.6) into a Schrodinger equation for someeffective potential coefficient (see, e.g., [30], [31], [32]). This requires additionalassumptions on the coefficients which in general can be avoided in the context of(1.18). Indeed, the change of variables

R � x �→ t =

∫ x

0

dx′ p(x′)−1, (1.19)

turns the equation −(pu′)′ + qu = zru on R into

− v′′ +Qv = zRv on

(−∫ 0

−∞dx′ p(x′)−1,

∫ +∞

0

dx′ p(x′)−1

),

v(t) = u(x(t)), Q(t) = p(x(t))q(x(t)), R(t) = p(x(t))r(x(t)).

(1.20)

However, assuming for instance, ± ∫±∞dx p(x) = ∞, the change of variables is

only unitary between the spaces L2(R; dx) and L2(R; dx/p(x)) and hence necessi-tates a change in the underlying measure.

The primary aim of this note is to sketch a few instances in which the integraloperator approach in (1.18) naturally, and in a straightforward manner, leadsto much more general spectral results and hence is preferable to the Liouville–Green approach. In particular, we are interested in generalized situations, wherethe coefficients q and r lie in certain classes of distributions. To the best of ourknowledge, this level of generality is new in this context.


In Section 2 we analyze basic spectral theory of T−1/2rT−1/2 in L2(R; dx)assuming Hypothesis 1.1 and appropriate additional assumptions on p, q, r. Themore general case where q and r lie in certain classes of distributions is treatedin detail in Section 3. There we heavily rely on supersymmetric methods andMiura transformations. This approach exploits the intimate relationship betweenspectral theory for Schrodinger operators factorized into first-order differentialoperators and that of an associated Dirac-type operator. Section 4 is devoted toapplications in the special case where q and r are periodic (for simplicity we takep = 1). We permit q to lie in a class of distributions and r to be a signed measure,which underscores the novelty of our approach. Three appendices provide amplebackground results: Appendix A is devoted to basic facts on relative boundednessand compactness of operators and forms; the supersymmetric formalism relatingSchrodinger and Dirac-type operators is presented in Appendix B, and details onsesquilinear forms and their associated operators are provided in Appendix C.

Finally, we briefly summarize some of the notation used in this paper: Let Hbe a separable complex Hilbert space, (·, ·)H the scalar product in H (linear in thesecond factor), and IH the identity operator in H. Next, let T be a linear operatormapping (a subspace of) a Banach space into another, with dom(T ), ran(T ), andker(T ) denoting the domain, range, and kernel (i.e., null space) of T . The closureof a closable operator S is denoted by S.

The spectrum, essential spectrum, point spectrum, discrete spectrum, abso-lutely continuous spectrum, and resolvent set of a closed linear operator in H willbe denoted by σ(·), σess(·), σp(·), σd(·), σac(·), and ρ(·), respectively.

The Banach spaces of bounded and compact linear operators inH are denotedby B(H) and B∞(H), respectively. Similarly, the Schatten–von Neumann (trace)ideals will subsequently be denoted by Bs(H), s ∈ (0,∞). The analogous notationB(X1,X2), B∞(X1,X2), etc., will be used for bounded and compact operators be-tween two Banach spaces X1 and X2. Moreover, X1 ↪→ X2 denotes the continuousembedding of the Banach space X1 into the Banach space X2. Throughout thismanuscript we use the convention that if X denotes a Banach space, X∗ denotesthe adjoint space of continuous conjugate linear functionals on X , also known asthe conjugate dual of X .

In the bulk of this note, H will typically represent the space L2(R; dx). Op-erators of multiplication by a function V ∈ L1

loc(R; dx) in L2(R; dx) will by aslight abuse of notation again be denoted by V (rather than the frequently used,but more cumbersome, notation MV ) and unless otherwise stated, will always as-sumed to be maximally defined in L2(R; dx) (i.e., dom(V ) =

{f ∈ L2(R; dx)

∣∣ V f ∈L2(R; dx)

}). Moreover, in subsequent sections, the identity operator IL2(R;dx) in

L2(R; dx) will simply be denoted by I for brevity.The symbol D(R) denotes the space of test functions C∞

0 (R) with its usual(inductive limit) topology. The corresponding space of continuous linear function-als on D(R) is denoted by D′(R) (i.e., D′(R) = C∞

0 (R)′).


2. General spectral theory of T−1/2rT−1/2

In this section we derive some general spectral properties of T−1/2rT−1/2 whichreproduce some known results that were originally derived in the CH context of(1.6), but now we prove them under considerably more general conditions on thecoefficients p, q, r, and generally, with great ease. In this section p, q, r will satisfyHypothesis 1.1 and appropriate additional assumptions. (The case where q, r liein certain classes of distributions will be treated in Section 3.)

For a quick summary of the notions of relatively bounded and compact op-erators and forms frequently used in this section, we refer to Appendix A.

Before analyzing the operator T−1/2rT−1/2 we recall three useful results:

We denote by T0 (minus) the usual Laplacian in L2(R; dx) defined by

T0f = −f ′′, (2.1)

f ∈ dom(T0) ={g ∈ L2(R; dx)

∣∣ g, g′ ∈ ACloc(R); g′′ ∈ L2(R; dx)}= H2(R),

where Hm(R), m ∈ N, abbreviate the usual Sobolev spaces of functions whosedistributional derivatives up to order m lie in L2(R; dx).

In the following it is useful to introduce the spaces of locally uniformly Lp-integrable functions on R,

Lploc unif(R; dx) =

{f ∈ Lp

loc(R; dx)

∣∣∣∣ supa∈R

(∫ a+1

a

dx |f(x)|p)

<∞}, (2.2)

p ∈ [1,∞). Equivalently, let

η ∈ C∞0 (R), 0 ≤ η ≤ 1, η|B(0;1) = 1, (2.3)

with B(x; r) ⊂ R the open ball centered at x0 ∈ R and radius r > 0, then

Lploc unif(R; dx) =

{f ∈ Lp

loc(R; dx)∣∣∣ supa∈R

‖η(· − a)f‖Lp(R;dx) <∞}, p ∈ [1,∞).

(2.4)We refer to Appendix A for basic notions in connection with relatively boundedlinear operators.

Theorem 2.1 ([125, Theorem 2.7.1], [127, p. 35]). Let V,w ∈ L2loc(R; dx). Then the

following conditions (i)–(iv) are equivalent:

(i) dom(w) ⊇ dom(T

1/20

)= H1(R). (2.5)

(ii) w ∈ L2loc unif(R; dx). (2.6)

(iii) For some C > 0,

‖wf‖2L2(R;dx) ≤ C[∥∥T 1/2

0 f‖2L2(R;dx) + ‖f‖2L2(R;dx)

],

f ∈ dom(T

1/20

)= H1(R).

(2.7)


(iv) For all ε > 0, there exists Cε > 0 such that:

‖wf‖2L2(R;dx) ≤ ε∥∥T 1/2

0 f∥∥2L2(R;dx)

+ Cε‖f‖2L2(R;dx),

f ∈ dom(T

1/20

)= H1(R).

(2.8)

Moreover, also the following conditions (v)–(viii) are equivalent:

(v) dom(V ) ⊇ dom(T0) = H2(R). (2.9)

(vi) V ∈ L2loc unif(R; dx). (2.10)

(vii) For some C > 0,

‖V f‖2L2(R;dx) ≤ C[‖T0f‖2L2(R;dx) + ‖f‖2L2(R;dx)

],

f ∈ dom(T0) = H2(R).(2.11)

(viii)For all ε > 0, there exists Cε > 0 such that:

‖V f‖2L2(R;dx) ≤ ε‖T0f‖2L2(R;dx) + Cε‖f‖2L2(R;dx),

f ∈ dom(T0) = H2(R).(2.12)

In fact, it is possible to replace T1/20 by any polynomial Pm

(T

1/20

)of degree m ∈ N

in connection with items (i)–(iv).

We emphasize the remarkable fact that according to items (iii), (iv) and (vii),(viii), relative form and operator boundedness is actually equivalent to infinitesi-mal form and operator boundedness in Theorem 2.1.

For completeness, we briefly sketch some of the principal ideas underlyingitems (i)–(iv) in Theorem 2.1, particularly, focusing on item (ii): That item (i)implies item (iii) is of course a consequence of the closed graph theorem. Exploitingcontinuity of f ∈ H1(R), yields for arbitrary ε > 0,

|f(x)|2 − |f(x′)|2 =

∫ x

x′dy[f(yf ′(y) + f ′(y)f(y)

](2.13)

≤ ε

∫Idy |f ′(y|2 + ε−1

∫Idy |f(y)|2, f ∈ H1(R), x, x′ ∈ I,

with I ⊂ R an arbitrary interval of length one. The use of the mean value theoremfor integrals then permits one to choose x′ ∈ I such that

|f(x′)|2 =

∫Idy |f(y)|2 (2.14)

implying

|f(x)|2 ≤ ε

∫Idx′ |f ′(x′)|2 +(1+ ε−1)

∫Idx′ |f(x′)|2, f ∈ H1(R), x ∈ I, (2.15)

and hence after summing over all intervals I of length one, and using boundednessof f ∈ H1(R),

|f(x)|2 ≤ ‖f‖2L∞(R;dx) ≤ ε‖f ′‖2L2(R;dx) + (1 + ε−1)‖f‖2L2(R;dx), (2.16)


f ∈ H1(R), x ∈ R. Multiplying (2.15) by |w(x)|2 and integrating with respect tox over I yields∫

Idx |w(x)|2|f(x)|2 ≤ εC0

∫Idx′ |f ′(x′)|2 + (1 + ε−1)C0

∫Idx′ |f(x′)|2, (2.17)

and summing again over all intervals I of length one implies

‖wf‖2L2(R;dx) ≤ εC0‖f ′‖2L2(R) +(1 + ε−1

)C0‖f‖2L2(R;dx), f ∈ H1(R), (2.18)

where

C0 := supa∈R

(∫ a+1

a

dx |w(x)|2)

<∞, (2.19)

illustrating the sufficiency part of condition w ∈ L2loc unif(R; dx) in item (ii) for

item (iv) to hold.

Next, consider ψ(x) = e1−x2

, ψa(x) = ψ(x− a), x, a ∈ R. Then∫ a+1

a

dx |w(x)|2 ≤∫R

dx[|w(x)||ψa(x)|

]2(2.20)

≤ C[∥∥T 1/2

0 ψa

∥∥2L2(R;dx)

+ ‖ψa‖2L2(R;dx)

](2.21)

≤ C[‖ψ′‖2L2(R;dx) + ‖ψ‖2L2(R;dx)

]= C, (2.22)

with C independent of a, illustrates necessity of the condition w ∈ L2loc unif(R; dx)

in item (ii) for item (iii) to hold.Given ε > 0, there exists η(ε) > 0, such that the obvious inequality

‖f ′‖2L2(R;dx) ≤ ε∥∥Tm/2

0

∥∥2L2(R;dx)

+ η(ε)‖f‖2L2(R;dx),

f ∈ dom(T

m/20

), m ∈ N, m ≥ 2,

(2.23)

holds. It suffices applying the Fourier transform and using |p| ≤ ε|p|m + η(ε),

m ∈ N, m ≥ 2 to extend this to polynomials in T1/20 . This illustrates the sufficiency

of the condition V ∈ L2loc unif(R; dx) in item (vi) for item (viii) to hold.

We note that items (i)–(iv) in Theorem 2.1 are mentioned in [127, p. 35]without proof, but the crucial hint that f ∈ H1(R) implies that f ∈ ACloc(R) ∩L∞(R; dx), is made there. We also remark that Theorem 2.7.1 in [125] is primar-ily concerned with items (v)–(viii) in Theorem 2.1. Nevertheless, its method ofproof also yields the results (2.1)–(2.8), in particular, it contains the fundamentalinequality (2.18).

Next, we also recall the following result (we refer to Appendix A for detailson the notion of relative compactness for linear operators):

Theorem 2.2 ([125, Theorem 3.7.5], [126, Sects. 15.7, 15.9]). Let w ∈ L2loc(R; dx).

Then the following conditions (i)–(iii) are equivalent:

(i) w is T1/20 -compact. (2.24)

(ii) w is T0-compact. (2.25)


(iii) lim|a|→∞

(∫ a+1

a

dx |w(x)|2)

= 0. (2.26)

In fact, it is possible to replace T1/20 by any polynomial Pm

(T

1/20

)of degree m ∈ N

in item (i).

We note that w ∈ L2loc(R; dx) together with condition (2.26) imply that

w ∈ L2loc unif(R; dx) (cf. [126, p. 378]).It is interesting to observe that the if and only if characterizations (2.1)–(2.8)

for relative (resp., infinitesimal) form boundedness mentioned by Simon [127, p.35], and those in (2.24)–(2.26) for relative (form) compactness by Schechter in thefirst edition of [126, Sects. 15.7, 15.9], were both independently published in 1971.

In the context of Theorems 2.1 and 2.2 we also refer to [5] for interestingresults on necessary and sufficient conditions on relative boundedness and rela-tive compactness for perturbations of Sturm–Liouville operators by lower-orderdifferential expressions on a half-line (in addition, see [26], [77]).

We will also use the following result on trace ideals. To fix our notation,we denote by f(X) the operator of multiplication by the measurable functionf on R, and similarly, we denote by g(P ) the operator defined by the spectraltheorem for a measurable function g (equivalently, the operator of multiplicationby the measurable function g in Fourier space L2(R; dp)), where P denotes theself-adjoint (momentum) operator defined by

Pf = −if ′, dom(P ) = H1(R). (2.27)

Theorem 2.3 ([128, Theorem 4.1]). Let f ∈ Ls(R; dx), g ∈ Ls(R; dx)), s ∈ [2,∞).Then

f(X)g(P ) ∈ Bs

(L2(R; dx)

)(2.28)

and

‖f(X)g(P )‖Bs(L2(R;dx)) ≤ (2π)−1/s‖f‖Ls(R;dx)‖g‖Ls(R;dx). (2.29)

If s = 2, f and g are both nonzero on a set of positive Lebesgue measure, andf(X)g(P ) ∈ B2

(L2(R; dx)

), then

f, g ∈ L2(R; dx). (2.30)

Given these preparations, we introduce the following convenient assumption:

Hypothesis 2.4. In addition to the assumptions in Hypothesis 1.1 suppose that theform domain of T is given by

dom(T 1/2

)= dom

(T

1/20

)= H1(R). (2.31)

Assuming for some positive constants c and C that

0 < c ≤ p ≤ C a.e. on R, (2.32)

an application of Theorem 2.1 (i), (ii) shows that (2.31) holds if q ∈ L1loc(R; dx)

satisfiesq ∈ L1

loc unif(R; dx). (2.33)


Indeed, since by the comment following Hypothesis 1.1, T is essentially self-adjointon dom(Tmin), T ≥ εI for some ε > 0, and dom

(T 1/2

)= H1(R), the sesquilinear

form QT associated with T is of the form

QT (f, g) =

∫R

dx p(x)f ′(x)g′(x) +∫R

dx q(x)f(x)g(x),

f, g ∈ dom(QT ) = dom(T 1/2

)= dom

(T

1/20

)= H1(R).

(2.34)

Hence, by Theorem 2.1 (i), (ii), this is equivalent to (2.33) keeping in mind that qis such that (1.11) holds.

Our first result then reads as follows:

Theorem 2.5. Assume Hypothesis 2.4.

(i) Then

|r|1/2T−1/2 ∈ B(L2(R; dx))

(2.35)

if and only if

r ∈ L1loc unif(R; dx). (2.36)

In particular, if (2.36) holds, introducing

T−1/2rT−1/2 =[|r|1/2T−1/2

]∗sgn(r)

[|r|1/2T−1/2], (2.37)

one concludes that

T−1/2rT−1/2 ∈ B(L2(R; dx)). (2.38)

(ii) Let r0 ∈ R. Then

|r − r0|1/2T−1/2 ∈ B∞(L2(R; dx)

)(2.39)

if and only if

lim|a|→∞

(∫ a+1

a

dx |r(x) − r0|)

= 0. (2.40)

In particular, if (2.40) holds, introducing

T−1/2(r − r0)T−1/2

=[|r − r0|1/2T−1/2

]∗sgn(r − r0)

[|r − r0|1/2T−1/2],

(2.41)

one concludes that

T−1/2(r − r0)T−1/2 ∈ B∞

(L2(R; dx)

). (2.42)

(iii) Let r0 ∈ R. Then

|r − r0|1/2T−1/2 ∈ B2

(L2(R; dx)

)(2.43)

if and only if ∫R

dx |r(x) − r0| <∞. (2.44)


In particular, if (2.44) holds, then

T−1/2(r − r0)T−1/2 ∈ B1

(L2(R; dx)

). (2.45)

Proof. (i) By hypothesis (2.31) and the closed graph theorem one concludes that[(T0 + I)1/2T−1/2

] ∈ B(L2(R; dx)). (2.46)(

and analogously,[(T0 + I)1/2T−1/2

]−1= T 1/2(T0 + I)−1/2 ∈ B(L2(R; dx)

)). The

equivalence of (2.35) and (2.36) then follows from (2.1) and (2.6) and the fact that

|r|1/2T−1/2 =[|r|1/2(T0 + I)−1/2

][(T0 + I)1/2T−1/2

]. (2.47)

The inclusion (2.38) immediately follows from (2.35) and (2.37).

(ii) The equivalence of (2.39) and (2.40) follows from (2.24) and (2.26). The inclu-sion (2.42) then follows from (2.41), (2.46), and (2.47) with r replaced by r − r0.

(iii) The equivalence of (2.43) and (2.44) follows from (2.29) and (2.30), employingagain (2.46) and the fact that (|p|2 + 1)−1/2 ∈ L2(R; dp). The relation (2.45)once more follows from (2.41), (2.46), and (2.47) with r replaced by r − r0, andthe fact that S ∈ B1(H) if and only if |S| ∈ B1(H) and hence if and only if|S|1/2 ∈ B2(H). �

In the following we use the obvious notation for subsets of M ⊂ R andconstants c ∈ R:

cM = {c x ∈ R |x ∈M}. (2.48)

Corollary 2.6. Assume Hypothesis 2.4.

(i) If (2.40) holds for some r0 ∈ R, then

σess

(T−1/2rT−1/2

)=

{r0σess

(T−1

), r0 ∈ R\{0},

{0}, r0 = 0.(2.49)

(ii) If (2.44) holds for some r0 ∈ R, then

σac

(T−1/2rT−1/2

)=

{r0σac

(T−1

), r0 ∈ R\{0},

∅, r0 = 0.(2.50)

Proof. For r0 ∈ R\{0} it suffices to use the decomposition

T−1/2rT−1/2 = T−1/2[r0 + (r − r0)]T−1/2 = r0T

−1+T−1/2(r − r0)T−1/2 (2.51)

and employ (2.42) together with Weyl’s theorem (cf., e.g., [53, Sect. IX.2], [121,Sect. XIII.4], [135, Sect. 9.2]) to obtain (2.49), and combine (2.45) and the Kato–Rosenblum theorem (cf., e.g., [90, Sect. X.3], [120, Sect. XI.3], [135, Sect. 11.1]) toobtain (2.50).

In the case r0 = 0 relation (2.49) holds since T−1/2rT−1/2 ∈ B∞(L2(R; dx)

)and L2(R; dx) is infinite dimensional. By the same argument one obtains (2.50)for r0 = 0. �


In connection with (2.49) we also recall that by the spectral mapping theoremfor self-adjoint operators A in H,

0 �= z ∈ σess

((A− z0IH)−1

), z0 ∈ ρ(A), if and only if z−1 + z0 ∈ σess(A) (2.52)

(cf., e.g., [121, Sect. XIII.4]). Finally, we mention that there exists a large bodyof results on determining essential and absolutely continuous spectra for Sturm–Liouville-type operators T associated with the differential expressions of the typeτ = − d

dxp(x)ddx + q(x), x ∈ R. We refer, for instance, to [45, XIII.7], [115, Chs. 2,

4], [116, Sect. 24], and the literature cited therein.

Remark 2.7. While it is well known that for T densely defined and closed in H,

T is bounded (resp., compact, Hilbert–Schmidt)

if and only if T ∗T is bounded (resp., compact, trace class),(2.53)

the following example, communicated to us by G. Teschl [130], shows that if S isbounded and self-adjoint in H with spectrum σ(S) = {−1, 1} then

T bounded is not equivalent to T ∗ST bounded (2.54)

assuming T ∗ST to be densely defined in H (and hence closable in H, since T ∗STis symmetric). Indeed, considering

T =

(A 00 A−1

), A = A∗, A ≥ IH, S =

(0 IHIH 0

), (2.55)

thenT ∗ST = S, (2.56)

and hence T ∗ST is bounded, but T is unbounded if A is chosen to be unbounded.Thus one cannot assert on abstract grounds that

T−1/2rT−1/2 =[|r|1/2T−1/2

]∗sgn(r)|r|1/2T−1/2 (2.57)

is bounded if and only if |r|1/2T−1/2 is. In fact, this is utterly wrong as we shalldiscuss in the following Section 3. Indeed, focusing directly on |r|1/2T−1/2 insteadof T−1/2rT−1/2 ignores crucial oscillations of r that permit one to considerably en-large the class of admissible weights r. In particular, thus far we relied on estimatesof the type∥∥|q|1/2f∥∥2

L2(R;dx)≤ C

[∥∥T 1/20 f

∥∥2L2(R;dx)

+ ‖f‖2L2(R;dx)

], f ∈ H1(R), (2.58)

equivalently,∫R

dx |q(x)||f(x)|2 ≤ ∥∥[T0 + I]1/2

f∥∥2L2(R;dx)

, f ∈ H1(R). (2.59)

Consequently, we ignored all oscillations of q (and hence, r). Instead, we shouldfocus on estimating∣∣∣∣ ∫

R

dx q(x)|f(x)|2∣∣∣∣ ≤ ∥∥[T0 + I

]1/2f∥∥2L2(R;dx)

, f ∈ H1(R), (2.60)

and this will be the focus of the next Section 3.


3. Distributional coefficients

In this section we extend our previous considerations where q, r ∈ L1loc unif(R; dx),

to the case where q and r are permitted to lie in a certain class of distributions.The extension to distributional coefficients will be facilitated by employing su-persymmetric methods and an underlying Miura transformation. This approachpermits one to relate spectral theory for Schrodinger operators factorized into aproduct of first-order differential operators with that of an associated Dirac-typeoperator.

We start with some background (cf., e.g., [70, Chs. 4–6], [103, Chs. 2, 3, 11],[112, Ch. 3]) and fix our notation in connection with Sobolev spaces. Introducing

L2s(R) = L2

(R;(1 + |p|2)sdp), s ∈ R, (3.1)

and identifying,

L20(R) = L2(R; dp) =

(L2(R; dp)

)∗=(L20(R)

)∗, (3.2)

one gets the chain of Hilbert spaces with respect to the pivot space L20(R) =

L2(R; dp),

L2s(R) ⊂ L2(R; dp) ⊂ L2

−s(R) =(L2s(R)

)∗, s > 0. (3.3)

Next, we introduce the maximally defined operator G0 of multiplication by the

function(1 + | · |2)1/2 in L2(R; dp),

(G0f)(p) =(1 + |p|2)1/2f(p),

f ∈ dom(G0) ={g ∈ L2(R; dp)

∣∣∣ (1 + | · |2)1/2g ∈ L2(R; dp)}.

(3.4)

The operator G0 extends to an operator defined on the entire scale L2s(R), s ∈ R,

denoted by G0, such that

G0 : L2s(R)→ L2

s−1(R),(G0

)−1: L2

s(R)→ L2s+1(R), bijectively, s ∈ R. (3.5)

In particular, while

I : L2(R; dp)→ (L2(R; dp)

)∗= L2(R; dp) (3.6)

represents the standard identification operator between L20(R) = L2(R; dp) and its

adjoint space,(L2(R; dp)

)∗=(L20(R)

)∗, via Riesz’s lemma, we emphasize that we

will not identify(L2s(R)

)∗with L2

s(R) when s > 0. In fact, it is the operator G20

that provides a unitary map

G20 : L2

s(R)→ L2s−2(R), s ∈ R. (3.7)

In particular,

G20 : L2

1(R)→ L2−1(R) =

(L21(R)

)∗is a unitary map, (3.8)

and we refer to (C.40) for an abstract analog of this fact.


Denoting the Fourier transform on L2(R; dp) by F , and then extended to the

entire scale L2s(R), s ∈ R, more generally, to S ′(R) by F (with F : S ′(R)→ S ′(R)

a homeomorphism), one obtains the scale of Sobolev spaces via

Hs(R) = FL2s(R), s ∈ R, L2(R; dx) = FL2(R; dp), (3.9)

and hence,

FG0F−1 = (T0 + I)1/2 : H1(R)→ L2(R; dx), bijectively, (3.10)

FG0F−1 =(T0 + I

)1/2: Hs(R)→ Hs−1(R), bijectively, s ∈ R, (3.11)

F(G0

)−1F−1 =(T0 + I

)−1/2: Hs(R)→ Hs+1(R), bijectively, s ∈ R. (3.12)

We recall that T0 was defined as

T0 = −d2/dx2, dom(T0) = H2(R), (3.13)

in (2.1), but now the extension T0 of T0 is defined on the entire Sobolev scaleaccording to (3.11),(

T0 + I): Hs(R)→ Hs−2(R) is a unitary map, s ∈ R, (3.14)

and the special case s = 1 again corresponds to (C.26), (C.40),(T0 + I

): H1(R)→ H−1(R) =

(H1(R)

)∗is a unitary map. (3.15)

In addition, we note that

H0(R) = L2(R; dx),(Hs(R)

)∗= H−s(R), s ∈ R, (3.16)

S(R) ⊂ Hs(R) ⊂ Hs′(R) ⊂ L2(R; dx) ⊂ H−s′(R) ⊂ H−s(R) ⊂ S ′(R),

s > s′ > 0.(3.17)

Moreover, we recall that Hs(R) is conveniently and alternatively introduced as thecompletion of C∞

0 (R) with respect to the norm ‖ · ‖s,Hs(R) = C∞

0 (R)‖·‖s

, s ∈ R, (3.18)

where for ψ ∈ C∞0 (R) and s ∈ R,

‖ψ‖s =(∫

R

dξ(1+|ξ|2s)|ψ(ξ)|2))1/2

, ψ(ξ) = (2π)−1/2

∫R

dx e−iξxψ(x). (3.19)

Equivalently,

Hs(R) ={u ∈ S ′(R)

∣∣∣∣ ‖u‖2Hs(R) =

∫Rn

dξ(1 + |ξ|2s)|u(ξ)|2 <∞

}, s ∈ R.

(3.20)Similarly,

Hsloc(R) =

{u ∈ D′(R)

∣∣ ‖ψ u‖Hs(R) <∞ for all ψ ∈ C∞0 (R)

}={u ∈ D′(R)

∣∣ ‖η(· − a)u‖Hs(R) <∞ for all a ∈ R}, s ∈ R

(3.21)


(cf. [70, p. 140]), and

Hsloc unif(R) =

{u ∈ Hs

loc(R)∣∣∣ supa∈R

‖η(· − a)u‖Hs(R) <∞}, s ∈ R, (3.22)

with η defined in (2.3).Moreover, as proven in [43, Sect. 2] (cf. also [83], [113], [114]) elements q ∈

H−1loc (R) ⊂ D′(R) can be represented by

q = q′2 for some q2 ∈ L2loc(R; dx). (3.23)

Similarly, if q ∈ Hs−1(R) for some s ≥ 0, [83, Lemma 2.1] proves the representation

q = v∞ + v ′s for some v∞ ∈ H∞(R), vs ∈ Hs(R), (3.24)

where

H∞(R) =⋂t≥0

Ht(R) ⊂ C∞(R). (3.25)

In particular, if q ∈ H−1(R) one has the representation

q = v∞ + q ′2 for some v∞ ∈ H∞(R), q2 ∈ L2(R; dx). (3.26)

Next, for q ∈ H−1loc unif(R), [78, Theorem 2.1] proves the representation

q = q1 + q′2 for some qj ∈ Ljloc unif(R; dx), j = 1, 2. (3.27)

The decomposition q = q1 + q′2 in (3.27) is nonunique. In fact, also the represen-tation

q = q∞ + q′2 for some q∞ ∈ L∞(R; dx), q2 ∈ L2loc unif(R; dx) (3.28)

is proved in [78, Theorem 2.1]. Finally, if q ∈ H−1loc (R) is periodic with period

ω > 0, [78, Remark 2.3] (see also [43, Proposition 1]) provides the representation

q = c+ q′2 for some c ∈ C, q2 ∈ L2loc unif(R; dx), q2 periodic with period ω > 0.

(3.29)Next, we turn to sequilinear forms Qq generated by a distribution q ∈ D′(R)

as follows: For f, g ∈ C∞0 (R), f (the complex conjugate of f) is a multiplier for q,

that is, fq = qf ∈ D′(R) and hence the distributional pairing

D′(R)〈qf, g〉D(R) = (fq)(g) = q(fg) = Qq(f, g), f, g ∈ C∞0 (R), (3.30)

is well defined and thus determines a sesquilinear form Qq(·, ·) defined on D(R) =C∞

0 (R). The distribution q ∈ D′(R) is called a multiplier from H1(R) to H−1(R)if (3.30) continuously extends from C∞

0 (R) to H1(R), that is, for some C > 0,

|Qq(f, g)| ≤ C‖f‖H1(R)‖g‖H1(R), f, g ∈ C∞0 (R), (3.31)

and hence one defines this extension Qq via

Qq(f, g) = limn→∞Qq(fn, gn), f, g ∈ H1(R), fn, gn ∈ C∞

0 (R),

assuming limn→∞ ‖f − fn‖H1(R) = 0, lim

n→∞ ‖g − gn‖H1(R) = 0.(3.32)


(This extension is independent of the particular choices of sequences fn, gn and by

polarization, (3.31) for f = g suffices to yield the extension Qq in (3.32).) The set ofall multipliers from H1(R) to H−1(R) is usually denoted by M

(H1(R), H−1(R)

),

equivalently, one could use the symbol B(H1(R), H−1(R)), the bounded linear

operators mapping H1(R) into H−1(R). Thus, for q ∈ M(H1(R), H−1(R)

), the

distributional pairing (3.30) extends to

H−1(R)〈qf, g〉H1(R) = Qq(f, g), f, g ∈ H1(R). (3.33)

Theorem 3.1 ([7], [103, Sects. 2.5, 11.4], [104], [106], [117]). Assume that q ∈ D′(R)generates the sesquilinear form Qq as in (3.30). Then the following conditions (i)–(iii) are equivalent:

(i) q is form bounded with respect to T0, that is, for some C > 0,

|Qq(f, f)| ≤ C‖f‖2H1(R) = C[‖f ′‖2L2(R;dx) + ‖f‖2L2(R;dx)

], f ∈ C∞

0 (R), (3.34)

equivalently,

q ∈M(H1(R), H−1(R)

). (3.35)

(ii) q is infinitesimally form bounded with respect to T0, that is, for all ε > 0,there exists Cε > 0, such that,

|Qq(f, f)| ≤ ε‖f ′‖2L2(R;dx) + Cε‖f‖2L2(R;dx), f ∈ H1(R). (3.36)

(iii) q is of the form

q = q1 + q′2, where qj ∈ Ljloc unif(R; dx), j = 1, 2. (3.37)

Equivalently (cf. (3.22), (3.27)),

q ∈ H−1loc unif(R). (3.38)

Of course, if (3.34) (equivalently, (3.36)) holds, it extends to Qq and allf ∈ H1(R).

Theorem 3.2 ([103, Sect. 11.4], [104], [106]). Assume that q ∈ D′(R). Then thefollowing conditions (i) and (ii) are equivalent:

(i) q is form compact with respect to T0, that is, the map

q : H1(R)→ H−1(R) is compact. (3.39)

(ii) q is of the form

q = q1 + q′2, where qj ∈ Ljloc unif(R; dx), j = 1, 2, (3.40)

and

lim|a|→∞

(∫ a+1

a

dx |q1(x)|)

= 0, lim|a|→∞

(∫ a+1

a

dx |q2(x)|2)

= 0. (3.41)


We should emphasize that the references [7], [103, Sects. 2.5, 11.4], [106], [117]in connection with Theorems 3.1 and 3.2, primarily focus on the multi-dimensionalsituation. In particular, the methods employed in Maz’ya and Verbitsky [104],[105], [106], [107], and Maz’ya and Shaposhnikova [103] rely on Bessel capacitymethods. A considerably simplified approach to this circle of ideas, based on theexistence of positive solutions of the underlying zero-energy Schrodinger operator(more generally, an equation of the type −div(A∇u)+qu = 0 in arbitrary open setsΩ ∈ Rn, with A satisfying an ellipticity condition) appeared in [81]. The specialone-dimensional case is explicitly treated in [7], [104], and [117].

Remark 3.3. If q ∈ D′(R) is real valued and one of the conditions (i)–(iii) inTheorem 3.1 is satisfied, then the form sum

QT (f, g) = QT0(f, g) + q(fg), f, g ∈ dom(QT ) = H1(R), (3.42)

defines a closed, densely defined, symmetric sesquilinear form QT in L2(R; dx),bounded from below. The self-adjoint operator T in L2(R; dx), bounded frombelow, and uniquely associated to the form QT then can be described as follows,

Tf = τf, τf = −(f ′ − q2f)′ − q2(f

′ − q2f) + (q1 − q22)f, (3.43)

f ∈ dom(T ) ={g ∈ L2(R; dx)

∣∣ g, (g′ − q2g) ∈ ACloc(R), τg ∈ L2(R; dx)}.

In particular, the differential expression τ formally corresponds to a Schrodingeroperator with distributional potential q ∈ H−1

loc unif(R),

τ = −(d2/dx2) + q(x), q = q1 + q′2, qj ∈ Ljloc unif(R; dx), j = 1, 2. (3.44)

This is a consequence of the direct methods established in [7], [78]–[80], [83], [123],[124], [136], and of the Weyl–Titchmarsh theory approach to Schrodinger operatorswith distributional potentials developed in [49] (see also [48], [50], and the detailedlist of references therein). In particular, since τ is assumed to be bounded frombelow, τ is in the limit point case at ±∞, rendering the maximally defined operatorT in (3.43) to be self-adjoint (see also [4] and [49]). We will provide further detailson dom(T ) in Remark 3.8.

Next, we turn to an elementary alternative approach to this circle of ideas inthe real-valued context, based on the concept of Miura transformations (cf. [24],[39], [48], [57], [58], [66], [67], [83], [84], [132, Ch. 5], and the extensive literaturecited therein) {

L2loc(R; dx)→ H−1

loc (R)

φ �→ φ2 − φ′ (3.45)

with associated self-adjoint Schrodinger operator T1 ≥ 0 in L2(R; dx) given by

T1 = A∗A, (3.46)

with A the closed operator defined in in L2(R; dx) by

Af = αf, αf = f ′ + φf,

f ∈ dom(A) ={g ∈ L2(R; dx)

∣∣ g ∈ ACloc(R), αg ∈ L2(R; dx)},

(3.47)


implying,

A∗f = α+f, α+f = −f ′ + φf,

f ∈ dom(A∗) ={g ∈ L2(R; dx)

∣∣ g ∈ ACloc(R), α+g ∈ L2(R; dx)}.

(3.48)

Closedness of A and the fact that A∗ is given by (3.48) was proved in [83] (theextension to φ ∈ L1

loc(R; dx), φ real-valued, was treated in [48]). In addition, itwas proved in [83] that

C∞0 (R) is an operator core for A and A∗. (3.49)

Thus, T1 acts as,

T1f = τ1f, τ1f = α+αf = −(f ′ + φf)′ + φ(f ′ + φf),

f ∈ dom(T1) ={g ∈ L2(R; dx)

∣∣ g, αg ∈ ACloc(R), τ1g ∈ L2(R; dx)}.

(3.50)

In particular, τ1 is formally of the type,

τ1 = −(d2/dx2) + V1(x), V1 = φ2 − φ′, φ ∈ L2loc(R; dx), (3.51)

displaying the Riccati equation connection between V1 and φ in connection withMiura’s transformation (3.45).

Theorem 3.4 ([83]). Assume that q ∈ H−1loc (R) is real valued. Then the following

conditions (i)–(iii) are equivalent:

(i) q = φ2 − φ′ for some real-valued φ ∈ L2loc(R; dx).

(ii) (−d2/dx2) + q ≥ 0 in the sense of distributions, that is,

(f ′, f ′)L2(R;dx) + q(ff) = H−1(R)〈(−f ′′ + qf), f〉H1(R) ≥ 0

for all f ∈ C∞0 (R).

(3.52)

(iii) [(−d2/dx2) + q]ψ = 0 has a positive solution 0 < ψ ∈ H1loc(R).

We note that multi-dimensional extensions this circle of ideas are studied ingreat depth in [81].

Theorem 3.5 ([83]). Assume that q ∈ Hs−1(R), s ≥ 0, is real valued. Then thefollowing conditions (i) and (ii) are equivalent:

(i) q = φ2 − φ′ for some real-valued φ ∈ Hs(R).(ii) (−d2/dx2) + q ≥ 0 in the sense of distributions (cf. (3.52)) and q = q1 + q′2

for some qj ∈ Lj(R; dx), j = 1, 2.

The following appears to be a new result:

Theorem 3.6. Assume that q ∈ H−1loc unif(R) is real valued. Then the following

conditions (i)–(iii) are equivalent:

(i) q = φ2 − φ′ for some real-valued φ ∈ L2loc unif(R; dx).

(ii) (−d2/dx2) + q ≥ 0 in the sense of distributions (cf. (3.52)).(iii) [(−d2/dx2) + q]ψ = 0 has a positive solution 0 < ψ ∈ H1

loc(R).


Proof. We will show that (ii) =⇒ (iii) =⇒ (i) =⇒ (ii).

Given item (ii), that is, q ∈ H−1loc unif(R) is real valued and (−d2/dx2)+ q ≥ 0,

one concludes the existence of 0 < ψ0 ∈ H1loc(R) such that −ψ′′

0 + qψ0 = 0 byTheorem 3.4 (iii). Thus, item (iii) follows.

Introducing

φ0 = −ψ′0/ψ0, (3.53)

one infers that

φ0 ∈ L2loc(R; dx) is real valued and q = φ2

0 − φ′0. (3.54)

Next, introducing A0 and A∗0 as in (3.47) and (3.48), with α replaced by α0 =

(d/dx)+φ0 (and analogously for α+), we now introduce the sesquilinear form QT1

and its closure, QT1 , by

QT1(f, g) = (A0f,A0g)L2(R;dx), f, g ∈ dom(QT1

)= C∞

0 (R),

QT1(f, g) = (A0f,A0g)L2(R;dx), f, g ∈ dom(QT1) = dom(A0),(3.55)

with 0 ≤ T1 = A∗0A0 the uniquely associated self-adjoint operator.

Since by hypothesis q ∈ H−1loc unif(R), (2.46) implies (cf. [78]) that q can be

written as

q = q1 + q′2 for some qj ∈ Ljloc unif(R; dx), j = 1, 2, (3.56)

and hence, we also introduce the sesquilinear form˙Q and its closure, Q (cf. [78]

for details),

˙Q(f, g) = (f ′, g′)L2(R;dx) − (f ′, q2g)L2(R;dx) − (q2f, g

′)L2(R;dx) (3.57)

+(|q1|1/2f, sgn(q1)|q1|1/2g)L2(R;dx)

, f, g ∈ dom( ˙Q)= C∞

0 (R),

Q(f, g) = (f ′, g′)L2(R;dx) − (f ′, q2g)L2(R;dx) − (q2f, g′)L2(R;dx) (3.58)

+(|q1|1/2f, sgn(q1)|q1|1/2g)L2(R;dx)

, f, g ∈ dom(Q)= H1(R).

Since

QT1(f, g) = Q(f, g) = (f ′, g′)L2(R;dx) + q(fg), f, g ∈ C∞0 (R), (3.59)

and C∞0 (R) is a form core for QT1 (cf. (3.49)) and Q, one concludes that QT1 = Q

and hence

dom(QT1) = dom(A0) = dom(Q)= H1(R). (3.60)

A comparison of (3.47) (with α replaced by α0) and (3.60) implies that φ0g ∈L2(R; dx) for g ∈ dom(A0) = H1(R), and hence,

dom(φ0) ⊇ H1(R). (3.61)

An application of Theorem 2.1 (i), (ii) then finally yields

φ0 ∈ L2loc unif(R; dx), (3.62)

which together with (3.54) implies item (i).


Finally, given φ ∈ L2loc unif(R; dx), φ real-valued, such that q = φ2 − φ′, one

computes, with α = (d/dx) + φ,

0 ≤ ‖αf‖2L2(R;dx) = ‖f ′‖2L2(R;dx) + q(|f |2) = H−1(R)〈(−f ′′ + qf), f〉H1(R),

f ∈ C∞0 (R),

(3.63)

and hence item (i) implies item (ii). �

Thus, Theorem 3.6 further illustrates the results by Bak and Shkalikov [7]and Maz’ya and Verbitsky [104], [105], [106] (specialized to the one-dimensionalsituation) recorded in Theorem 3.1 in the particular case where q is real valued.

In connection with Theorem 3.6 (i), we also recall the following useful result:

Lemma 3.7 ([80]). Assume that q ∈ H−1loc unif(R) is real valued and of the form

q = φ2 − φ′ for some real-valued φ ∈ L2loc(R; dx). Then, actually,

φ ∈ L2loc unif(R; dx). (3.64)

Remark 3.8. Combining (3.42)–(3.44), (3.50), (3.51), (3.55), and (3.61) (identify-ing φ and φ0 as well as T and T1) then yields the following apparent improvementover the domain characterizations (3.43), (3.50),

T1f = τ1f, τf = −(f ′ + φf)′ + φ(f ′ + φf),

f ∈ dom(T1) ={g ∈ L2(R; dx)

∣∣ g, αg ∈ ACloc(R),

g′, φg ∈ L2(R; dx), τ1g ∈ L2(R; dx)},

(3.65)

with (3.51) staying in place. In fact, (3.50) and (3.65) are, of course, equivalent;the former represents a minimal characterization of dom(T1).

Remark 3.9. Given q = φ2 − φ′, φ ∈ L2loc unif(R; dx) as in Theorems 3.4–3.6,

the question of uniqueness of φ for prescribed q ∈ H−1loc (R) arises naturally. This

has been settled in [83] and so we briefly summarize some pertinent facts. Sinceφ = −ψ′/ψ for some 0 < ψ ∈ H1

loc(R), uniqueness of φ is equivalent to uniquenessof ψ > 0 satisfying [(−d2/dx2) + q]ψ = 0. Thus, suppose 0 < ψ0 ∈ H1

loc(R) isa solution of [(−d2/dx2) + q]ψ = 0. Then, the general, real-valued solution of[(−d2/dx2) + q]ψ = 0 is of the type

ψ(x) = C1ψ0(x) + C2ψ0(x)

∫ x

0

dx′ ψ0(x′)−2, x ∈ R, Cj ∈ R, j = 1, 2. (3.66)

Next, introducing

c± = ± limx→±∞

∫ x

0

dx′ ψ0(x′)−2 ∈ (0,+∞], (3.67)

and defining c−1± = 0 if c± = +∞, all positive solutions 0 < ψ on R of [(−d2/dx2)+

q]ψ = 0 are given by

ψ(x) = ψ0(x)

[1 + c

∫ x

0

dx′ ψ0(x′)−2

], c ∈ [− c−1

+ , c−1−]. (3.68)


Consequently,

0 < ψ0 ∈ H1loc(R) is the unique solution of [(−d2/dx2) + q]ψ = 0

if and only if ± limx→±∞

∫ x

0

dx′ ψ0(x′)−2 =∞.

(3.69)

On the other hand, if at least one of ± limx±∞∫ x

0dx′ ψ0(x

′)−2 <∞, [(−d2/dx2)+

q]ψ = 0 has a one (real) parameter family of positive solutions on R lying inH1loc(R)

given by (3.68). Without going into further details, we note that Weyl–Titchmarshsolutions ψ±(λ, ·) corresponding to T in (3.43) for energies λ < inf(σ(T )), are

actually constant multiples of Hartman’s principal solutions T ψ±(λ, ·) = λψ±(λ, ·),that is, those that satisfy ± ∫ ±∞

dx′ [ψ±(λ, x′)]−2

=∞.

Theorem 3.10. Assume that q ∈ H−1loc (R) is real valued and suppose in addition

that (−d2/dx2)+q ≥ 0 in the sense of distributions (cf. (3.52)). Then the followingconditions (i)–(iv) are equivalent:

(i) q is form compact with respect to T0, that is, the map

q : H1(R)→ H−1(R) is compact. (3.70)

(ii) q is of the form q = φ2 − φ′, where φ ∈ L2loc unif(R; dx) is real valued and

lim|a|→∞

(∫ a+1

a

dxφ(x)2)

= 0. (3.71)

(iii) The operator of multiplication by φ is T1/20 -compact.

(iv) The operator of multiplication by φ is Pm

(T

1/20

)-compact, where Pm is a

polynomial of degree m ∈ N.

Proof. By Theorem 3.4, (−d2/dx2) + q ≥ 0 in the sense of distributions impliesthat q is of the form q = φ2 − φ′ for some real-valued φ ∈ L2

loc(R). By Lemma3.7, one actually concludes that φ ∈ L2

loc unif(R). The equivalence of items (i) and(ii) then follows from Theorem 3.2 since upon identifying q1 = φ2, q2 = φ, thetwo limiting relations in (3.41) are equivalent to (3.71). Equivalence of condition(3.71) and item (iii) is guaranteed by Theorem 2.2. �

At this point it is worth recalling a few additional details of the supersym-metric formalism started in (3.45)–(3.51), whose abstract roots can be found inAppendix B: Assuming φ ∈ L2

loc unif(R; dx) to be real valued (we note, however,that this supersymmetric formalism extends to the far more general situationwhere φ ∈ L1

loc(R; dx) is real valued, in fact, it extends to the situation where φ ismatrix valued, see [48] for a detailed treatment of these matters), one has

A = (d/dx) + φ, A∗ = −(d/dx) + φ, dom(A) = dom(A∗) = H1(R), (3.72)

T1 = A∗A = −(d2/dx2) + V1, V1 = φ2 − φ′, (3.73)

T2 = AA∗ = −(d2/dx2) + V2, V2 = φ2 + φ′, (3.74)


D =

(0 A∗

A 0

)in L2(R; dx) ⊕ L2(R; dx), (3.75)

D2 =

(A∗A 00 AA∗

)= T1 ⊕ T2 in L2(R; dx)⊕ L2(R; dx). (3.76)

As a consequence, one can show (cf. [48]) the Weyl–Titchmarsh solutions, ψ±, forD,T1, T2 satisfy

ψD,1,±(ζ, x) = ψT1,±(z, x), z = ζ2, ζ ∈ C\R, (3.77)

ψT2,±(z, x) = c1(z)(AψT1,±)(z, x), (3.78)

with c1(z) a normalization constant. Similarly, after interchanging the role of T1

and T2,

ψD,2,±(ζ, x) = ψT2,±(z, x), z = ζ2, ζ ∈ C\R, (3.79)

ψT1,±(z, x) = c2(z)(A∗ψT2,±)(z, x), (3.80)

again with c2(z) a normalization constant. Here,

ΨD,±(ζ, x) =(ψD,1,±(ζ, x)ψD,2,±(ζ, x)

)(3.81)

are the Weyl–Titchmarsh solutions for D =(

0 A∗A 0

).

The (generalized, or renormalized) Weyl–Titchmarsh m-functions for D, T1,T2 satisfy:

mD,±(ζ, x0) =1

ζmT1,±(z, x0) =

−ζmT2,±(z, x0)

, (3.82)

where x0 is a fixed reference point (typically, x0 = 0), and

mT1,±(z, x0) =ψ[1,1]T1,±(z, x0)

ψT1,±(z, x0)=

(AψT1,±)(z, x0)

ψT1,±(z, x0), (3.83)

mT2,±(z, x0) =ψ[1,2]T2,±(z, x0)

ψT2,±(z, x0)=

(−A∗ψT2,±)(z, x0)

ψT2,±(z, x0). (3.84)

Here, y[1,1] = Ay = [y′ + φy] is the quasi-derivative corresponding to T1 andy[1,2] = −A∗y = [y′ − φy] is the quasi-derivative corresponding to T2.

Thus, spectral properties of D instantly translate into spectral properties ofTj, j = 1, 2, and vice versa (the latter with the exception of the zero spectralparameter). In particular, φ ∈ L2

loc unif(R; dx) ⊂ L2loc(R; dx) in D is entirely “stan-

dard” (in fact, even φ ∈ L1loc(R; dx) in D is entirely standard, see, e.g., [28] and the

extensive literature cited therein), while the potentials Vj = (−1)jφ′+φ2, j = 1, 2,

involve the distributional coefficient φ′ ∈ H−1loc unif(R). (We also note that while in

this paper the Dirac operator D only involves the L2loc(R; dx)-coefficient φ, Dirac-

type operators with distributional potentials have been studied in the literature,see, for instance [3, App. J] and [27].) In particular, spectral results for the “stan-dard” one-dimensional Dirac-type operatorD imply corresponding spectral resultsfor Schrodinger operators bounded from below, with (real-valued) distributional


potentials. Some applications of this spectral correspondence between D and Tj ,j = 1, 2, to inverse spectral theory, local Borg–Marchenko uniqueness results, etc.,were treated in [48]. In Section 4 we will apply this spectral correspondence toderive some Floquet theoretic results in connection with the Schrodinger opera-tors Tj and hence for the distributional potentials [φ2 + (−1)jφ′] ∈ H−1

loc unif(R),j = 1, 2.

Remark 3.11. For simplicity we restricted ourselves to the special case p = 1 inTheorems 3.4–3.6 and Remarks 3.8 and 3.9. However, assuming

0 < p, p−1 ∈ L∞(R; dx), 0 < r, r−1 ∈ L∞(R; dx), (3.85)

the observations thus far in this section extend to the case where

τ1f = α+αf = −f ′′ +[φ2 − φ′]f

= −(f ′ + φf)′ + φ(f ′ + φf)(3.86)

is replaced by

τ1f = β+βf = r−1[− (pf ′)′ +

[pφ2 − (pφ)′

]f]

= r−1[− [p(f ′ + φf)]′ + φ[p(f ′ + φf)]

],

(3.87)

where

βf = (pr)−1/2[p(f ′ + φf)],

β+f = −(pr)−1{p[[(pr)1/2f

]′ − φ[(pr)1/2f

]]}.

(3.88)

Remark 3.12. We only dwelled on

dom(|T |1/2) = H1(R) (3.89)

to derive a number of if and only if results. For practitioners in this field, the suf-ficient conditions on q, r in terms of the Lj

loc unif(R; dx), j = 1, 2, and boundednessconditions on 0 < p, p−1, yielding form boundedness (i.e., self-adjointness) results,relative compactness, and trace class results, all work as long as one ensures

dom(|T |1/2) ⊆ H1(R). (3.90)

This permits larger classes of coefficients p, q, r for which one can prove these typesof self-adjointness and spectral results.

Before returning to our principal object, the Birman–Schwinger-type operatorT−1/2rT−1/2, but now in the context of distributional coefficients q and r, webriefly examine the well-known example of point interactions:

Example 3.13 (Delta distributions).

q1(x) = 0, q2(x) =

{1, x > x0,

0, x < x0,then q = q′2 = δx0 , x0 ∈ R. (3.91)


Introducing the operator

Aα,x0 =d

dx− α

2

{1, x > x0,

−1, x < x0,that is, φ(x) =

α

2sgn(x − x0), α, x0 ∈ R,

dom(Aα,x0) = H1(R), (3.92)

in L2(R; dx), one infers that

A∗α,x0

Aα,x0 = −Δα,x0 + (α2/4)I. (3.93)

Here −Δα,x0 = −d2/dx2+α δx0 in L2(R; dx) represents the self-adjoint realizationof the one-dimensional point interaction (cf. [3, Ch. I.3]), that is, the Schrodingeroperator with a delta function potential of strength (coupling constant ) α centeredat x0 ∈ R.

This extends to sums of delta distributions supported on a discrete set(Kronig–Penney model, etc.).

Next we apply this distributional approach to the Birman–Schwinger-typeoperator T−1/2rT−1/2. We outline the basic ideas in the following three steps:

Step 1. Assume p, p−1 ∈ L∞(R; dx), p > 0 a.e. on R.

Step 2. Suppose q = q1 + q′2, where qj ∈ Ljloc unif(R; dx), j = 1, 2, are real val-

ued. This uniquely defines a self-adjoint operator T in L2(R; dx), bounded frombelow, T ≥ cI for some c ∈ R, as the form sum T = −(d/dx)p(d/dx) + q of−(d/dx)p(d/dx) and the distribution q = q1 + q′2 ∈ D′(R). Then

dom(|T |1/2) = H1(R). (3.94)

If in addition, lim|a|→∞( ∫ a+1

a dx |q1(x) − c1|)= 0 for some constant c1 ∈

R, and lim|a|→∞( ∫ a+1

adx |q2(x)|2

)= 0, one again obtains results on essential

spectra.

Step 3. Suppose without loss of generality, that T ≥ cI, c > 0, and introducer = r1 + r′2, rj ∈ Lj

loc unif(R; dx) real-valued, j = 1, 2. This uniquely defines a

bounded self-adjoint operator T−1/2rT−1/2 in L2(R; dx) as described next: Firstwrite

T−1/2rT−1/2 (3.95)

=[(T0 + I)1/2T−1/2

]∗[(T0 + I)−1/2r(T0 + I)−1/2

][(T0 + I)1/2T−1/2

].

Next, one interprets (T0 + I)−1/2r(T0 + I)−1/2 as follows: Employing T0 and its

extension, T0, to the entire Sobolev scale Hs(R) in (3.9)–(3.15), in particular, we

will employ the mapping properties,(T0 + I

)−1/2: Hs(R) → Hs+1(R), s ∈ R.

Thus, using [(T0 + I)1/2T−1/2

],[(T0 + I)1/2T−1/2

]∗ ∈ B(L2(R; dx)), (3.96)


and[ (T0 + I

)−1/2︸︷︷︸∈B(H−1(R),L2(R;dx))

r︸︷︷︸∈B(H1(R),H−1(R))

(T0 + I

)−1/2︸︷︷︸∈B(L2(R;dx),H1(R))

]∈ B(L2(R; dx)

),

(3.97)finally yields

T−1/2rT−1/2 =[(T0 + I)1/2T−1/2

]∗[(T0 + I

)−1/2r(T0 + I

)−1/2]

× [(T0 + I)1/2T−1/2] ∈ B(L2(R; dx)

). (3.98)

Hence, our reformulated left-definite generalized eigenvalue problem becomesagain a standard self-adjoint spectral problem in L2(R; dx),

T−1/2rT−1/2 χ =1

zχ, z ∈ C\{0}, (3.99)

associated with the bounded, self-adjoint operator T−1/2rT−1/2 in L2(R; dx), yetthis time we permit distributional coefficients satisfying

p, p−1 ∈ L∞(R; dx), p > 0 a.e. on R, (3.100)

q = q1 + q′2, qj ∈ Ljloc unif(R; dx) real-valued, j = 1, 2, (3.101)

r = r1 + r′2, rj ∈ Ljloc unif(R; dx) real-valued, j = 1, 2, (3.102)

with T defined as the self-adjoint, lower-semibounded operator uniquely associatedwith the lower-bounded, closed sesquilinear form QT in L2(R; dx) given by (cf.(3.30))

QT (f, g) =(p1/2f ′, p1/2g′

)L2(R;dx)

+ q(fg) (3.103)

=(p1/2f ′, p1/2g′

)L2(R;dx)

− (f ′, q2g)L2(R;dx) − (q2f, g′)L2(R;dx)

+(|q1|1/2f, sgn(q1)|q1|1/2g)L2(R;dx)

, (3.104)

=(p−1/2(pf ′ − q2f), p

−1/2(pg′ − q2g))L2(R;dx)

(3.105)

+(|q1|1/2f, sgn(q1)|q1|1/2g)L2(R;dx)

− (p−1/2q2f, p−1/2q2g

)L2(R;dx)

,

f, g ∈ dom(QT ) = H1(R).

In particular, T corresponds to the differential expression τ = −(d/dx)p(d/dx) +q(x), x ∈ R, and hence is explicitly given by

Tf = τf, τf = −(pf ′ − q2f)′ − p−1q2(pf

′ − q2f) +(q1 − p−1q22

)f,

f ∈ dom(T ) ={g ∈ L2(R; dx)

∣∣ g, (pg′ − q2g) ∈ ACloc(R), τg ∈ L2(R; dx)}.

={g ∈ L2(R; dx)

∣∣ g, (pg′ − q2g) ∈ ACloc(R), τg ∈ L2(R; dx)

(pg′ − q2g) ∈ L2(R; dx)}. (3.106)

Without loss of generality we assume T ≥ cI for some c > 0.


4. The case of periodic coefficients

In this section we apply some of the results collected in Sections 2 and 3 to thespecial, yet important, case where all coefficients are periodic with a fixed period.For simplicity, we will choose p = 1 throughout, but we emphasize that includ-ing the nonconstant, periodic coefficient p can be done in a standard manner asdiscussed in Remark 3.11. It is not our aim to present a thorough treatment ofFloquet theory, rather, we intend to illustrate some of the scope underlying theapproach developed in this paper.

One recalls that q ∈ H−1loc (R) is called periodic with period ω > 0 if

H−1(R)〈q, f(· − ω)〉H1(R)q = H−1(R)〈q, f〉H1(R), f ∈ H1(R). (4.1)

By (3.29), if q ∈ H−1loc (R) is periodic, it can be written as q = q1 + q′2, where q1

is a constant and q2 ∈ L2loc unif(R; dx) is periodic with period ω. The analogous

statement applies, of course, to the coefficient r in the differential equation (1.7),assuming (3.102) to hold. Introducing the abbreviations Q = q−zr, Q1 = q1−zr1,andQ2 = q2−zr2 and the quasi-derivative y[1] = y′−Q2y we may now write (1.7) as

τy = −(y[1])′ −Q2y[1] + (Q1 −Q2

2)y = 0, (4.2)

or, equivalently, as the first-order system(yy[1]

)′=

(Q2 1

Q1 −Q22 −Q2

)(yy[1]

). (4.3)

Existence and uniqueness for the corresponding initial value problem as well asthe constancy of the modified Wronskian,

W (f, g)(x) = f(x)g[1](x)− f [1](x)g(x), (4.4)

were established in [49]. As a consequence, the monodromy map

M(z) : y �→ y(·+ ω) (4.5)

maps the two-dimensional space of solutions of equation (4.2) onto itself and hasdeterminant 1 (as usual this is seen most easily by introducing a standard basis

u1, u2 defined by the initial values u1(c) = u[1]2 (c) = 1 and u

[1]1 (c) = u2(c) = 0).

The trace of M(z), given by u1(c+ω) + u[1]2 (c+ω), is real which implies that the

eigenvalues ρ(z) and 1/ρ(z) of M(z) (the Floquet multipliers) are either both real,or else, are complex conjugates of each other, in which case they both lie on theunit circle. The proof of Theorem 2.7 in [49] may also be adapted to show that,

for each fixed point x, the functions u1(x), u2(x), u[1]1 (x), and u

[1]2 (x) are entire

functions of growth order 1/2 with respect to z. In particular, trC2(M(·)) is anentire function of growth order 1/2.

We start by focusing on the operator T as discussed in (3.42)–(3.44).


Throughout this section we make the following assumptions:

Hypothesis 4.1. Assume that q ∈ H−1loc (R) is real valued and periodic with period

ω > 0 (and hence, actually, q ∈ H−1loc unif(R)). Define T in L2(R; dx) according to

(3.42)–(3.44) and suppose that T ≥ 0.

Lemma 4.2. Assume Hypothesis 4.1. Then there exists φ0 ∈ L2loc unif(R; dx), real-

valued and periodic of period ω > 0, such that q = φ20 − φ′

0.

Proof. It suffices to note that (as in the standard case where q ∈ L1loc(R) is real

valued and periodic with period ω > 0) the Weyl–Titchmarsh solutions ψT,±(z, · )satisfy

ψT,±(z, x) > 0, z < 0, x ∈ R, (4.6)

which extends by continuity to z = 0, that is,

ψT,±(0, x) > 0, x ∈ R, (4.7)

although, ψT,±(0, ·) may no longer lie in L2 near ±∞ and hence cease to bea Weyl–Titchmarsh solution. (By oscillation theory, cf. [49], a zero of ψT,±(0, ·)would contradict T ≥ 0.) Using the Floquet property of ψT,±(z, · ), φ± defined by

φ±(x) = ψ′T,±(0, x)/ψT,±(0, x), x ∈ R, (4.8)

satisfiesφ± ∈ L2

loc(R), φ±(·) is periodic with period ω > 0, (4.9)

in particular,φ± ∈ L2

loc unif(R) and q = φ2± − φ′

±. (4.10)

(If inf(σ(T )) = 0, one has ψT,+(0, x) = ψT,−(0, x) and hence φ+ = φ−.) �Given Hypothesis 4.1, Lemma 4.2 guarantees the existence of a real-valued,

ω-periodic φ ∈ L2loc unif(R; dx) such that q = φ2−φ′ and hence we can identify the

operator T in L2(R; dx) with T1 = A∗A in (3.46) (resp., (3.72)), where A and A∗

defined as in (3.47) and (3.48) (resp., (3.72)). In addition, we define the periodicDirac-type operator D in L2(R; dx) ⊕ L2(R; dx) by (3.75).

Since φ ∈ L2([0, ω]; dx), for any ε > 0 and all g ∈ H1((0, ω)), one has

‖φg‖2L2([0,ω];dx) ≤ ε‖g′‖2L2([0,ω];dx)

+ ‖φ‖2L2([0,ω];dx)

[ω−1 + ‖φ‖2L2([0,ω];dx)ε

−1]‖g‖2L2([0,ω];dx)

(4.11)

(cf. [125, p. 19–20, 37]). Utilizing (4.11), one can introduce the reduced Dirac-typeoperator Dθ in L2([0, ω]; dx), θ ∈ [0, 2π], by

Dθ =

(0 A∗

θ

Aθ 0

)in L2([0, ω]; dx)⊕ L2([0, ω]; dx), (4.12)

where

Aθ = (d/dx) + φ, dom(Aθ) ={g ∈ H1((0, ω))

∣∣ g(ω) = eiθg(0)}, (4.13)

A∗θ = −(d/dx) + φ, dom(A∗

θ) ={g ∈ H1((0, ω))

∣∣ g(ω) = eiθg(0)}, (4.14)

and Aθ (and hence A∗θ) is closed in L2([0, ω]; dx), implying self-adjointness of Dθ.


Employing the identity (3.76), D2 = T1 ⊕ T2, and analogously for D2θ ,

D2θ =

(A∗

θAθ 00 AθA

∗θ

)= T1,θ ⊕ T2,θ in L2([0, ω]; dx)⊕ L2([0, ω]; dx), (4.15)

T1,θ = A∗θAθ, T2,θ = AθA

∗θ in L2([0, ω]; dx), (4.16)

and applying the standard direct integral formalism combined with Floquet theoryto D, Dθ (cf., [22, App. to Ch. 10], [46], [121, Sect. XIII.16]), where

L2(R; dx) � 1

2π

∫ ⊕

[0,2π]

dθ L2([0, ω]; dx), (4.17)

then yields the following result (with � abbreviating unitary equivalence):

Theorem 4.3. Assume Hypothesis 4.1. Then the periodic Dirac operator D (cf.(3.75)) satisfies

D � 1

2π

∫ ⊕

[0,2π]

dθ Dθ, (4.18)

with respect to the direct integral decomposition (4.17), and σp(D) = σsc(D) = ∅.Moreover, σ(D) is purely absolutely continuous of uniform spectral multiplicityequal to two, and σ(D) consists of a union of compact intervals accumulating at+∞ and −∞.

In addition, the spectra of Tj (cf. (3.73), (3.74)) satisfy σp(Tj) = σsc(Tj) = ∅,in fact, σ(Tj) is purely absolutely continuous of uniform spectral multiplicity equalto two, and σ(Tj) consists of a union of compact intervals accumulating at +∞,j = 1, 2.

We note in passing that the spectral properties of Tj, j = 1, 2, alternatively,also follow from them-function relations (3.83), (3.84). In fact, applying the resultsin [48], one can extend Theorem 4.3 to the case where φ ∈ L1

loc(R; dx) is real valuedand periodic of period ω > 0, but we will not pursue this any further in this paper.

The supersymmetric approach linking (periodic, quasi-periodic, finite-gap,etc.) Schrodinger and Dirac-type operators has been applied repeatedly in the lit-erature, see, for instance, [40], [57], [58], [66], [67], [94], and the extensive literaturecited therein. In addition, we note that spectral theory (gap and eigenvalue asymp-totics, etc.) for Schrodinger operators with periodic distributional potentials hasbeen thoroughly investigated in [41], [42], [43], [44], [78], [79], [82], [89], [95], [97],[113], [114].

We now investigate the eigenvalues associated with the differential equation(1.7) and quasi-periodic boundary conditions utilizing the operatorT−1/2rT−1/2 in L2([0, ω]; dx) when r is a measure. More precisely, let R : [0, ω]→R be a left-continuous real-valued function of bounded variation and μR the asso-ciated signed measure. We associate with R the following map

r : H1((0, ω))→ H−1((0, ω)) (4.19)


via the Lebesgue–Stieltjes integral,

H−1(R)〈rf, g〉H1(R) =

∫ ω

0

dμR(x) f(x)g(x), f, g ∈ H1((0, ω)). (4.20)

One notes that the map r defined in terms of (4.19), (4.20) is bounded.We also write R = R+ − R− where R± are both left-continuous and nonde-

creasing and thus give rise to positive finite measures on [0, ω].Thus,

K ∈ B(L2([0, ω]; dx), H1((0, ω)))implies K∗rK ∈ B(L2([0, ω]; dx)

). (4.21)

Similarly,

K ∈ B∞(L2([0, ω]; dx), H1((0, ω))

)implies K∗rK ∈ B∞

(L2([0, ω]; dx)

). (4.22)

Lemma 4.4. Suppose K ∈ B∞((L2([0, ω]; dx), H1((0, ω))

)is compact and that

C∞0 ((0, ω)) ⊂ ran(K). In addition, assume that R is a real-valued function of

bounded variation on [0, ω] and define r as in (4.19), (4.20). Then K∗rK has in-finitely many positive (resp., negative) eigenvalues unless R+ (resp., R−) is a purejump function with only finitely many jumps (if any).

Proof. Without loss of generality we may assume that ω = 1 and we may alsorestrict attention to R+ only. Accordingly, suppose that the measure associatedwith R+ has a continuous part or that R+ has infinitely many jumps, but, thatby way of contradiction, K∗rK has only finitely many (say, N ≥ 0) positiveeigenvalues. We will show below that there is a positive number � and N + 1 setsΩ1, . . . ,ΩN+1, which have a distance of at least � from each other and from theendpoints of [0, 1], for which

∫Ωj

dμR > 0.

For any ε, with 0 < ε < �/2, let Jε be the Friedrichs mollifier as introduced,for instance, in [1, Sect. 2.28]. Applying [1, Theorem 2.29], the functions

gj,ε = Jε ∗ χΩj , j = 1, . . . , N + 1, (4.23)

satisfy the following properties:

(i) gj,ε ∈ C∞0 ((0, 1)) ⊂ ran(K),

(ii) gj,ε are zero at points which are further than ε away from Ωj ,(iii) limε↓0 ‖gj,ε − χΩj‖L2([0,1];dx) = 0,(iv) |gj,ε(x)| ≤ 1.

Property (i) implies that there are functions fj,ε ∈ L2([0, 1]; dx) such that gj,ε =Kfj,ε since C∞

0 ((0, 1)) ⊂ ran(K). By property (iii), gj,ε → χΩj pointwise a.e. on(0, 1) as ε ↓ 0, and hence the dominated convergence theorem implies that

H−1(R)〈rKfj,ε,Kfj,ε〉H1(R) =

∫[0,1]

dμR(x) |gj,ε(x)|2 −→ε↓0

∫Ωj

dμR(x) > 0. (4.24)

Hence we may fix ε > 0 in such a way that∫[0,1]

dμR(x) |(Kfj,ε)(x)|2 > 0, j = 1, . . . , N + 1. (4.25)


Next, by property (ii) mentioned above, the supports of the gj,ε are pairwise dis-joint, implying ∣∣∣∣N+1∑

j=1

cjgj,ε

∣∣∣∣2 =

N+1∑j+1

|cj |2|gj,ε|2 (4.26)

for any choice of cj ∈ C, j = 1, . . . , N + 1.

Assume now that f =∑N+1

j=1 cjfj,ε, where at least one of the coefficients

cj �= 0. Then equations (4.25) and (4.26) imply

(f,K∗rKf)L2([0,1];dx) =

∫[0,1]

dμR(x) |(Kf)(x)|2

=

N+1∑j=1

|cj |2∫[0,1]

dμR(x) |(Kfj,ε)(x)|2 > 0.

(4.27)

We will now prove that for some choices of the coefficients cj , the expression(f,K∗rKf)L2([0,1];dx) cannot be positive so that one arrives at a contradiction to(4.27), proving that there must be infinitely many positive eigenvalues. To do so,we denote the nonzero eigenvalues and eigenfunctions of the compact, self-adjointoperator K∗rK by λk and ϕk, respectively. More specifically, assume that thepositive eigenvalues have labels k = 1, . . . , N , while the labels of the non-positiveeigenvalues are chosen from the non-positive integers. The spectral theorem, ap-plied to K∗rK, yields

0 < (f,K∗rKf)L2([0,1];dx) =

N∑k=−∞

λk|(ϕk, f)L2([0,1];dx)|2

≤N∑

k=1

λk|(ϕk, f)L2([0,1];dx)|2(4.28)

for any f ∈ L2([0, 1]; dx). If N = 0, this is the desired contradiction. If N ≥1, the inequality (4.28) shows that no non-zero element of L2([0, 1]; dx) can beorthogonal to all the eigenfunctions associated with positive eigenvalues. However,the underdetermined system

N+1∑j=1

cj(ϕk, fj,ε)L2([0,1];dx) = (ϕk, f)L2([0,1];dx) = 0, k = 1, . . . , N, (4.29)

has nontrivial solutions (c1, . . . , cN ) proving that f =∑N+1

j=1 cjfj,ε is orthogonal toall the eigenfunctions associated with positive eigenvalues so that we again arriveat a contradiction.

It remains to establish the existence of the sets Ωj with the required proper-ties. Recall that, by Lebesgue’s decomposition theorem, R = R1+R2 +R3, whereR1 is absolutely continuous, R2 is continuous but R′

2 = 0 a.e. on [0, 1], and R3 isa jump function and that these generate an absolutely continuous measure μ1, asingular continuous measure μ2, and a discrete measure μ3 (i.e., one supported on


a countable subset of R), respectively. By Jordan’s decomposition theorem, eachof these measures may be split into its positive and negative part μj,±, j = 1, 2, 3.We will denote the respective supports of these measures by Aj,±, j = 1, 2, 3. Notethat Aj,+ ∩ Aj,− is empty for each j by Hahn’s decomposition theorem. We alsodefine Rj,±(x) = μj,±([0, x]).

First, we assume that the support A1,+ of μ1,+ has positive Lebesgue mea-sure. Since the supports of μ2 and μ3 have zero Lebesgue measure, they are subsetsof a union of open intervals whose total length is arbitrarily small. Thus, we mayfind a set Ω ⊂ A1,+ of positive Lebesgue measure which avoids a neighborhood ofthe supports of μ2 and μ3 so that

∫ΩdμR > 0. Now define M = '(2N +3)/m(Ω)(,

with m(·) abbreviating Lebesgue measure and 'x( the smallest integer not smallerthan x. Dividing the interval [0, 1] uniformly into M subintervals, each will havelength not exceeding � = m(Ω)/(2N + 3). Consequently, at least 2N + 3 of theseintervals will intersect Ω in a set of positive Lebesgue measure and hence of pos-itive μR-measure. N + 1 of the latter ones will have a distance of at least � fromeach other and from the endpoints of [0, 1]. These intersections will be the soughtafter sets Ω1, . . . ,ΩN+1.

Next assume μ1,+ = 0 but μ2,+([0, 1]) = a2 > 0. Since A3,− is countablewe have μ2,+(A3,−) = 0. Also, of course, μ2,+(A2,−) = 0. By the regularity ofμ2,+ there is, for every positive ε, an open set W covering A2,− ∪ A3,− such that

μ2,+(W ) < ε. Set Ω = (0, 1)\W and ε = a2/2. Since W −W is countable we haveμ2,+(Ω) = μ2,+((0, 1)\W ) > a2/2. Since R2,+ is uniformly continuous there is aδ > 0 so that R2,+(y)−R2,+(x) < a2/(2(2N +3)) as long as 0 < y− x < δ. Thus,splitting Ω in intervals of length at most δ, we have that at least 2N + 3 of theseintervals have positive μ2,+-measure and N + 1 of these have a positive distancefrom each other and from the endpoints of [0, 1]. We denote these intervals by Ω′

1,. . . , Ω′

N+1. We now have μ2,+(Ω′k) > 0 but μ2,−(Ω′

k) = μ3,−(Ω′k) = 0. However,

it may still be the case that μ1,−(Ω′k) > μ2,+(Ω

′k). Regularity of μ1,− allows us to

find a set Ωk such that A2,+ ∩ Ω′k ⊂ Ωk ⊂ Ω′

k and μ1,−(Ωk) are arbitrarily small.This way we may guarantee that μ(Ωk) > 0 for k = 1, . . . , N + 1.

Finally, assume that R+ is a pure jump function, but with infinitely manyjumps. Then we may choose pairwise disjoint intervals Ωk about N + 1 of thejump discontinuities of R+ and we may choose them so small that their μj,−(Ωk)is smaller than the jump so that again μ(Ωk) > 0 for k = 1, . . . , N + 1. �

We emphasize that Lemma 4.4 applies, in particular, to the special case,where dμR(x) = r(x)dx is purely absolutely continuous on R:

Corollary 4.5. Suppose K ∈ B∞(L2([0, ω]; dx)

)is self-adjoint with ran(K) ⊇

H1((0, ω)). Assume in addition that r ∈ L1([0, ω]; dx) is real valued such that|r|1/2K ∈ B∞

(L2([0, ω]; dx)

). Then KrK := [|r|1/2K]∗ sgn(r)|r|1/2K has infin-

itely many positive (resp., negative) eigenvalues unless r+ = 0 (resp., r− = 0) a.e.on (0, ω).


Identifying Tθ in L2([0, ω]; dx) with T1,θ = A∗θAθ (in analogy to the identifi-

cation of T in L2(R; dx) with T1 = A∗A), recalling the construction of T , Tθ ac-cording to (C.26), (C.40), an application of Lemma 4.4, employing (C.44)–(C.49),then yields the following result:

Theorem 4.6. Assume Hypothesis 4.1, suppose that μR is a signed measure, andlet r be defined as in (4.19), (4.20). In addition, assume that r is periodic of periodω > 0.

(i) Suppose that Tθ ≥ cθIL2([0,ω];dx) for some cθ > 0. Then(Tθ

)−1/2r(Tθ

)−1/2

has infinitely many positive (resp., negative) eigenvalues unless R+ (resp.,R−) is a pure jump function with only finitely many jumps (if any ).

(ii) Suppose that T ≥ cIL2(R;dx) for some c > 0. Then σ((

T)−1/2

r(T)−1/2

)consists of a union of compact intervals accumulating at 0 unless R = 0 a.e.on (0, ω). In addition,

−ψ′′ + qψ = zrψ (4.30)

has a conditional stability set (consisting of energies z with at least onebounded solution on R) composed of a sequence of intervals on (0,∞) tendingto +∞ and/or −∞, unless R+ and/or R− is a pure jump function with onlyfinitely many jumps (if any ). Finally,

σp

((T)−1/2

r(T)−1/2

)= ∅. (4.31)

Proof. Lemma 4.4, identifying K and(Tθ

)−1/2(cf. (C.48) and our notational

convention (C.49)) proves item (i).As usual (see Eastham [46, Sect. 2.1] or Brown, Eastham, and Schmidt [25,

Sect. 1.4]), the conditional stability set S of equation (4.30) is given by

S = {λ ∈ R | | trC2(M(λ))| ≤ 2} (4.32)

since, if λ ∈ S and only then, the monodromy operator M(λ) has at least oneeigenvector associated with an eigenvalue of modulus 1. Since trC2(M(·)) is ananalytic, hence, continuous function, the set S0 = {λ ∈ R | | trC2(M(λ))| < 2} isan open set and thus a union of open intervals. Moreover, {λ ∈ R | trC2(M(λ)) = 2}(i.e., the set of periodic eigenvalues) and {λ ∈ R | trC2(M(λ)) = −2} (i.e., the setof anti-periodic eigenvalues) are discrete sets without finite accumulation points. Itfollows that S is obtained as the union of the closures of each of the open intervalsconstituting S0, equivalently, S is a union of closed intervals. One notes that theclosure of several disjoint components of S0 may form one closed interval in S.

Applying Lemma 4.4 to the case K =(Tθ

)−1/2one obtains a countable

number of eigenvalues ζn(θ), n ∈ Z\{0} which we may label so that n ζn(θ) > 0.These eigenvalues accumulate at zero (from either side). It is clear that equation(4.30) posed on the interval [0, ω] has a nontrivial solution satisfying the boundaryconditions ψ(ω) = eiθψ(0) and ψ[1](ω) = eiθψ[1](0) precisely when z = 1/ζn(θ) forsome n ∈ Z\{0}. In particular, the endpoints of the conditional stability intervals,which correspond to the values θ = 0 and θ = π, tend to both, +∞ and −∞.


Finally, eigenfunctions u ∈ L2(R; dx) of(T)−1/2

r(T)−1/2

are related to solu-

tions y ∈ H1(R) of (1.7) (with p = 1) via y =(T)−1/2

u. Since the basics of Floquettheory apply to (1.7) (cf. our comments at the beginning of this section and earlierin the current proof), the existence of Floquet multipliers ρ(z) and 1/ρ(z) prevents(1.7) from having an L2(R; dx) (let alone, H1(R)) solution. Hence, the existence of

an eigenfunction u ∈ L2(R; dx) of(T)−1/2

r(T)−1/2

would imply the contradiction

y ∈ H1(R), implying (4.31). �Theorem 4.6 considerably extends prior results by Constantin [30] (see also

[31], [32]) on eigenvalue asymptotics for left-definite periodic Sturm–Liouville prob-lems since no smoothness is assumed on q and r, in addition, q is permitted tobe a distribution and r is extended from merely being a function to a measure.Moreover, it also extends results of Daho and Langer [37], Marletta and Zettl [102],and Philipp [119]: While these authors consider the nonsmooth setting, our resultappears to be the first that permits periodic distributions, respectively, measuresas coefficients.

Remark 4.7. In the special case where the measure dμR(x) = r(x)dx is purelyabsolutely continuous on R, the fact that

T−1/2rT−1/2 � 1

2π

∫ ⊕

[0,2π]

T−1/2θ rT

−1/2θ (4.33)

with respect to the decomposition (4.17), together with continuity of the eigenval-

ues of T−1/2θ rT

−1/2θ with respect to θ, proves that σ(T−1/2rT−1/2) consists of a

union of compact intervals accumulating at 0 unless r = 0 a.e. on (0, ω).Moreover, employing the methods in [65, Sect. 2], Theorem 4.6 (i) immedi-

ately extends to any choice of self-adjoint separated boundary conditions replacingthe θ boundary conditions

g(ω) = eiθg(0), g′(ω) = eiθg′(0), θ ∈ [0, 2π], (4.34)

in A∗θAθ by separated ones of the type

sin(α)g′(0) + cos(α)g(0) = 0,

sin(β)g′(ω) + cos(β)g(ω) = 0, α, β ∈ [0, π].(4.35)

We emphasize that the following Appendices A, B, and C do not contain newresults. We offer them for the convenience of the reader with the goal of providinga fairly self-contained account, enhancing the readability of this manuscript.

Appendix A. Relative boundedness and compactnessof operators and forms

In this appendix we briefly recall the notion of relatively bounded (resp., compact)and relatively form bounded (resp., form compact) perturbations of a self-adjointoperator A in some complex separable Hilbert space H:


Definition A.1.

(i) Suppose that A is a self-adjoint operator in H. A closed operator B in His called relatively bounded (resp., relatively compact ) with respect to A (inshort, B is called A-bounded (resp., A-compact )), if

dom(B) ⊇ dom(A) and

B(A− zIH)−1 ∈ B(H) (resp., ∈ B∞(H)), z ∈ ρ(A).(A.1)

(ii) Assume that A is self-adjoint and bounded from below (i.e., A � cIH forsome c ∈ R). Then a densely defined and closed operator B in H is calledrelatively form bounded (resp., relatively form compact ) with respect to A (inshort, B is called A-form bounded (resp., A-form compact )), if

dom(|B|1/2) ⊇ dom

(|A|1/2) and

|B|1/2((A+ (1− c)IH))−1/2 ∈ B(H) (resp., ∈ B∞(H)).(A.2)

Remark A.2. (i) Using the polar decomposition of B (i.e., B = UB|B|, with UB apartial isometry), one observes that B is A-bounded (resp., A-compact) if and onlyif |B| is A-bounded (resp., A-compact). Similarly, by (A.2), B is A-form bounded(resp., A-form compact), if and only if |B| is.(ii) SinceB is assumed to be closed (in fact, closability of B suffices) in DefinitionA.1 (i), the first condition dom(B) ⊇ dom(A) in (A.1) already implies B(A −zIH)−1 ∈ B(H), z ∈ ρ(A), and hence the A-boundedness of B (cf. again [90,Remark IV.1.5], [135, Theorem 5.9]). By the same token, since A1/2 and |B|1/2 areclosed, the requirement dom

(|B|1/2) ⊇ dom(A1/2

)in Definition A.1 (ii), already

implies that |B|1/2((A + (1 − c)IH))−1/2 ∈ B(H) (cf. [90, Remark IV.1.5], [135,Theorem 5.9]), and hence the first condition in (A.2) suffices in the relatively formbounded context.

(iii) In the special case whereB is self-adjoint, condition (A.2) implies the existenceof α ≥ 0 and β ≥ 0, such that∣∣(|B|1/2f, sgn(B)|B|1/2f)H∣∣ ≤ ∥∥|B|1/2f∥∥2H ≤ α

∥∥|A|1/2f∥∥2H + β‖f‖2H,

f ∈ dom(|A|1/2). (A.3)

(iv) In connection with relative boundedness, (A.1) can be replaced by the condi-tion

dom(B) ⊇ dom(A), and there exist numbers a � 0, b � 0 such that

‖Bf‖H � a‖Af‖H + b‖f‖H for all f ∈ dom(A),(A.4)

or equivalently, by

dom(B) ⊇ dom(A), and there exist numbers a � 0, b � 0 such that

‖Bf‖2H � a2‖Af‖2H + b2‖f‖2H for all f ∈ dom(A).(A.5)


(v) If A is self-adjoint and bounded from below, the number α defined by

α = limμ↑∞

∥∥B(A+ μIH)−1∥∥B(H)

= limμ↑∞

∥∥|B|(A+ μIH)−1∥∥B(H)

(A.6)

equals the greatest lower bound (i.e., the infimum) of the possible values for a in(A.4) (resp., for a in (A.5)). This number α is called the A-bound of B. Similarly,we call

β = limμ↑∞

∥∥|B|1/2(|A|1/2 + μIH)−1∥∥

B(H)(A.7)

the A-form bound of B (resp., |B|). If α = 0 in (A.6) (resp., β = 0 in (A.7)) then Bis called infinitesimally bounded (resp., infinitesimally form bounded ) with respectto A.

We then have the following result:

Theorem A.3. Assume that A � 0 is self-adjoint in H.

(i) Let B be a closed, densely defined operator in H and suppose that dom(B) ⊇dom(A). Then B is A-bounded and hence (A.4) holds for some constantsa ≥ 0, b ≥ 0. In addition, B is also A-form bounded,

|B|1/2(A+ IH)−1/2 ∈ B(H). (A.8)

More specifically,∥∥|B|1/2(A+ IH)−1/2∥∥B(H)

� (a+ b)1/2, (A.9)

and hence, if B is A-bounded with A-bound α strictly less than one, 0 ≤ α < 1(cf. (A.6)), then B is also A-form bounded with A-form bound β strictly lessthan one, 0 ≤ β < 1 (cf. (A.7)). In particular, if B is infinitesimally boundedwith respect to A, then B is infinitesimally form bounded with respect to A.

(ii) Suppose that B is self-adjoint in H, that dom(B) ⊇ dom(A), and hence (A.4)holds for some constants a ≥ 0, b ≥ 0. Then

(A+ IH)−1/2B(A+ IH)−1/2 ∈ B(H), (A.10)∥∥(A+ IH)−1/2B(A+ IH)−1/2∥∥B(H)

� (a+ b). (A.11)

We also recall the following result:

Theorem A.4. Assume that A � 0 is self-adjoint in H.

(i) Let B be a densely defined closed operator in H and suppose that dom(B) ⊇dom(A). In addition, assume that B is A-compact. Then B is also A-formcompact,

|B|1/2(A+ IH)−1/2 ∈ B∞(H). (A.12)

(ii) Suppose that B is self-adjoint in H and that dom(B) ⊇ dom(A). In addition,assume that B is A-compact. Then

(A+ IH)−1/2B(A + IH)−1/2 ∈ B∞(H). (A.13)

For proofs of Theorems A.3 and A.4 under more general conditions on A andB, we refer to [63] and the detailed list of references therein.


Appendix B. Supersymmetric Dirac-type operators in a nutshell

In this appendix we briefly summarize some results on supersymmetric Dirac-typeoperators and commutation methods due to [39], [66], [131], and [132, Ch. 5] (seealso [71]).

The standing assumption in this appendix will be the following.

Hypothesis B.1. Let Hj, j = 1, 2, be separable complex Hilbert spaces and

A : H1 ⊇ dom(A)→ H2 (B.1)

be a densely defined, closed, linear operator.

We define the self-adjoint Dirac-type operator in H1 ⊕H2 by

Q =

(0 A∗

A 0

), dom(Q) = dom(A)⊕ dom(A∗). (B.2)

Operators of the type Q play a role in supersymmetric quantum mechanics (see,e.g., the extensive list of references in [24]). Then,

Q2 =

(A∗A 00 AA∗

)(B.3)

and for notational purposes we also introduce

H1 = A∗A in H1, H2 = AA∗ in H2. (B.4)

In the following, we also need the polar decomposition of A and A∗, that is, therepresentations

A = VA|A| = |A∗|VA = VAA∗VA on dom(A) = dom(|A|), (B.5)

A∗ = VA∗ |A∗| = |A|VA∗ = VA∗AVA∗ on dom(A∗) = dom(|A∗|), (B.6)

|A| = VA∗A = A∗VA = VA∗ |A∗|VA on dom(|A|), (B.7)

|A∗| = VAA∗ = AVA∗ = VA|A|VA∗ on dom(|A∗|), (B.8)

where

|A| = (A∗A)1/2, |A∗| = (AA∗)1/2, VA∗ = (VA)∗, (B.9)

VA∗VA = Pran(|A|) = Pran(A∗) , VAVA∗ = Pran(|A∗|) = Pran(A) . (B.10)

In particular, VA is a partial isometry with initial set ran(|A|) and final set ran(A)

and hence VA∗ is a partial isometry with initial set ran(|A∗|) and final set ran(A∗).In addition,

VA =

{A(A∗A)−1/2 = (AA∗)−1/2A on (ker(A))⊥,0 on ker(A).

(B.11)

Next, we collect some properties relating H1 and H2.


Theorem B.2 ([39]). Assume Hypothesis B.1 and let φ be a bounded Borel mea-surable function on R.

(i) One has

ker(A) = ker(H1) = (ran(A∗))⊥, ker(A∗) = ker(H2) = (ran(A))⊥, (B.12)

VAHn/21 = H

n/22 VA, n ∈ N, VAφ(H1) = φ(H2)VA. (B.13)

(ii) H1 and H2 are essentially isospectral, that is,

σ(H1)\{0} = σ(H2)\{0}, (B.14)

in fact,

A∗A[IH1 − Pker(A)] is unitarily equivalent to AA∗[IH2 − Pker(A∗)]. (B.15)

In addition,

f ∈ dom(H1) and H1f = λ2f, λ �= 0,

implies Af ∈ dom(H2) and H2(Af) = λ2(Af), (B.16)

g ∈ dom(H2) and H2 g = μ2g, μ �= 0,

implies A∗g ∈ dom(H1) and H1(A∗g) = μ2(A∗g), (B.17)

with multiplicities of eigenvalues preserved.(iii) One has for z ∈ ρ(H1) ∩ ρ(H2),

IH2 + z(H2 − zIH2)−1 ⊇ A(H1 − zIH1)

−1A∗, (B.18)

IH1 + z(H1 − zIH1)−1 ⊇ A∗(H2 − zIH2)

−1A, (B.19)

and

A∗φ(H2) ⊇ φ(H1)A∗, Aφ(H1) ⊇ φ(H2)A, (B.20)

VA∗φ(H2) ⊇ φ(H1)VA∗ , VAφ(H1) ⊇ φ(H2)VA. (B.21)

As noted by E. Nelson (unpublished), Theorem B.2 follows from the spectraltheorem and the elementary identities,

Q = VQ|Q| = |Q|VQ, (B.22)

ker(Q) = ker(|Q|) = ker(Q2) = (ran(Q))⊥ = ker(A)⊕ ker(A∗), (B.23)

IH1⊕H2 + z(Q2 − zIH1⊕H2)−1

= Q2(Q2 − zIH1⊕H2)−1 ⊇ Q(Q2 − zIH1⊕H2)

−1Q, z ∈ ρ(Q2),(B.24)

Qφ(Q2) ⊇ φ(Q2)Q, (B.25)

where

VQ =

(0 (VA)

∗

VA 0

)=

(0 VA∗

VA 0

). (B.26)

In particular,

ker(Q) = ker(A) ⊕ ker(A∗), Pker(Q) =

(Pker(A) 0

0 Pker(A∗)

), (B.27)


and we also recall that

S3QS3 = −Q, S3 =

(IH1 00 −IH2

), (B.28)

that is, Q and −Q are unitarily equivalent. (For more details on Nelson’s trick seealso [129, Sect. 8.4], [132, Subsect. 5.2.3].) We also note that

ψ(|Q|) =(ψ(|A|) 0

0 ψ(|A∗|))

(B.29)

for Borel measurable functions ψ on R, and

[Q|Q|−1] =

(0 (VA)

∗

VA 0

)= VQ if ker(Q) = {0}. (B.30)

Finally, we recall the following relationships between Q and Hj , j = 1, 2.

Theorem B.3 ([24], [131]). Assume Hypothesis B.1.

(i) Introducing the unitary operator U on (ker(Q))⊥ by

U = 2−1/2

(IH1 (VA)

∗

−VA IH2

)on (ker(Q))⊥, (B.31)

one infers that

UQU−1 =

(|A| 00 −|A∗|

)on (ker(Q))⊥. (B.32)

(ii) One has

(Q− ζIH1⊕H2)−1 =

(ζ(H1 − ζ2IH1)

−1 A∗(H2 − ζ2IH2)−1

A(H1 − ζ2IH1)−1 ζ(H2 − ζ2IH2)

−1

),

ζ2 ∈ ρ(H1) ∩ ρ(H2).

(B.33)

(iii) In addition,(f1f2

)∈ dom(Q) and Q

(f1f2

)= η

(f1f2

), η �= 0,

implies fj ∈ dom(Hj) and Hjfj = η2fj , j = 1, 2.

(B.34)

Conversely,

f ∈ dom(H1) and H1f = λ2f, λ �= 0,

implies

(f

λ−1Af

)∈ dom(Q) and Q

(f

λ−1Af

)= λ

(f

λ−1Af

).

(B.35)

Similarly,

g ∈ dom(H2) and H2 g = μ2g, μ �= 0,

implies

(μ−1A∗g

g

)∈ dom(Q) and Q

(μ−1A∗g

g

)= μ

(μ−1A∗g

g

).(B.36)


Appendix C. Sesquilinear forms and associated operators

In this appendix we describe a few basic facts on sesquilinear forms and linear op-erators associated with them following [64, Sect. 2]. Let H be a complex separableHilbert space with scalar product ( · , · )H (antilinear in the first and linear in thesecond argument), V a reflexive Banach space continuously and densely embeddedinto H. Then also H embeds continuously and densely into V∗. That is,

V ↪→ H ↪→ V∗. (C.1)

Here the continuous embedding H ↪→ V∗ is accomplished via the identification

H � v �→ ( · , v)H ∈ V∗, (C.2)

and recall our convention in this manuscript that if X denotes a Banach space,X∗ denotes the adjoint space of continuous conjugate linear functionals on X , alsoknown as the conjugate dual of X .

In particular, if the sesquilinear form

V〈 · , · 〉V∗ : V × V∗ → C (C.3)

denotes the duality pairing between V and V∗, then

V〈u, v〉V∗ = (u, v)H, u ∈ V , v ∈ H ↪→ V∗, (C.4)

that is, the V ,V∗ pairing V〈 · , · 〉V∗ is compatible with the scalar product ( · , · )Hin H.

Let T ∈ B(V ,V∗). Since V is reflexive, (V∗)∗ = V , one has

T : V → V∗, T ∗ : V → V∗ (C.5)

and

V〈u, T v〉V∗ = V∗〈T ∗u, v〉(V∗)∗ = V∗〈T ∗u, v〉V = V〈v, T ∗u〉V∗ . (C.6)

Self-adjointness of T is then defined by T = T ∗, that is,

V〈u, T v〉V∗ = V∗〈Tu, v〉V = V〈v, Tu〉V∗ , u, v ∈ V , (C.7)

nonnegativity of T is defined by

V〈u, Tu〉V∗ ≥ 0, u ∈ V , (C.8)

and boundedness from below of T by cT ∈ R is defined by

V〈u, Tu〉V∗ ≥ cT ‖u‖2H, u ∈ V . (C.9)

(By (C.4), this is equivalent to V〈u, Tu〉V∗ ≥ cT V〈u, u〉V∗ , u ∈ V .)Next, let the sesquilinear form a( · , · ) : V ×V → C (antilinear in the first and

linear in the second argument) be V-bounded, that is, there exists a ca > 0 suchthat

|a(u, v)| � ca‖u‖V‖v‖V , u, v ∈ V . (C.10)

Then A defined by

A :

{V → V∗,v �→ Av = a( · , v), (C.11)


satisfies

A ∈ B(V ,V∗) and V⟨u, Av

⟩V∗ = a(u, v), u, v ∈ V . (C.12)

Assuming further that a( · , · ) is symmetric, that is,

a(u, v) = a(v, u), u, v ∈ V , (C.13)

and that a is V-coercive, that is, there exists a constant C0 > 0 such that

a(u, u) ≥ C0‖u‖2V , u ∈ V , (C.14)

respectively, then,

A : V → V∗ is bounded, self-adjoint, and boundedly invertible. (C.15)

Moreover, denoting by A the part of A in H defined by

dom(A) ={u ∈ V | Au ∈ H} ⊆ H, A = A

∣∣dom(A)

: dom(A)→ H, (C.16)

then A is a (possibly unbounded) self-adjoint operator in H satisfying

A ≥ C0IH, (C.17)

dom(A1/2

)= V . (C.18)

In particular,

A−1 ∈ B(H). (C.19)

The facts (C.1)–(C.19) are a consequence of the Lax–Milgram theorem and thesecond representation theorem for symmetric sesquilinear forms. Details can befound, for instance, in [38, Sects. VI.3, VII.1], [53, Ch. IV], and [99].

Next, consider a symmetric form b( · , · ) : V × V → C and assume that b isbounded from below by cb ∈ R, that is,

b(u, u) ≥ cb‖u‖2H, u ∈ V . (C.20)

Introducing the scalar product ( · , · )Vb: V × V → C (and the associated norm

‖ · ‖Vb) by

(u, v)Vb= b(u, v) + (1− cb)(u, v)H, u, v ∈ V , (C.21)

turns V into a pre-Hilbert space (V ; ( · , · )Vb), which we denote by Vb. The form

b is called closed in H if Vb is actually complete, and hence a Hilbert space. Theform b is called closable in H if it has a closed extension. If b is closed in H, then

|b(u, v) + (1− cb)(u, v)H| � ‖u‖Vb‖v‖Vb

, u, v ∈ V , (C.22)

and

|b(u, u) + (1 − cb)‖u‖2H| = ‖u‖2Vb, u ∈ V , (C.23)

show that the form b( · , · ) + (1 − cb)( · , · )H is a symmetric, V-bounded, andV-coercive sesquilinear form. Hence, by (C.11) and (C.12), there exists a linearmap

Bcb :

{Vb → V∗

b ,

v �→ Bcbv = b( · , v) + (1− cb)( · , v)H,(C.24)


withBcb ∈ B(Vb,V∗

b) and

Vb

⟨u, Bcbv

⟩V∗

b

= b(u, v) + (1− cb)(u, v)H, u, v ∈ V , (C.25)

in particular, Bcb is bounded, self-adjoint, and boundedly invertible. Introducingthe linear map

B = Bcb + (cb − 1)I : Vb → V∗b , (C.26)

where I : Vb ↪→ V∗b denotes the continuous inclusion (embedding) map of Vb into

V∗b , B is bounded and self-adjoint, and one obtains a self-adjoint operator B in H

by restricting B to H,

dom(B) ={u ∈ V ∣∣ Bu ∈ H} ⊆ H, B = B

∣∣dom(B)

: dom(B)→ H, (C.27)

satisfying the following properties:

B ≥ cbIH, (C.28)

dom(|B|1/2) = dom

((B − cbIH)1/2

)= V , (C.29)

b(u, v) =(|B|1/2u, UB|B|1/2v

)H (C.30)

=((B − cbIH)1/2u, (B − cbIH)1/2v

)H + cb(u, v)H (C.31)

= Vb

⟨u, Bv

⟩V∗

b

, u, v ∈ V , (C.32)

b(u, v) = (u,Bv)H, u ∈ V , v ∈ dom(B), (C.33)

dom(B) = {v ∈ V | there exists an fv ∈ H such that

b(w, v) = (w, fv)H for all w ∈ V}, (C.34)

Bu = fu, u ∈ dom(B),

dom(B) is dense in H and in Vb. (C.35)

Properties (C.34) and (C.35) uniquely determine B. Here UB in (C.31) is thepartial isometry in the polar decomposition of B, that is,

B = UB|B|, |B| = (B∗B)1/2 ≥ 0. (C.36)

The operator B is called the operator associated with the form b.The norm in the Hilbert space V∗

b is given by

‖�‖V∗b= sup{|V

b〈u, �〉V∗

b| | ‖u‖V

b� 1}, � ∈ V∗

b , (C.37)

with associated scalar product,

(�1, �2)V∗b= V

b

⟨(B + (1− cb)I

)−1�1, �2

⟩V∗

b

, �1, �2 ∈ V∗b . (C.38)

Since ∥∥(B + (1− cb)I)v∥∥V∗

b

= ‖v‖Vb, v ∈ V , (C.39)

the Riesz representation theorem yields(B + (1− cb)I

) ∈ B(Vb,V∗b) and

(B + (1 − cb)I

): Vb → V∗

b is unitary. (C.40)


In addition,

Vb

⟨u,(B + (1− cb)I

)v⟩V∗

b

=((B + (1− cb)IH)1/2u, (B + (1− cb)IH)1/2v

)H

= (u, v)Vb, u, v ∈ Vb. (C.41)

In particular, ∥∥(B + (1− cb)IH)1/2u∥∥H = ‖u‖V

b, u ∈ Vb, (C.42)

and hence

(B + (1− cb)IH)1/2 ∈ B(Vb,H) and (B + (1− cb)IH)1/2 : Vb → H is unitary.(C.43)

The facts (C.20)–(C.43) comprise the second representation theorem of sesquilinearforms (cf. [53, Sect. IV.2], [54, Sects. 1.2–1.5], and [90, Sect. VI.2.6]).

We briefly supplement (C.20)–(C.43) with some considerations that hint at

mapping properties of(B+(1−cb)I

)±1/2on a scale of spaces, which, for simplicity,

we restrict to the triple of spaces Vb, H, and V∗b in this appendix. We start by

defining (Bcb + (1− cb)I

)1/2:

{Vb → H,

v �→ (B + (1 − cb)IH)1/2v,(C.44)

and similarly,

(Bcb + (1− cb)I

)1/2:

⎧⎪⎨⎪⎩H → V∗

b ,

f �→ b( · , (B + (1− cb)IH)−1/2f

)+(1− cb)

( · , (B + (1− cb)IH)−1/2f)H.

(C.45)

Then both maps in (C.44) and (C.45) are bounded and boundedly invertible. Inparticular,(

Bcb + (1− cb)I)1/2 ∈ B(Vb,H),

(Bcb + (1 − cb)I

)−1/2 ∈ B(H,Vb),(Bcb + (1− cb)I

)1/2 ∈ B(H,V∗b),

(Bcb + (1− cb)I

)−1/2 ∈ B(V∗b ,H),

(C.46)

and(Bcb + (1− cb)I

)1/2(Bcb + (1 − cb)I

)1/2=(B + (1 − cb)I

) ∈ B(Vb,V∗b),(

Bcb + (1− cb)I)−1/2(

Bcb + (1 − cb)I)−1/2

=(B + (1− cb)I

)−1 ∈ B(V∗b ,Vb).(C.47)

Due to self-adjointness of B as a bounded map from Vb to V∗b in the sense of (C.7),

one finally obtains that((Bcb + (1− cb)I

)±1/2)∗

=(Bcb + (1− cb)I

)±1/2,((

Bcb + (1− cb)I)±1/2

)∗=(Bcb + (1− cb)I

)±1/2.

(C.48)


Hence, we will follow standard practice in connection with chains of (Sobolev)spaces and refrain from painstakingly distinguishing the ˆ- and ˇ-operations andsimply resort to the notation (

B + (1 − cb)I)±1/2

(C.49)

for the operators in (C.46) in the bulk of this paper.A special but important case of nonnegative closed forms is obtained as fol-

lows: Let Hj , j = 1, 2, be complex separable Hilbert spaces, and T : dom(T ) →H2, dom(T ) ⊆ H1, a densely defined operator. Consider the nonnegative formaT : dom(T )× dom(T )→ C defined by

aT (u, v) = (Tu, T v)H2, u, v ∈ dom(T ). (C.50)

Then the form aT is closed (resp., closable) in H1 if and only if T is. If T isclosed, the unique nonnegative self-adjoint operator associated with aT in H1,whose existence is guaranteed by the second representation theorem for forms,then equals T ∗T ≥ 0. In particular, one obtains in addition to (C.50),

aT (u, v) = (|T |u, |T |v)H1, u, v ∈ dom(T ) = dom(|T |). (C.51)

Moreover, since

b(u, v) + (1− cb)(u, v)H =((B + (1− cb)IH)1/2u, (B + (1− cb)IH)1/2v

)H,

u, v ∈ dom(b) = dom(|B|1/2) = V ,

(C.52)

and (B + (1 − cb)IH)1/2 is self-adjoint (and hence closed) in H, a symmetric, V-bounded, and V-coercive form is densely defined in H×H and closed in H (a factwe will be using in the proof of Theorem 2.3). We refer to [90, Sect. VI.2.4] and[135, Sect. 5.5] for details.

Next we recall that if aj are sesquilinear forms defined on dom(aj), j = 1, 2,bounded from below and closed, then also

(a1 + a2) :

{(dom(a1) ∩ dom(a2))× (dom(a1) ∩ dom(a2))→ C,

(u, v) �→ (a1 + a2)(u, v) = a1(u, v) + a2(u, v)(C.53)

is bounded from below and closed (cf., e.g., [90, Sect. VI.1.6]).

Finally, we also recall the following perturbation theoretic fact: Suppose a isa sesquilinear form defined on V × V , bounded from below and closed, and let bbe a symmetric sesquilinear form bounded with respect to a with bound less thanone, that is, dom(b) ⊇ V ×V , and that there exist 0 � α < 1 and β � 0 such that

|b(u, u)| � α|a(u, u)|+ β‖u‖2H, u ∈ V . (C.54)

Then

(a+ b) :

{V × V → C,

(u, v) �→ (a+ b)(u, v) = a(u, v) + b(u, v)(C.55)


defines a sesquilinear form that is bounded from below and closed (cf., e.g., [90,Sect. VI.1.6]). In the special case where α can be chosen arbitrarily small, the formb is called infinitesimally form bounded with respect to a.

Acknowledgment

We gratefully acknowledge valuable correspondence with Rostyslav Hryniv, MarkMalamud, Roger Nichols, Fritz Philipp, Barry Simon, Gunter Stolz, and GeraldTeschl. In addition, we are indebted to Igor Verbitsky for helpful discussions.Finally, we sincerely thank the anonymous referee for numerous helpful commentsimproving the presentation of our results.

References

[1] R.A. Adams and J.J.F. Fournier, Sobolev Spaces, second edition, Academic Press,2003.

[2] S. Alama, M. Avellaneda, P.A. Deift, and R. Hempel, On the existence of eigenval-ues of a divergence-form operator A + λB in a gap of σ(A), Asymptotic Anal. 8,311–344 (1994).

[3] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models inQuantum Mechanics, with an Appendix by P. Exner, 2nd ed., AMS Chelsea Pub-lishing, Providence, RI, 2005.

[4] S. Albeverio, A. Kostenko, and M. Malamud, Spectral theory of semibounded Sturm–Liouville operators with local interactions on a discrete set, J. Math. Phys. 51,102102 (2010), 24 pp.

[5] T.G. Anderson and D.B. Hinton, Relative boundedness and compactness theory forsecond-order differential operators, J. Inequal. & Appl. 1, 375–400 (1997).

[6] F.V. Atkinson and A.B. Mingarelli, Asymptotics of the number of zeros and of theeigenvalues of general weighted Sturm–Liouville problems, J. reine angew. Math.375/376, 380–393 (1987).

[7] J.-G. Bak and A.A. Shkalikov, Multipliers in dual Sobolev spaces and Schrodingeroperators with distribution potentials, Math. Notes 71, 587–594 (2002).

[8] R. Beals, Indefinite Sturm–Liouville problems and half-range completeness, J. Diff.Eq. 56, 391–407 (1985).

[9] R. Beals, D.H. Sattinger, and J. Szmigielski, Multipeakons and the classical momentproblem, Adv. Math. 154, 229–257 (2000).

[10] R. Beals, D.H. Sattinger, and J. Szmigielski, Periodic peakons and Calogero–Francoise flows, J. Inst. Math. Jussieu 4, No. 1, 1–27 (2005).

[11] J. Behrndt, On the spectral theory of singular indefinite Sturm–Liouville operators,J. Math. Anal. Appl. 334, 1439–1449 (2007).

[12] J. Behrndt, Spectral theory of elliptic differential operators with indefinite weights,Proc. Roy. Soc. Edinburgh 143A, 21–38 (2013).

[13] J. Behrndt, Q. Katatbeh, and C. Trunk, Non-real eigenvalues of singular indefiniteSturm–Liouville operators, Proc. Amer. Math. Soc. 137, 3797–3806 (2009).


[14] J. Behrndt, R. Mows, and C. Trunk, Eigenvalue estimates for singular left-definiteSturm–Liouville operators, J. Spectral Th. 1, 327–347 (2011).

[15] J. Behrndt and F. Philipp, Spectral analysis of singular ordinary differential oper-ators with indefinite weights, J. Diff. Eq. 248, 2015–2037 (2010).

[16] J. Behrndt and C. Trunk, On the negative squares of indefinite Sturm–Liouvilleoperators, J. Diff. Eq. 238, 491–519 (2007).

[17] C. Bennewitz, On the spectral problem associated with the Camassa–Holm equation,J. Nonlinear Math. Phys. 11, 422–434 (2004).

[18] C. Bennewitz, B. M. Brown, and R. Weikard, Inverse spectral and scattering theoryfor the half-line left-definite Sturm–Liouville problem, SIAM J. Math. Anal. 40,2105–2131 (2009).

[19] C. Bennewitz, B.M. Brown, and R. Weikard, A uniqueness result for one-dimensional inverse scattering, Math. Nachr. 285, 941–948 (2012).

[20] C. Bennewitz, B.M. Brown, and R. Weikard, Scattering and inverse scattering fora left-definite Sturm–Liouville problem, J. Diff. Eq. 253, 2380–2419 (2012).

[21] C. Bennewitz and W.N. Everitt, On second-order left-definite boundary value prob-lems, in Ordinary Differential Equations and Operators, (Proceedings, Dundee,1982), W.N. Everitt and R.T. Lewis (eds.), Lecture Notes in Math., Vol. 1032,Springer, Berlin, 1983, pp. 31–67.

[22] A.M. Berthier, Spectral Theory and Wave Operators for the Schrodinger Equation,Research Notes in Mathematics, Vol. 71, Pitman, Boston, 1982.

[23] P. Binding and A. Fleige, Conditions for an indefinite Sturm–Liouville Riesz ba-sis property, in Recent Advances in Operator Theory in Hilbert and Krein Spaces,J. Behrndt, K.-H. Forster, and C. Trunk (eds.), Operator Theory: Advances andApplications, Birkhauser, Basel, Vol. 198, 2009, pp. 87–95.

[24] D. Bolle, F. Gesztesy, H. Grosse, W. Schweiger, and B. Simon, Witten index, axialanomaly, and Krein’s spectral shift function in supersymmetric quantum mechanics,J. Math. Phys. 28, 1512–1525 (1987).

[25] B.M. Brown, M.S.P. Eastham, and K.M. Schmidt, Periodic Differential Operators,Birkhauser, 2013.

[26] R.C. Brown and D.B. Hinton, Relative form boundedness and compactness for asecond-order differential operator, J. Comp. Appl. Math. 171, 123–140 (2004).

[27] R. Carlone, M. Malamud, and A. Posilicano, On the spectral theory of Gesztesy–Seba realizations of 1-D Dirac operators with point interactions on a discrete set,J. Diff. Eq. 254, 3835–3902 (2013).

[28] S. Clark and F. Gesztesy, Weyl–Titchmarsh M-function asymptotics and Borg-typetheorems for Dirac operators, Trans. Amer. Math. Soc. 354, 3475–3534 (2002).

[29] S. Clark and F. Gesztesy, On Povzner–Wienholtz-type self-adjointness results formatrix-valued Sturm–Liouville operators, Proc. Roy. Soc. Edinburgh 133A, 747–758(2003).

[30] A. Constantin, A general-weighted Sturm–Liouville problem, Scuola Norm. Sup. 24,767–782 (1997).

[31] A. Constantin, On the spectral problem for the periodic Camassa–Holm equation,J. Math. Anal. Appl. 210, 215–230 (1997).


[32] A. Constantin, On the inverse spectral problem for the Camassa–Holm equation, J.Funct. Anal. 155, 352–363(1998).

[33] A. Constantin, On the scattering problem for the Camassa–Holm equation, Proc.Roy. Soc. London A 457, 953–970 (2001).

[34] A. Constantin, V.S. Gerdjikov, and R.I. Rossen, Inverse scattering transform forthe Camassa–Holm equation, Inverse Probl. 22, 2197–2207 (2006).

[35] A. Constantin and J. Lenells, On the inverse scattering approach to the Camassa–Holm equation, J. Nonlinear Math. Phys. 10, 252–255 (2003).

[36] A. Constantin and H.P. McKean, A shallow water equation on the circle, Commun.Pure Appl. Math. 52, 949–982 (1999).

[37] K. Daho and H. Langer, Sturm–Liouville operators with an indefinite weight func-tion: The Periodic case, Radcvi Mat. 2, 165–188 (1986).

[38] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods forScience and Technology, Volume 2, Functional and Variational Methods, Springer,Berlin, 2000.

[39] P.A. Deift, Applications of a commutation formula, Duke Math. J. 45, 267–310(1978).

[40] P. Djakov and B. Mityagin, Multiplicities of the eigenvalues of periodic Dirac op-erators, J. Diff. Eq. 210, 178–216 (2005).

[41] P. Djakov and B. Mityagin, Spectral gap asymptotics of one-dimensional Schrodin-ger operators with singular periodic potentials, Integral Transforms Special Fcts. 20,nos. 3-4, 265–273 (2009).

[42] P. Djakov and B. Mityagin, Spectral gaps of Schrodinger operators with periodicsingular potentials, Dyn. PDE 6, no. 2, 95–165 (2009).

[43] P. Djakov and B. Mityagin, Fourier method for one-dimensional Schrodinger oper-ators with singular periodic potentials, in Topics in Operator Theory, Vol. 2: Sys-tems and Mathematical Physics, J.A. Ball, V. Bolotnikov, J.W. Helton, L. Rod-man, I.M. Spitkovsky (eds.), Operator Theory: Advances and Applications, Vol.203, Birkhauser, Basel, 2010, pp. 195–236.

[44] P. Djakov and B. Mityagin, Criteria for existence of Riesz bases consisting of rootfunctions of Hill and 1d Dirac operators, J. Funct. Anal. 263, 2300–2332 (2012).

[45] N. Dunford and J.T. Schwartz, Linear Operators Part II: Spectral Theory, Inter-science, New York, 1988.

[46] M.S.P. Eastham, The Spectral Theory of Periodic Differential Equations, ScottishAcademic Press, Edinburgh and London, 1973.

[47] J. Eckhardt, Direct and inverse spectral theory of singular left-definite Sturm–Liouville operators, J. Diff. Eq. 253, 604–634 (2012).

[48] J. Eckhardt, F. Gesztesy, R. Nichols, and G. Teschl, Supersymmetry and Schro-dinger-type operators with distributional matrix-valued potentials, arXiv:1206.4966,J. Spectral Theory, to appear.

[49] J. Eckhardt, F. Gesztesy, R. Nichols, and G. Teschl, Weyl–Titchmarsh theory forSturm–Liouville operators with distributional potentials, Opuscula Math. 33, 467–563 (2013).


[50] J. Eckhardt, F. Gesztesy, R. Nichols, and G. Teschl, Inverse spectral theory forSturm–Liouville operators with distributional coefficients, J. London Math. Soc. (2)88, 801–828 (2013).

[51] J. Eckhardt and G. Teschl, On the isospectral problem of the dispersionlessCamassa–Holm equation, Adv. Math. 235, 469–495 (2013).

[52] J. Eckhardt and G. Teschl, Sturm–Liouville operators with measure-valued coeffi-cients, J. Analyse Math. 120, 151–224 (2013).

[53] D.E. Edmunds andW.D. Evans, Spectral Theory and Differential Operators, Claren-don Press, Oxford, 1989.

[54] W.G. Faris, Self-Adjoint Operators, Lecture Notes in Mathematics, Vol. 433,Springer, Berlin, 1975.

[55] J. Fleckinger and M.L. Lapidus, Eigenvalues of elliptic boundary value problemswith an indefinite weight function, Trans. Amer. Math. Soc. 295, 305–324 (1986).

[56] G. Freiling, V. Rykhlov, and V. Yurko, Spectral analysis for an indefinite singularSturm–Liouville problem, Appl. Anal. 81, 1283–1305 (2002).

[57] F. Gesztesy, On the modified Korteweg-de Vries equation, in Differential Equationswith Applications in Biology, Physics, and Engineering, J.A. Goldstein, F. Kappel,and W. Schappacher (eds.), Marcel Dekker, New York, 1991, pp. 139–183.

[58] F. Gesztesy, Quasi-periodic, finite-gap solutions of the modified Korteweg-de Vriesequation, in Ideas and Methods in Mathematical Analysis, Stochastics, and Appli-cations, S. Albeverio, J.E. Fenstad, H. Holden, and T. Lindstrøm (eds.), Vol. 1,Cambridge Univ. Press, Cambridge, 1992, pp. 428–471.

[59] F. Gesztesy, A complete spectral characterizaton of the double commutation method,J. Funct. Anal. 117, 401–446 (1993).

[60] F. Gesztesy and H. Holden, Algebro-geometric solutions of the Camassa–Holm hi-erarchy, Rev. Mat. Iberoamericana 19, 73–142 (2003).

[61] F. Gesztesy and H. Holden, Soliton Equations and Their Algebro-Geometric Solu-tions. Vol. I: (1 + 1)-Dimensional Continuous Models, Cambridge Studies in Ad-vanced Mathematics, Vol. 79, Cambridge Univ. Press, 2003.

[62] F. Gesztesy and H. Holden, Real-valued algebro-geometric solutions of the Camassa–Holm hierarchy, Phil. Trans. Roy. Soc. A 366, 1025–1054 (2008).

[63] F. Gesztesy, M. Malamud, M. Mitrea, and S. Naboko, Generalized polar decompo-sitions for closed operators in Hilbert spaces and some applications, Integral Eq.Operator Th. 64, 83–113 (2009).

[64] F. Gesztesy, M. Mitrea. Nonlocal Robin Laplacians and some remarks on a paperby Filonov on eigenvalue inequalities. J. Diff. Eq. 247, 2871–2896 (2009).

[65] F. Gesztesy and R. Nichols, Weak convergence of spectral shift functions for one-dimensional Schrodinger operators, Math. Nachr. 285, 1799–1838 (2012).

[66] F. Gesztesy, W. Schweiger, and B. Simon, Commutation methods applied to themKdV -equation, Trans. Amer. Math. Soc. 324, 465–525 (1991).

[67] F. Gesztesy and R. Svirsky, (m)KdV -Solitons on the background of quasi-periodicfinite-gap solutions, Memoirs Amer. Math. Soc. 118 (563), 1–88 (1995).

[68] I.M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differ-ential Operators, Israel Program for Scientific Translations, Jerusalem, 1965.


[69] T. Godoy, J.-P. Gossez, S. Paczka, On the asymptotic behavior of the principaleigenvalues of some elliptic problems, Ann. Mat. Pura Appl. 189, 497–521 (2010).

[70] G. Grubb, Distributions and Operators, Graduate Texts in Mathematics, Vol. 252,Springer, New York, 2009.

[71] V. Hardt, A. Konstantinov, and R. Mennicken, On the spectrum of the product ofclosed operators, Math. Nachr. 215, 91–102 (2000).

[72] P. Hartman Differential equations with non-oscillatory eigenfunctions, Duke Math.J. 15, 697–709 (1948).

[73] R. Hempel, A left-definite generalized eigenvalue problem for Schrodinger operators,Habilitation, Dept. of Mathematics, University of Munich, Germany, 1987.

[74] P. Hess, On the relative completeness of the generalized eigenvectors of elliptic eigen-value problems with indefinite weight functions, Math. Ann. 270, 467–475 (1985).

[75] P. Hess, On the asymptotic distribution of eigenvalues of some nonselfadjoint prob-lems, Bull. London Math. Soc. 18, 181–184 (1986).

[76] P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with anindefinite weight function, Commun. Partial Diff. Eq. 5, 999–1030 (1980).

[77] D. B. Hinton and S.C. Melescue, Relative boundedness-compactness inequalities fora second order differential operator, Math. Ineq. & Appls. 4, 35–52 (2001).

[78] R.O. Hryniv and Ya.V. Mykytyuk, 1D Schrodinger operators with periodic singularpotentials, Methods Funct. Anal. Topology 7, no. 4, 31–42 (2001).

[79] R.O. Hryniv and Ya.V. Mykytyuk, 1D Schrodinger operators with singular Gordonpotentials, Methods Funct. Anal. Topology 8, no. 1, 36–48 (2002).

[80] R.O. Hryniv and Ya.V. Mykytyuk, Self-adjointness of Schrodinger operators withsingular potentials, Meth. Funct. Anal. Topology 18, 152–159 (2012).

[81] B.J. Jaye, V.G. Maz’ya, and I.E. Verbitsky, Existence and regularity of positivesolutions of elliptic equations of Schrodinger type, J. Analyse Math. 118, 577–621(2012).

[82] T. Kappeler and C. Mohr, Estimates for periodic and Dirichlet eigenvalues of theSchrodinger operator with singular potentials, J. Funct. Anal. 186, 62–91 (2001).

[83] T. Kappeler, P. Perry, M. Shubin, and P. Topalov, The Miura map on the line, Int.Math. Res. Notices, 2005, No. 50.

[84] T. Kappeler and P. Topalov, Global fold structure of the Miura map on L2(T), Int.Math. Res. Notices 2004, No. 39, 2039–2068.

[85] I.M. Karabash, A functional model, eigenvalues, and finite singular critical pointsfor indefinite Sturm–Liouville operators, in Topics in Operator Theory. Volume 2.Systems and Mathematical Physics, J.A. Ball, V. Bolotnikov, J.W. Helton, L. Rod-man, I.M. Spitkovsky (eds.), Operator Theory: Advances and Applications, Vol.203, Birkhauser, Basel, 2010, pp. 247–287.

[86] I.M. Karabash and M.M. Malamud, Indefinite Sturm–Liouville operators (sgn x)(− d2

dx2 + q(x))

with finite-zone potentials, Operators and Matrices 1, 301–368

(2007).

[87] I.M. Karabash, A.S. Kostenko, and M.M. Malamud, The similarity problem forJ-nonnegative Sturm–Liouville operators, J. Diff. Eq. 246, 964–997 (2009).


[88] I. Karabash and C. Trunk, Spectral properties of singular Sturm–Liouville operatorswith indefinite weight sgn x, Proc. Roy. Soc. Edinburgh 139A, 483–503 (2009).

[89] M. Kato, Estimates for the eigenvalues of Hill’s operator with distributional coeffi-cients, Tokyo J. Math. 33, 361–364 (2010).

[90] T. Kato, Perturbation Theory for Linear Operators, corr. printing of the 2nd ed.,Springer, Berlin, 1980.

[91] Q. Kong, H. Wu, and A. Zettl, Left-definite Sturm–Liouville problems, J. Diff. Eq.177, 1–26 (2001).

[92] Q. Kong, H. Wu, and A. Zettl, Singular left-definite Sturm–Liouville problems, J.Diff. Eq. 206, 1–29 (2004).

[93] Q. Kong, H. Wu, A. Zettl, and M. Moller, Indefinite Sturm–Liouville problems,Proc. Roy. Soc. Edinburgh 133A, 639–652 (2003).

[94] E. Korotyaev, Inverse problem for periodic “weighted” operators, J. Funct. Anal.170, 188–218 (2000).

[95] E. Korotyaev, Characterization of the spectrum of Schrodinger operators with peri-odic distributions, Int. Math. Res. Notices 2003, No. 37, 2019–2031.

[96] E. Korotyaev, Inverse spectral problem for the periodic Camassa–Holm equation, J.Nonlinear Math. Phys. 11, 499–507 (2004).

[97] E. Korotyaev, Sharp asymptotics of the quasimomentum, Asymptot. Anal. 80, 269–287 (2012).

[98] A. Kostenko, The similarity problem for indefinite Sturm–Liouville operators withperiodic coefficients, Operators and Matrices 5, 707–722 (2011).

[99] J.L. Lions, Espaces d’interpolation et domaines de puissances fractionnaires d’ope-rateurs, J. Math. Soc. Japan 14, 233–241 (1962).

[100] B.V. Loginov and O.V. Makeeva, The pseudoperturbation method in generalizedeigenvalue problems, Dokl. Math. 77, 194–197 (2008).

[101] M. Marletta and A. Zettl, Counting and computing eigenvalues of left-definiteSturm–Liouville problems, J. Comp. Appl. Math. 148, 65–75 (2002).

[102] M. Marletta and A. Zettl, Floquet theory for left-definite Sturm–Liouville problems,J. Math. Anal. Appl. 305, 477–482 (2005).

[103] V.G. Maz’ya and T.O. Shaposhnikova, Theory of Sobolev Multipliers, Springer,Berlin, 2009.

[104] V.G. Maz’ya and I.E. Verbitsky, Boundedness and compactness criteria for the one-dimensional Schrodinger operator, in Function Spaces, Interpolation Theory andRelated Topics, M. Cwikel, M. Englis, A. Kufner, L.-E. Persson, G. Sparr (eds.), deGruyter, Berlin, 2002, pp. 369–382.

[105] V.G. Maz’ya and I.E. Verbitsky, The Schrodinger operator on the energy space:boundedness and compactness criteria, Acta Math. 188, 263–302 (2002).

[106] V.G. Maz’ya and I.E. Verbitsky, Infinitesimal form boundedness and Trudinger’ssubordination for the Schrodinger operator, Invent. Math. 162, 81–136 (2005).

[107] V.G. Maz’ya and I.E. Verbitsky, Form boundedness of the general second-orderdifferential operator, Commun. Pure Appl. Math. 59, 1286–1329 (2006).

[108] H.P. McKean, Addition for the acoustic equation, Commun. Pure Appl. Math. 54,1271–1288 (2001).


[109] H.P. McKean, Fredholm determinants and the Camassa–Holm hierarchy, Commun.Pure Appl. Math. 56, 638–680 (2003).

[110] H.P. McKean, The Liouville correspondence between the Korteweg–de Vries and theCamassa–Holm hierarchies, Commun. Pure Appl. Math. 56, 998–1015 (2003).

[111] H. P. McKean, Breakdown of the Camassa–Holm equation, Commun. Pure Appl.Math. 57, 416–418 (2004).

[112] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, CambridgeUniversity Press, Cambridge, 2000.

[113] V.A. Mikhailets and V.M. Molyboga, One-dimensional Schrodinger operators withsingular periodic potentials, Meth. Funct. Anal. Topology 14, no. 2, 184–200 (2008).

[114] V.A. Mikhailets and V.M. Molyboga, Spectral gaps of the one-dimensional Schro-dinger operators with singular periodic potentials, Meth. Funct. Anal. Topology 15,no. 1, 31–40 (2009).

[115] E. Muller-Pfeifer, Spectral Theory of Ordinary Differential Operators, Ellis Hor-wood, Chichester, 1981.

[116] M.A. Naimark, Linear Differential Operators, Part II, Ungar, New York, 1968.

[117] M.I. Neiman-zade and A.A. Shkalikov, Strongly elliptic operators with singular co-efficients, Russ. J. Math. Phys. 13, 70–78 (2006).

[118] W.V. Petryshyn, On the eigenvalue problem Tu − λSu = 0 with unbounded andnonsymmetric operators T and S, Phil. Trans. Roy. Soc. London A 262, 413–458(1968).

[119] F. Philipp, Indefinite Sturm–Liouville operators with periodic coefficients, Operatorsand Matrices 7, 777–811 (2013).

[120] M. Reed and B. Simon, Methods of Modern Mathematical Physics. III: ScatteringTheory, Academic Press, New York, 1979.

[121] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV: Analysis ofOperators, Academic Press, New York, 1978.

[122] F. Rellich, Halbbeschrankte gewohnliche Differentialoperatoren zweiter Ordnung,Math. Ann. 122, 343–368 (1951).

[123] A.M Savchuk and A.A. Shkalikov, Sturm–Liouville operators with singular poten-tials, Math. Notes 66, 741–753 (1999).

[124] A.M. Savchuk and A.A. Shkalikov, Sturm–Liouville operators with distribution po-tentials, Trans. Moscow Math. Soc. 2003, 143–192.

[125] M. Schechter,Operator Methods in Quantum Mechanics, North–Holland, New York,1981.

[126] M. Schechter, Principles of Functional Analysis, 2nd ed., Graduate Studies in Math-ematics, Vol. 36, Amer. Math. Soc., Providence, RI, 2002.

[127] B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms,Princeton University Press, Princeton, NJ, 1971.

[128] B. Simon, Trace Ideals and Their Applications, 2nd ed., Mathematical Surveys andMonographs, Vol. 120, Amer. Math. Soc., Providence, RI, 2005.

[129] G. Teschl, Mathematical Methods in Quantum Mechanics; With Applications toSchrodinger Operators, Graduate Studies in Mathematics, Amer. Math. Soc., Vol.99, RI, 2009.


[130] G. Teschl, private communication.

[131] B. Thaller, Normal forms of an abstract Dirac operator and applications to scatter-ing theory, J. Math. Phys. 29, 249–257 (1988).

[132] B. Thaller, The Dirac Equation, Springer, Berlin, 1992.

[133] C. Tretter, Linear operator pencils A − λB with discrete spectrum, Integral Eq.Operator Th. 37, 357–373 (2000).

[134] H. Volkmer, Sturm–Liouville problems with indefinite weights and Everitt’s inequal-ity, Proc. Roy. Soc. Edinburgh 126A, 1097–1112 (1996).

[135] J. Weidmann, Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics,Vol. 68, Springer, New York, 1980.

[136] J. Weidmann, Spectral Theory of Ordinary Differential Operators, Lecture Notes inMath., Vol. 1258, Springer, Berlin, 1987.

[137] A. Zettl, Sturm–Liouville Theory, Mathematical Surveys and Monographs, Vol. 121,Amer. Math. Soc., Providence, RI, 2005.

[138] D.R. Yafaev, On the spectrum of the perturbed polyharmonic operator, in Topics inMathematical Physics, Vol. 5. Spectral Theory, M.Sh. Birman (ed.), ConsultantsBureau, New York, 1972, pp. 107–112.

Fritz GesztesyDepartment of MathematicsUniversity of MissouriColumbia, MO 65211, USAe-mail: [email protected]: http://www.math.missouri.edu/personnel/faculty/gesztesyf.html

Rudi WeikardDepartment of MathematicsUniversity of Alabama at BirminghamBirmingham, AL 35294, USAe-mail: [email protected]: http://www.math.uab.edu/∼rudi/


Remarks on Spaces of Compact Operatorsbetween Reflexive Banach Spaces

G. Godefroy

Abstract. We observe that if X and Y are two reflexive separable spaces suchthat the canonical map J : X⊗Y → X⊗Y is injective, then every compactoperator from X to Y ∗ is in the norm closure of finite rank operators, andevery bounded operator T ∈ L(X, Y ∗) is uniform limit on compact sets ofa sequence (Rn) of finite rank operators such that ‖Rn‖ ≤ ‖T‖. This wouldapply in particular to the case X = Y , i.e., to a reflexive Pisier space ifsuch a space exists. We show that if Z ⊂ L(X) is a subspace which strictlycontains the space K(X) of compact operators on a reflexive Banach spaceX, then K(X) is not 1-complemented in Z, and it is locally 1-complementedin Z exactly when Z is contained in the closure of K(X) with respect tothe uniform convergence on compact subsets of X. Several consequences arespelled out.

Mathematics Subject Classification (2010). 46A32, 46B20.

Keywords. Spaces of compact operators, approximation properties, smoothnorms.

1. Introduction

The purpose of this note is to apply some general results from geometry of Banachspaces (mainly, from duality and isometric theory) to the space of compact oper-ators between reflexive Banach spaces on the real field. In Section 2, we considerpairs of reflexive Banach spaces such that the canonical map from the projectivetensor product to the injective tensor product is one-to-one. We show that in thiscase, the corresponding spaces of operators behave “as if” X or Y has the ap-proximation property, although it is not always so. Quite unexpectedly, James’techniques on norm-attaining linear functionals can be applied in this context (see[9] for an early use of such arguments). The results of Section 3 rely on the use ofthe Frechet smoothness of the norm of L(X) at certain operators of rank 1, in thespirit of [8] and [18]. The main result of this section is Lemma 3.2 which connects


190 G. Godefroy

the metric position of an operator T ∈ L(X) with respect to the space K(X) andthe uniform approximation of T on compact sets by compact operators.

2. Injectivity of the canonical map between tensor products

We start with the following lemma, whose roots go back to [15]. We refer to [4]for various applications of Petunin-Plichko’s result.

Lemma 2.1. Let X and Y be separable reflexive Banach spaces. Then (X⊗Y )∗ =L(X,Y ∗) and the restriction map to K(X,Y ∗) ⊂ L(X,Y ∗) defines a quotient mapQ from X⊗Y onto K(X,Y ∗)∗.

Proof. It is classical that (X⊗Y )∗ = L(X,Y ∗) and this isometric identificationcan actually be used as definition of the projective tensor product. Since X isreflexive, any operator T ∈ K(X,Y ∗) attains its norm, and it easily follows thatK(X,Y ∗) consists of norm-attaining elements of (X⊗Y )∗. The spaces K(X,Y ∗)and X⊗Y are separable. The Lemma is therefore a special case of ([2], Lemma2.5), which simply follows from the fact that the restriction to K(X,Y ∗) of theunit sphere of X⊗Y is a separable James boundary (in the sense of [5]) of the unitball of K(X,Y ∗)∗. �

It is interesting to notice that this quotient map, which we obtain here withvery general arguments relying ultimately on James’ theorem [10] and Simons’inequality [17], can also be derived with an algebraic approach, adapted to theparticular Banach spaces under consideration: indeed it is this same quotient mapQ which appears in the commutative diagram of [8] (see the proof of Proposition1.1 there), relying on the previous work [3].

We now prove the following:

Theorem 2.2. If X and Y are separable reflexive spaces such that the canonical mapJ : X⊗Y → X⊗Y is injective, then every compact operator from X to Y ∗ is in thenorm closure of finite rank operators, and every bounded operator T ∈ L(X,Y ∗)is uniform limit on compact sets of a sequence (Rn) of finite rank operators suchthat ‖Rn‖ ≤ ‖T ‖.Proof. Since the map J is one-to-one, the simple tensors x∗⊗y∗ separate the spaceX⊗Y and thus their linear span X∗ ⊗ Y ∗ ⊂ K(X,Y ∗) separate X⊗Y . It followsnow from Lemma 2.1 that Q is a canonical isometry from X⊗Y onto K(X,Y ∗)∗,and that the space of compact operators from X to Y ∗ is the norm-closure of thespace of finite rank operators, in other words that K(X,Y ∗) = X∗⊗Y ∗. Now wecan dualize the equation K(X,Y ∗)∗ = X⊗Y and get K(X,Y ∗)∗∗ = (X⊗Y )∗ =L(X,Y ∗).

If X and Y are reflexive, the space K(X,Y ∗)∗∗ consists of all operatorsT ∈ L(X,Y ∗) which are uniform limits on compact sets of compact operators ([8],Cor. 1.2) and the control of the norm follows from ([8], Theorem 1.5). Finally wecan replace compact operators by finite rank ones since K(X,Y ∗) = X∗⊗Y ∗. �

Spaces of Compact Operators 191

The simplest case where Theorem 2.2 applies is when X or Y has the ap-proximation property (see [12], page 3). In this case, Theorem 2.2 is well knownand it goes back to Grothendieck’s fundamental works. When Y = X∗, the mapJ is one-to-one (if and) only if X has the approximation property (see [12], The-orem 0.3) and this is a special case of Theorem 2.2. However, the assumptions ofTheorem 2.2 are also satisfied when X and Y have type 2 (see [13], Theorem 6.6).Hence, if for instance X is a subspace of Lp (2 < p < ∞) failing A.P., then thespaces K(X,X∗) and L(X,X∗) behave “as if” X or X∗ has the A.P. although itis not so, but factorization does the job in this case.

Along these lines, we recall that it is not known whether there exists a re-flexive Pisier space, that is, a reflexive space X such that X⊗X = X⊗X , bothalgebraically and topologically, in other words such that J is one-to-one and onto.Such a space X , if it exists, fails the approximation property ([14]) but satis-fies of course the conclusions of Theorem 2.2. Actually, dualizing the equationX⊗X = X⊗X shows that every bounded operator from X to X∗ is nuclear.

3. Isometric properties of the space of compact operators

Our next observation is an extension of ([8], Remark 5.8) and shows that thereflexivity of K(X) is irrelevant to such isometric considerations (see Question5.9 in [8]). We state it in the case of reflexive spaces, but the argument can beextended to any Asplund space X with the Radon-Nikodym property.

Proposition 3.1. Let X be a reflexive Banach space, and let Z be a subspace ofL(X) which strictly contains K(X). Then K(X) is not 1-complemented in Z.

Proof. It suffices to show that if T ∈ L(X) is a non-compact operator, then thespace K(X) is not 1-complemented in the space ET = K(X)⊕RT .

Let x (resp. x∗) be strongly exposed in the unit ball of X (resp. X∗) byx∗0 ∈ BX∗ (resp. by x0 ∈ BX). Then x⊗ x∗ is strongly exposed in the unit ball of

X⊗X∗ by x∗0 ⊗ x0 ([16] or [8], Lemma 5.1). It follows that x∗

0 ⊗ x0 ∈ L(X) is apoint of Frechet smoothness of the norm of L(X), with tangent linear form x⊗x∗.

Assume that π : ET → K(X) is a linear projection with norm 1, and letT0 = π(T ). For any S ∈ K(X) and any scalar λ, one has ‖S + λ(T − T0)‖ ≥ ‖S‖.When S = x∗

0⊗x0, this inequality and smoothness of the norm shows the equation〈x⊗x∗, T −T0〉 = 0 and thus 〈x∗, T (x)−T0(x)〉 = 0. Since this last equation holdsfor all strongly exposed points x and x∗ and the unit ball of any reflexive space isthe norm closed convex hull of its strongly exposed points, it follows that T = T0,but this contradicts T �∈ K(X). �

In view of the above statement, it is natural to ask when K(X) is locally1-complemented in Z. We recall that Y is locally 1-complemented in Z if forevery finite-dimensional subspace F of Z and any ε > 0, there is a linear operatorL : F → Y with ‖L‖ ≤ 1+ ε and L(y) = y for all y ∈ Y ∩F , and this holds if andonly if Y ⊥ is the kernel of a contractive projection on Z∗ ([11]). In the notation

192 G. Godefroy

of [7], this means that Y is an ideal in Z, in the Banach space sense since thisterminology does not request the presence of an algebraic structure. The followinglemma, where we use the notation of Proposition 3.1, answers our question. Itsproof is relevant to the ball topology which is defined and studied in [6].

Lemma 3.2. Let X be a reflexive Banach space, and let Z be a subspace of L(X)which contains K(X). Then the space K(X) is locally 1-complemented in Z ifand only if Z is contained in the closure of K(X) with respect to the uniformconvergence on compact subsets of X.

Proof. For T ∈ L(X), we denote as before ET = K(X)⊕RT .By Corollary 1.2 in [8], the space K(X)∗∗ is canonically isometric to the

closure of K(X) in the space L(X) equipped with the topology of uniform con-vergence on compact subsets of X . If Z is a subspace of K(X)∗∗, it follows fromthe local reflexivity principle that K(X) is locally 1-complemented in Z. We notein passing that the local reflexivity principle provides operators L which are closeto being isometries, and also that if K(X) is locally complemented in ET and ES ,it is locally complemented in span[ET ∪ ES ].

Conversely, let us assume that K(X) is locally 1-complemented in ET . Forany finite-dimensional subspace F ⊂ ET with T ∈ F and any ε > 0, there is alinear operator LF,ε : F → K(X) with ‖LF,ε‖ ≤ 1+ ε and LF,ε = Id on F ∩K(X).We set LF,ε(T ) = SF,ε. The natural order (F, ε) ≤ (G, δ) when F ⊂ G and δ ≤ εdefines a filter F on the set of pairs (F, ε).

We use the notation of the proof of Proposition 3.1. The operator x∗0 ⊗ x0 ∈

K(X) is a point of Frechet smoothness of the norm of L(X), with tangent linearform x⊗ x∗. For any operator S ∈ ET , we have therefore that

〈x∗, S(x)〉 = limt→0

t−1[‖x∗0 ⊗ x0 + tS‖ − 1].

Frechet smoothness means that this limit is uniform on bounded subsets ofET . Pick now S ∈ K(X) and t > 0. It follows from the properties of LF,ε that

lim supF

t−1[‖x∗0 ⊗ x0 + tSF,ε‖ − 1] ≤ t−1[‖x∗

0 ⊗ x0 + tT ‖ − 1].

Since the set (SF,ε) is uniformly bounded, we can take the limit when t→ 0+

and we getlim sup

F〈x∗, SF,ε(x)〉 ≤ 〈x∗, T (x)〉

but since we can reproduce the argument with −(x∗ ⊗ x), it follows that

limF〈x∗, SF,ε(x)〉 = 〈x∗, T (x)〉.

This limit holds if x∗ and x are strongly exposed in their respective unit balls.Since the unit ball of any reflexive Banach space is the closed convex hull of itsstrongly exposed points, it follows that T = limF SF,ε in the weak operator topol-ogy. Since the weak operator topology and the topology of uniform convergenceon compact subsets of X have the same closed convex bounded sets, the resultfollows. �

Spaces of Compact Operators 193

Note that by the proof of Lemma 3.2, the metric properties of the net which isobtained from T by local complementation (e.g. by the local reflexivity principle)actually imply weak* convergence to T . This behaviour reflects the smoothness ofthe Banach space K(X).

In what follows, we identify K(X)∗∗ with the closure of K(X) in L(X)equipped with the topology of uniform convergence on compact subsets of X (see[8], Cor. 1.2).

Lemma 3.3. Let X be a reflexive Banach space. The space K(X)∗∗ is a two-sidedideal of the algebra L(X), which coincide with L(X) if and only if X has thecompact approximation property.

Proof. The product of L(X) is separately continuous for the compact convergence,and it follows that K(X)∗∗ is a two-sided ideal in L(X). This ideal coincide withL(X) if and only if it contains IX , and this exactly means that X has the compactapproximation property. �

Let us gather in a single statement what we know about this ideal.

Theorem 3.4. Let X be a reflexive Banach space. The following statements areequivalent:

(1) X has the compact approximation property.(2) K(X)∗∗ = L(X).(3) K(X) is locally 1-complemented in L(X).(4) K(X) is locally 1-complemented in K(X)⊕R IX .(5) K(X)∗∗ contains an invertible operator.

Proof. It follows from Lemma 3.2 that (4) implies (1), and from Lemma 3.3 that(5) implies (2). The other implications are clear. �

We can summarize our observations as follows: the collection of spaces Zsuch that K(X) ⊂ Z ⊂ L(X) and K(X) is locally 1-complemented in Z (in otherwords, such that K(X) is an ideal in Z in the Banach space sense) admits amaximal space, namely K(X)∗∗, and this maximal space is a two-sided ideal inthe algebra L(X). Note that by Lemma 3.3, if K(X) is a maximal closed two-sidedideal in the algebra L(X), exactly one of the following properties holds: X has thecompact approximation property, or K(X) is reflexive. And by Lemma 3.2, thereis no subspace Z ⊂ L(X) strictly containing K(X) in which K(X) is an idealin the Banach space sense (i.e., in which K(X) is locally 1-complemented) if andonly if K(X) is reflexive.

It is not known whether there exists an infinite-dimensional Banach spacesuch that K(X) is reflexive. Note that a reflexive space X with the Argyros–Haydon property L(X) = K(X) ⊕R IX and failing the compact approximationproperty would be such that K(X) is reflexive. It is not known at present whethera reflexive space can have the Argyros–Haydon property, however reflexive spaceswhich enjoy the invariant subspace property hereditarily have been constructed [1].

194 G. Godefroy

References

[1] S. Argyros and P. Motakis, A reflexive space with the hereditary invariant subspaceproperty, to appear.

[2] P. Bandyopadhyay and G. Godefroy, Linear structures in the set of norm-attainingfunctionals on a Banach space, J. of Convex Anal. 13 (2006), 489–497.

[3] M. Feder and P.D. Saphar, Spaces of compact operators and their dual spaces, IsraelJ. Math. 21 (1975), 38–49.

[4] G. Godefroy, The use of norm attainment, Bulletin of the Belgian Math. Society, toappear.

[5] G. Godefroy, Boundaries of a convex set and interpolation sets, Math. Annalen 277(1987), 173–184.

[6] G. Godefroy and N.J. Kalton, The ball topology and its applications, Contemp. Math.85, Amer. Math. Soc. (1989), 195–237.

[7] G. Godefroy, N.J. Kalton and P.D. Saphar, Unconditional ideals in Banach spaces,Studia Math. 104, 1 (1993), 13–59.

[8] G. Godefroy and P.D. Saphar, Duality in spaces of operators and smooth norms onBanach spaces, Illinois J. of Math. 32,4 (1988), 672–695.

[9] J.R. Holub, Reflexivity of L(E,F ), Proceedings of the Amer. Math. Soc. 39 (1973),175–177.

[10] R.C. James, Weakly compact sets, Trans. Amer. Math. Soc. 113 (1964), 129–140.

[11] N.J. Kalton, Locally complemented spaces and Lp-spaces for 0 < p < 1, Math. Nachr.115 (1984), 71–97.

[12] G. Pisier, Factorization of linear operators and geometry of Banach spaces, CBMSregional conference series in mathematics 60 (1986).

[13] G. Pisier, The operator Hilbert space OH, complex interpolation and tensor norms,Memoir of the AMS. 122 (1996), no. 585.

[14] G. Pisier, Un theoreme sur les operateurs entre espaces de Banach qui se factorisentpar un espace de Hilbert, Annales Scient. Ecole Norm. Sup. 13 (1980), 23–43.

[15] Y. I, Petunin and A.N. Plichko, Some properties of the set of functionals that attaina supremum on the unit sphere, Ukrain. Mat. Z. 26 (1974), 102–106.

[16] W. Ruess and C. Stegall, Exposed and denting points in duals of operator spaces,Israel J. Math. 53, 2 (1986), 163–190.

[17] S. Simons, A convergence theorem with boundary, Pacific J. Math. 40 (1972), 703–708.

[18] W. Werner, The type of a factor with separable predual is determined by its geometry,in Interaction between functional analysis, harmonic analysis and probability, editedby N.J. Kalton, S. Montgomery-Smith and E. Saab, M. Dekker Lecture notes 175(1996).

G. GodefroyCNRS-Universite Paris 6Institut de Mathematiques de Jussieu-Paris Rive GaucheCase 247. 4, Place JussieuF-75252 Paris Cedex 05, Francee-mail: [email protected]


Harmonic Analysis and StochasticPartial Differential Equations:The Stochastic Functional Calculus

Brian Jefferies

Abstract. It has been recognised recently that there is a close connection be-tween existence and regularity results for stochastic partial differential equa-tions and functional calculus techniques in harmonic analysis. The connectionis made more explicit in this paper with the notion of a stochastic functionalcalculus.

In the deterministic setting, suppose that A1, A2 are bounded linearoperators acting on a Banach space E. A pair (μ1, μ2) of continuous proba-bility measures on [0, 1] determines a functional calculus f −→ fμ1,μ2(A1, A2)for analytic functions f by weighting all possible orderings of operator prod-ucts of A1 and A2 via the probability measures μ1 and μ2. For example,f −→ fμ,μ(A1, A2) is the Weyl functional calculus with equally weighted op-erator products.

Replacing μ1 by Lebesgue measure λ on [0, t] and μ2 by stochastic in-tegration with respect to a Wiener process W , we show that there exists afunctional calculus f −→ fλ,W ;t(A+B) for bounded holomorphic functions fif A is a densely defined Hilbert space operator with a bounded holomorphicfunctional calculus and B is small compared to A relative to a square func-tion norm. By this means, the solution of the stochastic evolution equationdXt = AXtdt+BXtdWt, X0 = x, is represented as t −→ eA+B

λ,W ;tx, t ≥ 0. Weshow how to extend some of our results to Lp-spaces, 2 ≤ p < ∞ and applythem to the regularity of solutions of the Zakai equation.

Mathematics Subject Classification (2010). Primary 47A60; Secondary 47D06,60H15.

Keywords. functional calculus, stochastic evolution equation, H∞-functionalcalculus, square function, Feynman’s operational calculus.

The author would like to thank Z. Brzezniak for invaluable discussions.


196 B. Jefferies

1. Introduction

In a recent series of papers by J. van Neerven, M. Veraar and L. Weis, harmonicanalysis techniques have been used to establish stochastic maximal regularity inLp-spaces [22] for p ≥ 2. Combined with fixed point arguments, the existence,uniqueness and regularity results for solutions to general nonlinear stochastic PDEsare obtained from stochastic maximal regularity estimates.

It is already apparent from the papers of F. Flandoli [6] and Z. Brzezniak[1] that square function estimates facilitate the use of fixed point arguments forthe solution of stochastic PDEs in Hilbert space. On the other hand, in harmonicanalysis, the work of A. McIntosh and A. Yagi [19] showed that square functionestimates in Hilbert space determine the existence of anH∞-functional calculus forsectorial operators, which can be used to solve irregular boundary value problemsin Hilbert space.

In the semigroup approach to linear evolution problems, the solution of theequation

du(t)

dt= Au(t) +Bu(t), u(0) = u0,

for a function u : [0,∞) → X with values in a Banach space X is sought in theform

u(t) = et(A+B)u0, t ≥ 0,

where the linear operator B is distinguished as a lower-order perturbation of A. Inthe case that X is Hilbert space and A+B is selfadjoint with a spectral measurePA+B, then

et(A+B) =

∫R

etλ dPA+B(λ)

and the operator A+B has a rich functional calculus associated with the spectralmeasure PA+B. Similarly, in the stochastic setting, we find that the solution t �−→eA+Bdt,dWt;t

x of the stochastic evolution equation

dXt = AXt dt+BXt dWt, X0 = x, (1.1)

is defined and there is an H∞-functional calculus

f �−→ fλ,W ;t(A+B) (1.2)

for A + B. The relevant properties are that A should have an H∞-functionalcalculus and B should be small compared to A relative to a “square functionnorm”.

The notation fλ,W ;t(A+B) is inspired by Feynman’s operational calculus inwhich operator ordering in a functional calculus is indexed by continuous measures[11]. For example, if μ1 and μ2 are continuous Borel probability measures on [0, 1],A1, A2 are bounded linear operators and P 1,1(x1, x2) = x1x2 for x1, x2 ∈ R, then

P 1,1μ1,μ2

(A1, A2) = (μ1 ⊗ μ2)({t2 < t1})A1A2 + (μ1 ⊗ μ2)({t1 < t2})A2A1. (1.3)

Feynman’s idea was to attach time indices to each operator so that in operatorexpressions, operators with smaller time indices act before those with larger time

The Stochastic Functional Calculus 197

indices, as in formula (1.3) above. If μ1 = μ2 is the Lebesgue measure λ on theinterval [0, t], t > 0, then

eA1+A2

λ,λ;t = et(A1+A2).

The expression eA1+A2

λ,λ;t is shorthand for fλ,λ;t(A1, A2), where f is the exponential

function f(x1, x2) = ex1+x2 , x1, x2 ∈ R, in two real variables. More generally, ifμ = μ1 = μ2 is any continuous Radon measure on R, then

eA1+A2μ,μ;t = eμ([0,t])(A1+A2)

for every t > 0, because the linear operators A1, A2 are equally weighted by themeasure μ in the time-ordering of operator products [12, Proposition 5.5]. Therelation of the mapping (1.2) with Feynman’s operational calculus is explainedmore fully in [10]. In the case that A and B are bounded linear operators actingon an arbitrary Banach space E, there exists an operator-valued random evolutiont �−→ eA+B

λ,W ;t, t ≥ 0, such that Xt = eA+Bλ,W ;tx, t ≥ 0, is a strong solution of (1.1) [10,

Corollary 4.8]. For a Hilbert space E = H, the Ito isometry gives the bound [10,Theorem 5.1]

‖eA+Bλ,W ;tx‖L2(P,H) ≤ ‖x‖.‖etA‖

∞∑n=0

(t12 ‖B‖)n√

n!, t ≥ 0. (1.4)

In the setting of equation (1.2) the operators A and B are generally closedand unbounded operators on the Banach space in which we expect solutions tolie, so expressions like formula (1.3) are problematic. Nevertheless, in the nota-tion fdt,dWt;t(A+B), we take the time-ordering measure associated with A to beLebesgue measure dt and the time-ordering with respect to B is given by stochas-tic integration with respect to Brownian motion Ws, s ≥ 0 on the interval [0, t]for t > 0 – more (or least!) colourfully, we have time-ordering of B with respectto white noise dWt.

A systematic study of the existence, uniqueness and regularity of solutionsof parabolic stochastic evolution equations in UMD Banach spaces that includesequation (1.1) as a special case is given in [2, 21]. The emphasis here is on makingsense of expressions like fdt,dWt;t(A + B) for bounded holomorphic functions fand studying the joint functional calculus properties of A and B in the stochasticsetting related to Feynman’s operational calculus.

The Zakai equation arising in filtering theory is a typical evolution equationthat possesses a stochastic functional calculus, where in equation (1.1), we have

Au(x) =

d∑i,j=1

aij(x)∂2u

∂xi∂xj+

d∑i=1

qi(x)∂u

∂xi+ r(x)u(x), x ∈ Rd,

Bu(x) =

d∑i=1

bi(x)∂u

∂xi+ c(x)u(x), x ∈ Rd.

The existence, uniqueness and regularity of solutions of the Zakai equation in UMDBanach spaces is treated in [2].

198 B. Jefferies

The present paper builds on [10]. Multiple stochastic integration of Hilbertspace-valued functions is reviewed in Section 2. Stochastic equations in Banachspaces are discussed in Section 3 with emphasis on the van Neerven–Veraar–Weisanalysis. The construction of the exponential stochastic exponential eA+B

λ,W ;t re-quires norm estimates for the multiple stochastic integrals of Banach space-valuedfunctions and we discuss such estimates for the case of M -type 2 Banach spacesin Section 4. The case of Lp-space for p ≥ 2 is covered. The main result, Theo-rem 5.4 gives the construction of the stochastic functional calculus in M -type 2Banach spaces. Applications to the Zakai equation are given in Hilbert space inTheorem 5.7.

2. Multiple stochastic integrals

The perturbation series expansion for eA+Bdt,dWt;t

is written in terms of multiplestochastic integrals with respect to the Brownian motion process, which we nowdefine.

Let W denote Brownian motion in R with respect to the probability measurespace (Ω,S,P) such that W0 = 0 almost surely. In the case that Ω is taken to bethe set of all continuous functions ω : [0,∞) → R, the σ-algebra S is the Borelσ-algebra of Ω for the compact-open topology and Wt(ω) = ω(t) for every ω ∈ Ωand t ≥ 0. There exists a unique Borel probability measure P on Ω – the Wienermeasure, such that for every 0 < t1 < · · · < tk, Borel subsets B1, . . . , Bk of R andk = 1, 2, . . . , the measure of the elementary event

E = {ω ∈ Ω : ω(t1) ∈ B1, . . . , ω(tk) ∈ Bk}is given by

P(E) =

∫Bk

· · ·∫B1

ptk−tk−1(xk − xk−1) · · · pt2−t1(x2 − x1)pt1(x1) dx1 . . . dxk,

where pt(x) = (2πt)−12 e−x2/(2t), t > 0, x ∈ R, is the associated transition function.

Then Wiener measure P has the property that Wt, t ≥ 0, is a process with station-ary and independent increments such that Wt is a Gaussian random variable withmean zero and variance t for t > 0, properties which define a Brownian motionWt, t ≥ 0, with W0 = 0 P-a.e. over a general probability measure space (Ω,S,P).

For a Banach space E and 1 ≤ p < ∞, the space of E-valued pth-Bochnerintegrable functions with respect to P is denoted by Lp(P, E) = Lp(Ω,S,P, E). Thelinear space L0(P, E) = L0(Ω,S,P, E) of equivalence classes of strongly measurableE-valued functions has the (metrisable) topology of convergence in probability.

For the purpose of expanding solutions of linear stochastic equations like (1.1)as a “stochastic Dyson series”, we need to consider multiple Wiener–Ito integrals ofdeterministic functions. We follow the account in [17, Section 10.3] with suitablemodifications for vector-valued functions. Wiener–Ito chaos in Banach spaces istreated in [18, Section 4].


Let H be a Hilbert space with inner product 〈·, ·〉H . Let T > 0 and k =1, 2, . . . . The case k = 1 corresponds to the Wiener integral. Let D1 = (0, T ] and

Dk = {(t1, . . . , tk) ∈ (0, T ]k : ∃i, j = 1, . . . , k, i �= j, such that ti = tj},k = 2, 3, . . . . Let A1, . . . , An be a partition of (0, T ] into disjoint intervals of theform (s, t] for 0 ≤ s < t ≤ T and suppose that

f =∑

1≤j1,...,jk≤n

αj1,...,jkχAj1×···×Ajk

(2.1)

is an H-valued function such that αj1,...,jk = 0 whenever two indices j1, . . . , jk areequal and f vanishes on Dk. Then

Ik(f) =

∫[0,T ]k

f(t1, . . . , tk) dWt1 . . . dWtk

is defined by

Ik(f) =∑

1≤j1,...,jk≤n

αj1,...,jkW (Aj1) · · ·W (Ajk).

Here W ((s, t]) denotes the random variable Wt − Ws for 0 ≤ s < t ≤ T . LetD((0, T ]k, H) denote the linear space of H-valued step functions f of the aboveform. Then Ik is well defined and Ik : D((0, T ]k, H) → L0(Ω,S,P, H) is a linearmap. Moreover, the maps Ik, k = 1, 2, . . . , enjoy the following properties.

1) The integral Ik(f) is invariant under the symmetrisation of the function f ,

that is, if f ∈ D((0, T ]k, H) is the symmetrisation

f(t1, . . . , tk) =1

k!

∑σ∈Sk

f(tσ(1), . . . , tσ(k)), t1, . . . , tk ∈ (0, T ]

of f ∈ D((0, T ]k, H) over the set Sk of all permutations of (1, . . . , k), then

Ik(f) = Ik(f).2) If k and k′ are positive integers such that k �= k′ and f ∈ D((0, T ]k, H),

g ∈ D((0, T ]k′, H), then E(〈Ik(f), Ik′ (g)〉H) = 0.

3) If f ∈ D((0, T ]k, H) and g ∈ D((0, T ]k, H), then

E(〈Ik(f), Ik(g)〉H) = k!〈f , g〉L2((0,T ]k,H).

The inner product on the right-hand side is taken in the Hilbert spaceL2((0, T ]k, H).

By property 3), we have a version of the Ito isometry

E(‖Ik(f)‖2H) = E(‖Ik(f)‖2H) = k!‖f‖2L2((0,T ]k,H) ≤ k!‖f‖2L2((0,T ]k,H), (2.2)

so that the mapping Ik can be extended to a bounded linear operator

Ik : L2((0, T ]k, H)→ L2(Ω,S,P, H).

200 B. Jefferies

We also write Ik(f) as∫[0,T ]k

f(s)W k(ds). In the case that 0 ≤ s < t ≤ T and

f ∈ L2((0, T ]k, H) is zero off the simplex

Δk(s, t) = {(s1, . . . , sk) ∈ [s, t]k : s < s1 < · · · < sk < t},then

Ik(f) =

∫ t

s

∫ tk

s

· · ·∫ t2

s

f(t1, . . . , tk) dWt1 · · · dWtk , (2.3)

where the right-hand side is interpreted as an iterated stochastic integral [17,pp. 299–300]. The equality is easily seen to be valid for all f ∈ D((0, T ]k, H)vanishing off Δk(s, t) and the linear subspace of all such functions is dense in theclosed subspace of L2((0, T ]k, H) consisting of all H-valued functions belonging toL2((0, T ]k, H) which are zero almost everywhere outside Δk(s, t) ⊂ (0, T ]k. TheIto isometry (2.2) for the integral (2.3) takes the form

E(‖Ik(f)‖2H) =

∫ t

s

∫ tk

s

· · ·∫ t2

s

‖f(t1, . . . , tk)‖2H dt1 · · · dtk. (2.4)

To check that the identity (2.4) is valid, we write

fσ(t1, . . . , tk) = f(tσ(1), . . . , tσ(k)), for t1, . . . , tk ∈ (0, T ] and σ ∈ Sk.

Then for σ, σ′ ∈ Sk, σ �= σ′, the functions fσ and fσ′ are supported by disjointopen simplexes in (0, T ]k, so 〈fσ, fσ′〉L2((0,T ]k,H) = 0 and we have

E(‖Ik(f)‖2H) = E(‖Ik(f)‖2H) [by property 1)],

= k!‖f‖2L2((0,T ]k,H), [by property 3)]

=1

k!

∥∥∥∥ ∑σ∈Sk

fσ

∥∥∥∥2L2((0,T ]k,H)

= ‖f‖2L2((0,T ]k,H),

[because ‖fσ‖L2((0,T ]k,H) = ‖f‖L2((0,T ]k,H) for σ ∈ Sk].

The calculation for a constant function supported by Δk(0, t) is instructive.Using Ito’s formula to compute

∫Δk(t)

W k(ds1, . . . , dsk) for k = 1, 2, . . . , we have∫ t

0

W (ds1) = Wt, [k = 1]∫ t

0

∫ s2

0

W (ds1)W (ds2) =

∫ t

0

Ws2 W (ds2)

=1

2W 2

t −1

2t, [k = 2]∫ t

0

∫ s3

0

∫ s2

0

W (ds1)W (ds2)W (ds3) =

∫ t

0

(1

2W 2

s3 −1

2s3

)W (ds3)

=1

3!W 3

t −1

2tWt, [k = 3]


...∫Δk(t)

W k(ds1, . . . , dsk) =1

k!hk(Wt/

√t)tk/2,

where hn(x) = (−1)nex2/2dn/dxne−x2/2, x ∈ R, is Hermite polynomial of degreen = 0, 1, 2, . . . , see [17, Theorem 10.3.2].

Note that by symmetry, the equality∫Δk(σ;t)

W k(ds1, . . . , dsk) =

∫Δk(t)

W k(ds1, . . . , dsk)

holds for each σ ∈ Sk with χΔk(σ;t) :=

(χΔk(t)

)σ, so by equation (2.2) the equality∥∥∥∥∥

∫Δk(t)

W k(ds1, . . . , dsk)

∥∥∥∥∥2

L2(P)

=tk

k!.

holds for each k = 1, 2 . . . . This may also be obtained by applying the Ito isometryconsecutively to the representation (2.3) or observing that

‖hk(Wt/√t)‖2L2(P) =

1√2π

∫R

hk(x)2e−x2/2 dx = k!.

Let T > 0. Every element F of L2(P, H) has a unique expansion (Wienerpolynomial chaos) as the sum of E(F ) and multiple stochastic integrals Ik(fk) ofsymmetric functions fk : (0, T ]k → H , k = 1, 2, . . . [17, Theorem 10.3.3].

3. Stochastic equations in Banach spaces

A comprehensive treatment of stochastic integration of Banach space-valued deter-ministic functions appears in [20]. Muliple Wiener–Ito integrals for Banach space-valued functions are treated in [18, Section 3]. A full treatment requires a discus-sion of γ-radonifying operators and their tensor products. In some situations it ispossible to get by with simpler arguments which we now describe.

3.1. Stochastic integration of vector-valued functions

We first mention some terminology related to stochastic integration. Let R+ =[0,∞).

Let (Ω,F ,P) be a probability measure space. A filtration is a family {Ft : t ∈R+} of sub σ-algebras of F such that Fs ⊆ Ft, ∀s < t. A filtration {Ft : t ∈ R+}is called a standard filtration if

(1) Ft = Ft+ := ∩s>tFs ∀t (right continuity)(2) F0 contains all the P-null sets (completeness)

Given an increasing family {Ft : t ∈ R+} of σ-algebras, a process X : R× Ω→ Cis adapted to Ft or progressively measurable if Xt is Ft measurable for all t ∈ R+.

Let Wt, t ≥ 0, be a Brownian motion process on the probability space(Ω,F ,P).

202 B. Jefferies

Definition 3.1. Let E be a Banach space. An E-valued random process Φt, t ≥ 0,is said to be stochastically integrable in E, if for each ξ ∈ E′, the scalar-valuedprocess 〈Φt, ξ〉, t ≥ 0 is stochastically integrable with respect to Wt, t ≥ 0, andthere exists an E-valued random process Ψt, t ≥ 0, such that

〈Ψt, ξ〉 =∫ t

0

〈Φs, ξ〉 dWs a.e. (3.1)

for every ξ ∈ E′ and t ≥ 0. We sometimes write

Ψ = Φ.W

in accordance with the notion that the integral Φ.W of a vector-valued processΦ with espect to W ought to be another vector-valued process. If M is an semi-martingale, then Φ.M should be a weak semimartingale, that is, 〈Φ, ξ〉.M is asemimartingale for each ξ ∈ E′.

Remark 3.2. It can happen that a Pettis integrable vector-valued random variabledoes not possess a conditional expectation with respect to a sub-σ-algebra [8], sothere is a distinction between weak and strong semimartingales for vector-valuedprocesses, even in infinite-dimensional Hilbert space, see [9] for a discussion ofconditional expectation of Pettis integrable vector-valued random variables.

Let T > 0 and k = 1, 2, . . . . An E-valued function s �−→ Φs, s ∈ [0, T ]k, issaid to be k-stochastically integrable or W k-integrable in E if for each ξ ∈ E′, thescalar-valued function t �−→ 〈Φs, ξ〉, s ∈ [0, T ]k belongs to L2([0, T ]k), and thereexists an E-valued random process Ψt, t ∈ [0, T ], such that

〈Ψt, ξ〉 =∫[0,t]k

〈Φs, ξ〉W k(ds1, . . . , dsk) a.e. (3.2)

for every ξ ∈ E′ and t ≥ 0. We shall mainly be concerned with E-valued functionsof the form Φs = χΔk(T )(s)f(s) for s ∈ [0, T ]k.

If a deterministic function φ : (0, T )→ E is stochastically integrable in E andit is weakly L2, it follows that for every Borel subset A of (0, T ), there exists anE-valued Gaussian random variable XA such that

〈XA, ξ〉 =∫ T

0

χA(t)〈φ(t), ξ〉 dWt

for every ξ ∈ E′ [20]: it suffices that an E-valued random variable X(0,T ) exists.

3.2. The van Neerven–Veraar–Weis approach to the stochastic integrationof vector-valued functions

For a given Banach space E, we want to find conditions to integrate an E-valuedprocess Φ with respect to a semimartingale or just a Brownian motion processW . Bilinear stochastic equations in Banach spaces with respect to bounded linearoperators A and B are treated in [10, Corollary 4.8] by a projective tensor product.For Hilbert spaces, the Ito isometry is used, see [10, Theorem 5.1].


In order to solve and find estimates for the solution of a stochastic PDE, weneed to treat unbounded linear operators on a Banach space, in which case theargument of [10, Theorem 5.1] fails. In [21], van Neerven, Weis and Veraar showhow this is done in UMD Banach spaces, such as Lp spaces for 1 < p < ∞. Avariety of bilinear stochastic PDE are solved in [2].

Let E be a real Banach space and let H be a separable Hilbert space. Acontinuous real linear map T : H → E is said to be γ-radonifying if the standardGaussian cylindrical measure γ is mapped by T into the restriction to cylindersets of a regular Borel probabilty measure (Radon probability) on E. If H hasan orthonormal basis 〈en〉∞n=1 and ϕJ (h) = 〈(h, en)〉n∈J for any finite subset J ofpositive integers, then γ ◦ ϕ−1

J is the standard Gaussian probability measure onϕJ (H).

Many continuous linear maps are γ-radonifying. As mentioned in [10, Remark4.4], nuclear maps are γ-radonifying, as are absolutely summing maps [28]. If E isa Hilbert space, then T is γ-radonifying iff T is a Hilbert–Schmidt operator iff T isabsolutely summing. The mapping T is γ-radonifying iff h �−→ ‖Th‖E, h ∈ H , isa measurable seminorm on H in the sense of L. Gross [15]. The standard exampleis the map

T : h �−→∫

h dt, h ∈ L2([0, T ])

with values in C0([0, T ]). Here the absolutely continuous function∫h dt ∈ C0([0, T ])

is the indefinite integral of h vanishing at t = 0 and γ ◦T−1 is Wiener measure onall continuous sample paths ω : [0, T ]→ R satisfying ω(0) = 0. As is well known,γ ◦ T−1 is concentrated on all Holder continuous paths of order 0 < α < 1

2 .

Now suppose that E is an arbitrary Banach space. We say that the E-valuedfunction Φ : [0, T ]→ E is stochastically integrable (in the sense of [21]) if the linearmap

T : h �−→∫ T

0

Φ(t)h(t) dt, h ∈ L2([0, T ]),

is γ-radonifying in E. The vector-valued integral here is a Pettis integral. ThenΦ.W is the L2(P, E)-valued process defined by

〈(Φ.W )(t), ξ〉 =∫ t

0

〈Φ(s), ξ〉 dWs, ξ ∈ E∗, t ≥ 0.

If T is γ-radonifying, and 〈fn〉∞n=1 is an orthonormal basis of L2([0, T ]) then∫A

〈Φ(s), ξ〉 dWs =

∫ T

0

∞∑n=1

(∫A

〈Φ(s), ξ〉fn(s) ds)fn(t) dWt

converges in L2(P) for each ξ ∈ E∗ and Borel set A ⊆ [0, T ]. The sum

∞∑n=1

(∫A

〈Φ(s), ξ〉.fn(s) ds

)∫ T

0

fn(t) dWt

204 B. Jefferies

is an E-valued Gaussian series, see [20, Theorem 2.3], where the consistency withDefinition 3.1 is also shown. For each t > 0, let Ft be the σ-algebra generated bythe random variables {Ws : 0 ≤ s ≤ t }.

A function φ : R+ → L2(P) ⊗ E is said to be an elementary progressivelymeasurable function if there exist times 0 < t1 < · · · < tN , vectors xmn ∈ E andsets Amn ∈ Ftn−1 , n = 1, . . . , N , m = 1, . . . ,M such that

φ(t) =

N∑n=1

M∑m=1

xmnχAmn

.χ(tn−1,tn]

(t), t ∈ R+.

Then φ has values in every space Lp(P) ⊗ E for 1 ≤ p ≤ ∞, φ is W -integrable inLp(P)⊗ E ⊗ Lp(P) for every 1 ≤ p <∞ and we have∫

R+

φ⊗ dW =

N∑n=1

M∑m=1

(xmnχAmn

)⊗ (Wtn −Wtn−1). (3.3)

Let E denote the linear subspace of L∞(P)⊗E⊗Lp(P) consisting of all vectors∫R+

φ ⊗ dW with φ : R+ → L∞(P) ⊗ E an elementary progressively measurable

function. For each 1 ≤ p < ∞, let J : L∞(P) ⊗ E ⊗ Lp(P) → Lp(P, E) be thelinear map defined by J(g ⊗ x ⊗ f)(ω) = xg(ω).f(ω) for almost all ω ∈ Ω. Themap J multiplies the Lp function f by the bounded function g and leaves x ∈ Eunchanged, that is, J is a bilinear multiplication operator.

Definition 3.3. A Banach space E is called a UMD space (or, E has the uncondi-tional martingale difference property) if for any 1 < p < ∞, there exists Cp > 0such that for any E-valued martingale difference {ξj}nj=1 and n = 1, 2, . . . , theinequality

E

∥∥∥∥ n∑j=1

εjξj

∥∥∥∥pE

≤ CpE

∥∥∥∥ n∑j=1

ξj

∥∥∥∥pE

holds for every εj ∈ {±1}, j = 1, . . . , n. By a martingale difference sequence{ξj}nj=1, we mean that the sum

ξ1 +

k∑j=1

ξj , k = 1, . . . , n,

is an E-valued martingale.

The following result is from [7, Theorems 2 and 2’].

Theorem 3.4. Let E be a UMD space and 1 < p < ∞. The multiplication map Jis continuous from E into Lp(P, E) for the relative topology of Lp(P⊗ P, E) on E.

By this means we can prove that elements of a wide class of E-valued pro-cesses are W -integrable in the sense of Definition 3.1, provided that E has theUMD property, see [21]. In many examples, such as [10, Theorem 4.7], the UMDproperty is not needed.


4. The stochastic Dyson series in M-type 2 Banach spaces

Although the theory of stochastic integration is well developed in UMD Banachspaces, we shall employ a one-sided Ito inequality valid in M-type 2 Banach spacessuch as Lp-spaces with p ≥ 2, in which stochastic maximal regularity is valid [22].

Let 1 ≤ p ≤ 2. A Banach space E is said to be of type p if there exists C > 0such that

E

∥∥∥∥ n∑j=1

εjxj

∥∥∥∥pE

≤ Cpn∑

j=1

‖xj‖pE

for any symmetric identical independently distributed random variables ε1, . . . , εnwith values ±1, vectors x1, . . . , xn ∈ E and n = 1, 2, . . . . The smallest C with thisproperty is denoted by Kp(E).

In a type 2 Banach space E, the inequality

E

∥∥∥∥∫ T

0

f(s) dWs

∥∥∥∥2E

≤ T2(E)2∫ T

0

‖f(t)‖2E dt

holds for all E-valued Borel simple functions f : [0, T ]→ E [25, Proposition 5.2].Moreover, if all uniformly bounded strongly measurable functions f : [0, T ] → Eare stochastically integrable, then E necessarily has type 2 [25, Proposition 6.1].

For multiple stochastic integrals, we require a stronger property. Let 1 ≤ p ≤2. A Banach space E is said to be of M-type p if there exists C > 0 such that forany E-valued martingale {Mj}nj=1, n = 1, 2, . . . , the inequality

supj

E ‖Mj‖p ≤ Cpn∑

j=1

E ‖Mj −Mj−1‖p

holds with M−1 = 0. The smallest C with this property is denoted by Lp(E).

According to [24, pp. 221–222], if a Banach space E is of M-type p, then Eis of type p and reflexive, but there is a Banach space of type 2 which is of M-typep for no p > 1. If a UMD Banach space is of type p, then it is of M-type p.

For an M-type 2 Banach space E, the Ito isometry (2.2) becomes the one-sided inequality

E(‖Ik(f)‖2E) ≤ k!T2(E)2k‖f‖2L2((0,T ]k,E), (4.1)

with T2(E) = L2(E)T2(E), see [1, Corollary 3.4].

We are interested mainly in Lp-spaces with 2 ≤ p < ∞, which are both M -type 2 and UMD Banach spaces. Because we are only integrating deterministicE-valued functions, we could get by with only assuming that E is a Banach spaceof type 2 by appealing to a deep multilinear decoupling inequality of S. Kwapien[16] analogous to Theorem 3.4, which is valid for UMD Banach spaces, see [26,Proposition 1] and [17, Theorem 6.4.1].

206 B. Jefferies

4.1. Sectorial operators

Let 0 < ω < π/2. The sectors Sω± are defined by

Sω− = {−z : z ∈ C, | arg z| ≤ ω } ∪ {0}, Sω+ = {z : z ∈ C, | arg z| ≤ ω } ∪ {0}.Suppose that A : D(A) −→ E is a closed densely defined linear operator acting inthe Banach space E. The spectrum of A is denoted by σ(A). If 0 ≤ ω < π/2, thenA is said to be of type ω−, if σ(A) ⊂ Sω− and for each ν > ω, there exists Cν > 0such that

‖(zI −A)−1‖ ≤ Cν |z|−1, z /∈ Sν−. (4.2)

An operator A is of type ω− if and only if it is the generator of an analyticsemigroup ezA in the region | arg z| < π/2− ω so that for each ν > ω, there existsCν > 0 such that ‖ezA‖ ≤ Cν for all z ∈ C with | arg z| < π/2 − ν [23, §2.5].An operator A is of type ω+ if and only if −A is the generator of an analyticsemigroup in the region | arg z| < π/2− ω.

Let T > 0. Let E be a Banach space, A an operator of type ω−, ω < π/2and let V be a separable Banach space with norm ‖ · ‖V such that D(A) ⊂ V ⊂ Ewith continuous inclusions and B : V → E is bounded. Suppose that there existsc1 > 0 such that ∫ T

0

‖etAx‖2V dt ≤ c21‖x‖2E (4.3)

for all x ∈ D(A).

Lemma 4.1. Let c1 > 0. The inequality (4.3) holds if and only if∫ T

0

∫ t

0

‖e(t−s)Ag(s)‖2V dsdt ≤ c21

∫ T

0

‖g(t)‖2E dt (4.4)

for all E-valued simple functions g.

Proof. The inequality (4.3) holds for all x ∈ E because there exists c > 0 suchthat ‖AetAx‖ ≤ c‖x‖/t for all t > 0. Moreover, if the bound (4.3) holds, then∫ T

0

∫ t

0

‖e(t−s)Ag(s)‖2V dsdt =

∫ T

0

∫ T

s

‖e(t−s)Ag(s)‖2V dtds

=

∫ T

0

∫ T−s

0

‖etAg(s)‖2V dtds

≤∫ T

0

∫ T

0

‖etAg(s)‖2V dtds

≤ c21

∫ T

0

‖g(s)‖2E ds, by (4.3).

Now suppose that (4.4) holds. By taking g = χR.x, x ∈ E, we obtain∫R

∫ T−s

0‖etAx‖2V dtds

|R| ≤ c21‖x‖2E


for all finite unions R of intervals. Because s �→ ∫ T−s

0 ‖etAx‖2V dt is continuous,this is only possible if equation (4.3) holds. �

Theorem 4.2. Let E be a Banach space of M-type 2. Suppose that the estimate(4.3) holds for all x ∈ E and ‖Bx‖E ≤ c2‖x‖V for all x ∈ V . If c1c2T2(E) < 1,then the stochastic Dyson series

etAu0 +

∞∑k=1

∫ t

0

∫ sk

0

· · ·∫ s2

0

[e(t−sk)ABe(sk−sk−1)A · · ·Bes1Au0

]dWs1 . . . dWsk

(4.5)converges absolutely in L2(P;E) for every 0 < t ≤ T and every u0 ∈ E.

Proof. Suppose that the estimate (4.3) holds for all x ∈ E and ‖Bx‖E ≤ c2‖x‖Vfor all x ∈ V . The estimate (4.4) in Lemma 4.1 is also valid for all square integrableE-valued functions g by continuity. Then by the Ito bound (4.1), we have

E

∥∥∥∥∫ t

0

∫ sk

0

· · ·∫ s2

0


]dWs1 . . . dWsk

∥∥∥∥2E

≤ T2(E)2k∫ t

0

∫ sk

0

· · ·∫ s2

0

∥∥∥e(t−sk)ABe(sk−sk−1)A · · ·Bes1Au0

∥∥∥2E

ds1 . . . dsk

≤ C2T2(E)2k∫ t

0

∫ sk

0

· · ·∫ s2

0

∥∥∥Be(sk−sk−1)A · · ·Bes1Au0

∥∥∥2Eds1 . . . dsk

≤ C2T2(E)2kc22∫ t

0

∫ sk

0

· · ·∫ s2

0

∥∥∥e(sk−sk−1)ABe(sk−1−sk−2)A · · ·Bes1Au0

∥∥∥2V

ds1 . . . dsk

≤ C2T2(E)2k

(c1c2)2

∫ t

0

∫ sk−1

0

· · ·∫ s2

0

∥∥∥Be(sk−1−sk−2)A · · ·Bes1Au0

∥∥∥2Eds1 . . . dsk−1

...

≤ C2T2(E)2k(c1c2)2(k−1)

∫ t

0

∥∥Bes1Au0

∥∥2Eds1

≤ C2(c1c2T2(E))2k‖u0‖2E.

Here we have used the bound ‖esA‖ ≤ C for all s ≥ 0. If c1c2T2(E) < 1, then thesum (4.5) converges in L2(P;E) for every 0 < t ≤ T and every u0 ∈ E. �

Suppose that the conditions of Theorem 4.2 hold. For each u0 ∈ E and0 < t ≤ T , the E-valued random variable defined by the series (4.5) is denoted by

eA+Bλ,W ;tu0. We define eA+B

λ,W ;0u0 = u0. The mapping u0 �−→ eA+Bλ,W ;tu0 is an element

of the space L(E,L2(P, E)) of random linear operators [29] which we denote by

208 B. Jefferies

eA+Bλ,W ;t. It is easy to see that

t �−→ eA+Bλ,W ;t, 0 ≤ t ≤ T,

is a continuous map from the closed interval [0, T ] into L(E,L2(P, E)).The following corollary follows from the observation that the stochastic Dyson

series (4.5) is the solution obtained from the contraction mapping principle for thestochastic equation (4.6) below, see [6, Lemma 2.2]. By a mild solution, we meanan E-valued solution Xt, t ≥ 0, of the stochastic equation

Xt = etAx+

∫ t

0

e(t−s)ABXs dWs.

A general treatment of stochastic equations in Hilbert space is given in [4].The following consequence of Theorem 4.2 may be compared with [1, Theo-

rem 4.6], which is written in terms of the interpolation space

V = DA(1/2, 2) =

{x ∈ E : ‖x‖2DA(1/2,2) =

∫ ∞

0

‖AetAx‖2E dt <∞}.

When E is a Hilbert space and −A is a positive selfadjoint operator, then

DA(1/2, 2) = D((−A) 12 ).

Our point of departure is to obtain an explicit representation of the fixed pointsolution of the linear stochastic equation (4.6) as a “stochastic Dyson series”.

Corollary 4.3. Let E be a Banach space of M-type 2. Suppose that the conditionsof Theorem 4.2 hold. Then for each x ∈ E, the E-valued process

t �−→ eA+Bλ,W ;tx, 0 ≤ t ≤ T,

is the unique mild solution of the stochastic equation

dXt = AXt dt+BXt dWt, X0 = x. (4.6)

We can check that t �−→ eA+Bλ,W ;tx, t ≥ 0, is a strong solution of the stochastic

equation (4.6) for x ∈ V if etAV ⊆ V for t ≥ 0, see [1, Proposition 4.5]. Thepossibility of different choices of the space V are studied in [6, §3.1] in the Hilbertspace case.

For the definition of fractional powers of operators used in the next result,see [23], [14, Appendix]. The differential operator B is usually half the order ofthe elliptic operator A.

Corollary 4.4. Let E be a Banach space of M-type 2. Suppose that there existsc1 > 0 such that ∫ ∞

0

‖etAx‖2V dt ≤ c21‖x‖2E (4.7)

for all x ∈ E and ‖Bx‖E ≤ c2‖x‖V for all x ∈ V . If c1c2T2(E) < 1, then there

exists M > 0 such that ‖eA+Bλ,W ;tx‖L2(P,E) ≤M‖x‖E for all t ≥ 0.

Furthermore, suppose that A is a one-to-one operator of type ω− and the

norm ‖ · ‖V is defined by ‖x‖V = ‖(−A) 12 x‖E. Then for every t > 0, there exists


Lt > 0 such that ‖(−A) 12 eA+B

λ,W ;tx‖L2(P,E) ≤ Lt‖x‖E for all x ∈ E and t �−→ eA+Bλ,W ;tx

is a predictable continuous process with values in L2(P,D((−A) 12 )) for t > 0.

Proof. Under condition (4.7), the bound giving the convergence of (4.5) is uniform

in T > 0, from which the uniform bound for t �−→ eA+Bλ,W ;tx, t > 0, is obtained.

For the last statement, it suffices to apply Lemma 4.1 to note that

E

∥∥∥∥∫ t

0

∫ sk

0

· · ·∫ s2

0

[(−A) 1

2 e(t−sk)ABe(sk−sk−1)A · · ·Bes1Au0

]dWs1 . . . dWsk

∥∥∥∥2E

≤ T2(E)2k∫ t

0

∫ sk

0

· · ·∫ s2

0

∥∥∥(−A) 12 e(t−sk)ABe(sk−sk−1)A · · ·Bes1Au0

∥∥∥2E

ds1 . . . dsk

= T2(E)2k∫ t

0

∫ sk

0

· · ·∫ s2

0

∥∥∥e(t−sk)ABe(sk−sk−1)A · · ·Bes1Au0

∥∥∥2V

ds1 . . . dsk

≤ T2(E)2kc21

∫ t

0

∫ sk

0

· · ·∫ s2

0

∥∥∥Be(sk−sk−1)A · · ·Bes1Au0

∥∥∥2E

ds1 . . . dsk,

and then continue as in the proof of Theorem 4.2. The first term of (4.5) is treatedby noting that etAx ∈ D(A) for every x ∈ E and t > 0 [23, §2.5]. �

The condition c1c2T2(E) < 1 can be relaxed if we only require the sum (4.5)to converge absolutely for small times [6]. The solution of (4.6) is then obtainedby piecing together the solutions obtained from the stochastic Dyson series (4.5),

so that t �−→ eA+Bλ,W ;tx, t ≥ 0, has an exponential growth estimate.

5. Stochastic functional calculus

The significance of Corollary 4.4 above is that the bound (4.7) required for the

existence of the solution t �−→ eA+Bλ,W ;tx, t ≥ 0, of the stochastic equation (4.6) is a

type of square function estimate for the operator A. It has been known since thework of A. McIntosh [19] that such estimates are associated with the existence ofan H∞-functional calculus for A. Furthermore, it has been shown in [5, Theorem6.5] that the regularity of solutions of simple stochastic equations involving theoperator A in Hilbert space implies that A has an H∞-functional calculus.

A good reference for many of the results we need for an operator acting inHilbert space is [14, Chap. 2]. We now set down the basic definitions.

5.1. H∞ functional calculus

Let 0 < ω < π/2 and suppose that T : D(T )→ E is an operator of type ω− actingin the Banach space E as defined at the beginning of Section 4.

Then the bounded linear operator f(T ) is defined by the Riesz–Dunfordformula

f(T ) =1

2πi

∫C

(zI − T )−1f(z) dz. (5.1)

210 B. Jefferies

for any function f satisfying the bounds

|f(z)| ≤ Kν|z|s

1 + |z|2s , z ∈ S◦ν .

The contour C can be taken to be {z ∈ C : *(z) ≤ 0, |+(z)| = − tan θ.*(z) },with ω < θ < ν. The integral (5.1) converges as a Bochner integral in the uniformnorm due to the estimate (4.2) for the resolvent z �−→ (zI − T )−1 of T .

The operator T of type ω− is said to have a bounded H∞-functional calculusif for each ω < ν < π/2, there exists an algebra homomorphism f �−→ f(T )from H∞(S◦

ν−) to L(H) agreeing with (5.1) and a positive number Cν such that‖f(T )‖ ≤ Cν‖f‖∞ for all f ∈ H∞(S◦

ν ). The following Hilbert space result is from[19], see also [14, Theorem 11.9].

Theorem 5.1. Suppose that T is a one-to-one operator of type ω− in a Hilbertspace H. Then T has a bounded H∞-functional calculus if and only if for everyω < ν < π/2, there exists cν > 0 such that T and its adjoint T ∗ satisfy the squarefunction estimates ∫ ∞

0

‖ψt(T )u‖2dtt≤ cν‖u‖2, u ∈ H, (5.2)∫ ∞

0

‖ψt(T∗)u‖2 dt

t≤ cν‖u‖2, u ∈ H, (5.3)

for some function (every function) ψ ∈ H∞(S◦ν−), which satisfies∫ ∞

0

ψ2(−t)dtt

= 1, and (5.4)

|ψ(z)| ≤ Kν|z|s

1 + |z|2s , z ∈ S◦ν−, (5.5)

for some s > 0. Here ψt(z) = ψ(tz) for z ∈ S◦ν−.

For the function ψ(z) = Cz12 ez with C > 0 chosen such that (5.4) holds,∫ ∞

0

‖ψt(T )u‖2dtt

= C2

∫ ∞

0

‖(−tT ) 12 etTu‖2 dt

t

= C2

∫ ∞

0

‖(−T ) 12 etTu‖2dt.

With this choice for ψ, the bound (5.2) is equivalent to the bound (4.7) with

‖x‖V = ‖(−T ) 12 x‖ for x ∈ D((−T ) 1

2 ).

5.2. Random resolvents

Suppose that T : D(T ) → E is a closed linear map defined in the Banach spaceE. Then the resolvent R(ζ), ζ ∈ ρ(T ), of T is the bounded linear map defined by


R(ζ) = (ζI − T )−1 for all ζ ∈ C belonging to the set ρ(T ) for which the inverse isdefined. If T is the generator of a C0-semigroup etT , t ≥ 0, then we also have

(ζI − T )−1 =

∫ ∞

0

e−ζtetT dt (5.6)

for all ζ ∈ C in some right half-plane. We adopt the right-hand side of equation(5.6) as the definition of a resolvent in the setting of stochastic disentangling.

Let E be an M-type 2 Banach space.

1) A is an operator of type ω− for 0 < ω < π/2.2) There exists a real separable Banach space V with norm ‖ · ‖V such that

D(A) ⊂ V ⊂ E and B : V → E is a bounded linear operator with ‖Bx‖E ≤cB‖x‖V for all x ∈ V .

3) Let Aθ = eiθA for 0 ≤ |θ| < π/2−ω. For each 0 ≤ |θ| < π/2−ω, there existsmθ > 0 such that ∫ ∞

0

‖etAθx‖2V dt ≤ m2θ‖x‖2E (5.7)

for all x ∈ E.4) There exists 0 < δ < π/2− ω such that sup|θ|≤δ mθcBT2(E) < 1.

According to Corollary 4.4, the random process t �−→ eA+Bλ,W ;t, t ≥ 0, is uniformly

bounded in L(E,L2(P, E)) by a constant K. If the pair (A,B) of linear operatorssatisfies conditions 1)–4) above, then so does the pair (βA,

√βB) for any β >

0, so the mapping (β, t) �−→ eβA+√βB

λ,W ;t , β, t ≥ 0, is also uniformly bounded in

L(E,L2(P, E)) by K. Consequently, the following definition makes sense.

Definition 5.2. Let E be a Banach space of M-type 2 and suppose that the con-ditions 1)–4) above hold. The stochastic resolvent Rλ,W ;t(z;A + B), t ≥ 0, of

the process t �−→ eA+Bλ,W ;t, t ≥ 0, is the L(E,L2(P, E))-valued mapping t �−→

Rλ,W ;t(z;A+B), t ≥ 0, given by

Rλ,W ;t(z;A+B)x =

∫ ∞

0

e−zβeA+Bλ,W ;βtx dβ (5.8)

for all x ∈ E, t ≥ 0 and *z > 0.

We denote by the same symbol Rλ,W ;t(z;A + B) the analytic continuationof (5.8) as an element of L(E,L2(P, E)) to the left half-plane. We obtain anL(E,L2(P, E))-valued function of time t because we are considering disentanglingover an interval [0, t] as in Section 2.

Appealing to the orthogonality property 2) of multiple stochastic integrals,we see that (4.5) is a weakly orthogonal expansion in E-valued random variables.According to formula (5.8), the stochastic resolvent Rλ,W ;t(z;A+B)x also has aweakly orthogonal expansion in E-valued random variables. We use this expansionin order to establish the following bound.

212 B. Jefferies

Lemma 5.3. Let E be a Banach space of M-type 2. Suppose that conditions 1)–4)above hold. Then the L2(P, E)-valued function z �−→ Rλ,W ;t(z;A + B)x is holo-morphic in C \ Sδ− for all t > 0 and x ∈ E and for each π/2− δ < μ < π/2 thereexists Cμ > 0 such that

‖Rλ,W ;t(z;A+B)x‖L2(P,E) ≤ Cμ

|z| ‖x‖, z ∈ C \ Sμ− (5.9)

for all x ∈ E and t > 0.

Proof. Let√z denote the square root of z with positive real part. Under conditions

1)–4), replacing A by zA and B by√zB in the expansion (4.5), we obtain a

uniformly bounded L2(P, E)-valued holomorphic function z �−→ ezA+

√zB

λ,W ;t x in S◦δ+

for each t > 0 and x ∈ E.

For each 0 < μ < π/2, let Ξ±μ = {se±iμ : s ≥ 0}. Then for 0 < ν < δ, by thevector version of Cauchy’s Theorem we have

Rλ,W ;t(z, A+B) =

∫Ξ−ν

e−zζeζA+√ζB

λ,W ;t x dζ (5.10)

if *(ze−iν) > 0 and

Rλ,W ;t(z, A+B) =

∫Ξν

e−zζeζA+√ζB

λ,W ;t x dζ (5.11)

if *(zeiν) > 0. Because π/2 − δ < μ < π/2, we can choose 0 < ν < δ such thatπ/2− ν < μ < π/2. Then the bound (5.9) follows for all z ∈ C \ Sμ− with +z ≥ 0

from the representation (5.10) and the uniform boundedness of z �−→ ezA+

√zB

λ,W ;t x

in S◦δ+. For +z < 0, the representation (5.11) is used. �

For any holomorphic function ϕ in a sector S◦ν− with π/2− δ < ν < π/2 and

satisfying the bound

|ϕ(z)| ≤ Mν|z|s

1 + |z|2s , z ∈ S◦ν−, (5.12)

for some Mν , s > 0, we may define the integral

ϕλ,W ;t(A+B)x =1

2πi

∫C

ϕ(z)Rλ,W ;t(z;A+B)x dz, x ∈ E, (5.13)

in L2(P, E) for the contour C = {z ∈ C : |+(z)| = − tanμ.*(z), *(z) ≤ 0}taken anticlockwise around Sδ− for π/2 − δ < μ < ν. By Lemma 5.3 and theestimate (5.12), the contour integral converges as a Bochner integral in L2(P, E)and ϕλ,W ;t(A + B)x admits a weakly orthogonal expansion in E-valued randomvariables. In the case that B = 0, we obtain the Riesz–Dunford formula (5.1).

The following result says that the random part ϕλ,W ;t(A + B) − ϕ(A) ofϕλ,W ;t(A+B) has an H∞-bound under the assumptions 1)–4) above.


Theorem 5.4. Let E be a Banach space of M-type 2 and that conditions 1)–4)above hold. Then for every π/2− δ < ν < π/2, there exists Cν > 0 such that

(E‖ϕλ,W ;t(A+B)x − ϕ(A)x‖2) 12 ≤ Cν‖ϕ‖∞‖x‖

for every holomorphic function ϕ on S◦ν− satisfying the bound (5.12) and every

t > 0.

Proof. For each 0 < μ < π/2, let Ξ±μ = {se±iμ : s ≥ 0} and

Γμ,1 = {seiμ : −∞ ≤ s ≤ 0}, Γμ,2 = {−se−iμ : 0 ≤ s <∞}.Then for 0 < ν < δ, by the vector version of Cauchy’s Theorem Rλ,W ;t(z, A+

B) is given by equation (5.10) if *(ze−iν) > 0 and equation (5.11) if *(ze−iν) > 0.Let ϕ be a uniformly bounded holomorphic function in a sector S◦

ν− with π/2−δ <ν < π/2. Let π/2− δ < μ < ν. Then

2πiϕλ,W ;t(A+B)x =

∫Γμ,1

ϕ(z)Rλ,W ;t(z, A+B)x dz

+

∫Γμ,2

ϕ(z)Rλ,W ;t(z, A+B)x dz,

if the integrals converge. The Laplace transform

Lϕ(ζ) ={− ∫

Γμ,1e−zζϕ(z) dz, *(ζeiμ) < 0∫

Γμ,2e−zζϕ(z) dz, *(ζe−iμ) < 0

of ϕ is defined for π/2− ν < | arg ζ| < π.

From equation (4.5), the random part of Rλ,W ;t(ζ, A +B)x is given by

Rλ,W ;t(ζ, A+B)x = Rλ,W ;t(ζ, A +B)x− (ζI −A)−1x

In order to estimate

E

∥∥∥∥∫Γμ,2

ϕ(ζ)Rλ,W ;t(ζ, A +B)x dζ

∥∥∥∥2, (5.14)

we apply the Ito bound (4.1) and consider the sum

∞∑n=1

T2(E)2n∫ t

0

∫ tn

0

· · ·∫ t2

0

∥∥∥∥ ∫Ξ−θ

Lϕ(ζ)eζA(t−tn)(ζ12B)eζA(tn−tn−1)

· · · ζ 12B)eζAt1xdζ

∥∥∥∥2dt1 . . . dtn (5.15)

for π/2− μ < θ < π/2− ω. For each such θ, there exists Kθ > 0 such that

|Lϕ(ζ)| ≤ Kθ

|ζ| ‖ϕ‖∞, ζ ∈ Ξ−θ,

214 B. Jefferies

for every uniformly bounded holomorphic function in a sector S◦ν−. It suffices to

show that the sum( ∞∑n=1

T2(E)2n∫ t

0

∫ tn

0

· · ·∫ t2

0

(∫Ξ−θ

‖ϕ‖∞|ζ| ‖eζA(t−tn)(ζ

12B)eζA(tn−tn−1)

· · · ζ 12B)eζAt1x‖|dζ|

)2

dt1 . . . dtn

) 12

(5.16)

converges. The notation |dζ| means arclength measure. Then an application of theFubini–Tonelli Theorem shows that (5.14) is equal to (5.15) and is estimated bythe expression (5.16). Here we don’t actually appeal to the bound (5.12) which isonly needed to make sense of ϕ(A).

Applying Minkowski’s inequality, (5.16) is estimated by

‖ϕ‖∞∫Ξ−θ

( ∞∑n=1

T2(E)2n∫ t

0

∫ tn

0

· · ·∫ t2

0

(‖eζA(t−tn)(ζ

12B)eζA(tn−tn−1)

· · · ζ 12B)eζAt1x‖2dt1 . . . dtn

) 12 |dζ||ζ|

= ‖ϕ‖∞∫ ∞

0

( ∞∑n=1

T2(E)2n∫ t

0

∫ tn

0

· · ·∫ t2

0

(‖esA−θ(t−tn)(s

12B)esA−θ(tn−tn−1)

· · · s 12B)esA−θt1x‖2dt1 . . . dtn

) 12 ds

s

= ‖ϕ‖∞∫ ∞

0

( ∞∑n=1

T2(E)2n∫ st

0

∫ sn

0

· · ·∫ s2

0

(‖eA−θ(st−sn)BeA−θ(sn−sn−1)

· · ·BeA−θs1x‖2ds1 . . . dsn) 1

2 ds

s, [sj = stj for j = 1, . . . , n]

= ‖ϕ‖∞∫ ∞

0

( ∞∑n=1

T2(E)2n∫ r

0

∫ sn

0

· · ·∫ s2

0

(‖eA−θ(r−sn)BeA−θ(sn−sn−1)

· · ·BeA−θs1x‖2ds1 . . . dsn) 1

2 dr

r, [r = st]

We would like to know that this integral is finite. Split it into r ≥ 1 and r < 1.Applying the Cauchy–Schwarz inequality for r ≥ 1, we obtain

‖ϕ‖∞(∫ ∞

1

∞∑n=1

T2(E)2n∫ r

0

∫ sn

0

· · ·∫ s1

0

(‖eA−θ(r−sn)BeA−θ(sn−sn−1)

· · ·BeA−θs1x‖2ds1 . . . dsndr) 1

2

.


Each term∫ ∞

1

∫ r

0

∫ sn

0

· · ·∫ s1

0

‖eA−θ(r−sn)BeA−θ(sn−sn−1) · · ·BeA−θs1x‖2ds1 . . . dsndr

in the sum is bounded by∫ ∞

0

∫ r

0

∫ sn

0

· · ·∫ s1

0

‖eA−θ(r−sn)BeA−θ(sn−sn−1) · · ·BeA−θs1x‖2ds1 . . . dsndr.(5.17)

For every t > 0 and y ∈ E, the vector etA−θy is an element of D(A). ButD(A) ⊂ V ⊂ E with continuous embeddings, so there exists C > 0 such that(5.17) is bounded by

C2

∫ ∞

0

∫ r

0

∫ sn

0

· · ·∫ s1

0

‖eA−θ(r−sn)BeA−θ(sn−sn−1) · · ·BeA−θs1x‖2V ds1 . . . dsndr.(5.18)

Applying the inequality (5.7) and Lemma 4.1, the integral (5.17) is bounded by

C2m2−θ

∫ ∞

0

∫ sn

0

· · ·∫ s1

0

‖BeA−θ(sn−sn−1) · · ·BeA−θs1x‖2ds1 . . . dsn

≤ C2m2−θc

2B

∫ ∞

0

∫ sn

0

· · ·∫ s1

0

‖eA−θ(sn−sn−1)B · · ·BeA−θs1x‖2V ds1 . . . dsn.

Repeating the process, we obtain the bound

C2(m−θcB)2n

∫ ∞

0

‖eA−θs1x‖2V ds1 ≤ C2(m−θcB)2nm−θ‖x‖2.

By condition 4), m−θcBT2(E) < 1 and so the integral over r ≥ 1 converges.For r < 1, we can similarly estimate∫ r

0

∫ sn

0

· · ·∫ s1

0

‖eA−θ(r−sn)BeA−θ(sn−sn−1) · · ·BeA−θs1x‖2ds1 . . . dsnto get a bound

C′‖ϕ‖∞∫ 1

0

( ∞∑n=1

(m−θcBT2(E))2n−2

∫ r

0

‖x‖2dsn) 1

2 dr

r

which is finite. Combining the estimates for r ≥ 1 and r < 1, we obtain therequired bound for (5.16) and together with a similar argument for the integralover Γμ,1, this finishes the proof of the theorem. �

Remark 5.5. The above result also holds if we replace 4) by the condition

4′) sup‖x‖≤1,|θ|≤δ

∫ ∞

0

‖BetAθx‖2E dt < 1/T2(E)2.

In order to apply square function estimates for Lp-spaces for 1 < p < ∞given in [3] for the existence of an H∞ functional calculus, we need to establishthe convergence of (4.5) using multilinear square function estimates, which weleave to a later paper.

216 B. Jefferies

Combined with the characterisation of operators acting in Hilbert space withan H∞-functional calculus [19], we have the following result establishing the ex-istence of a stochastic functional calculus for “A+ B” in Hilbert space H , whereT2(H) = 1.

Theorem 5.6. Suppose that A is a one-to-one operator of type ω− in a Hilbertspace H such that A has an H∞-functional calculus on Sω−. Let V = D((−A) 1

2 )

with ‖x‖V = ‖(−A) 12x‖ for x ∈ V .

Then for every ω < ν < π/2, there exists bν > 0 such that for every boundedlinear map B : V → H with operator norm ‖B‖L(V,H) < bν, there exists a linearmap

ϕ �−→ ϕλ,W ;t(A+B)

from H∞(Sν−) with values in the linear space L(H,L2(P, H)) such that

(E‖ϕλ,W ;t(A+B)x‖2) 12 ≤ Cν‖ϕ‖∞‖x‖, t > 0,

for every uniformly bounded holomorphic function ϕ on S◦ν−.

The element ϕλ,W ;t(A + B) of L(H,L2(P, H)) is given by equation (5.13)for every uniformly bounded holomorphic function ϕ on S◦

ν− satisfying the bound(5.12). Furthermore, the number bν is given by

bν =

(sup

‖x‖≤1,|θ|≤π2 −ν

∫ ∞

0

‖(−A) 12 ete

iθAx‖2 dt)− 1

2

. (5.19)

Proof. Let ω < ν < π/2 and ψ(z) = (−z) 12 ez, for all z ∈ C \ [0,∞). Then for each

0 ≤ θ < π/2 − ν, the function z �−→ ψ(eiθz), z ∈ Sν−, satisfies the bound (5.5).Because A has an H∞-functional calculus on Sω−, the square function estimate(5.2) holds and there exists cν,θ > 0 such that∫ ∞

0

‖ψt(A)u‖2 dtt

=

∫ ∞

0

‖(−A) 12 ete

iθAx‖2 dt≤ cν,θ‖x‖2

for all x ∈ H . Because A has anH∞-functional calculus, the square function norms(5.2) and (5.3) are equivalent to the Hilbert space norm [19], [14, Theorem 11.9]and depend continuously on functions ψ uniformly satisfying the bound (5.5). Itfollows that

(x, θ) �−→∫ ∞

0

‖(−A) 12 ete

iθAx‖2 dt, 0 ≤ θ < π/2− ω, x ∈ H

is a continuous function. By the uniform boundedness principle,

sup‖x‖≤1,|θ|≤π

2 −ν

∫ ∞

0

‖(−A) 12 ete

iθAx‖2

is finite for each ω < ν < π/2 and conditions 1)–4) above are satisfied with δ = νand the given value bν .


The random linear operator ϕλ,W ;t(A + B) ∈ L(H,L2(P, H)) is defined bycontinuous extension from functions satisfying the bound (5.5). The nonrandompart of ϕλ,W ;t(A + B) has a limit by the convergence lemma of [19] and for therandom part of ϕλ,W ;t(A + B), from the proof of Theorem 5.4 it is clear that wecan appeal to dominated convergence. �

Finally, we state the relevance to the space-time regularity of strong solutionsof the Zakai equation

DtU(t, x) = A(x,D)U(t, x) +B(x,D)U(t, x)DtW (t), t ∈ [0, T ], x ∈ Rd,

U(0, x) = u0(x) for x ∈ Rd. Here

A(x,D) =

d∑i,j=1

aij(x)DiDj, B(x,D) =

d∑i=1

bi(x)Di + c(x).

This equation arises in filtering theory, and has been studied by many authors, see[2] and the references therein. It can be written as an abstract stochastic evolutionequation of the form

dXt = AXt dt+BXt dWt, t ∈ [0, T ], X0 = u0. (5.20)

Here the linear operator A is closed and densely defined on L2(Rd), the operator Bis a generator of a C0-group on L2(Rd), and W is a real-valued Brownian motionon some probability space (Ω,F ,P).

Theorem 5.7. Let A be the operator

d∑i,j=1

∂

∂xj

(aij(x)

∂

∂xj

).

with domain H2(Rd). The operator B is given by

Bu(x) =

d∑i=1

bi(x)∂u

∂xi(x), u ∈ H1(Rd).

If the coefficients aij(x) are real valued and belong to Cγ for some γ ∈ (0, 1) andsatisfy the joint ellipticity condition

d∑i,j=1

(cos ν.aij(x) − 1

2bi(x)bj(x)

)ξiξj ≥ ρ|ξ|2, ξ ∈ Rd, x ∈ Rd,

for some 0 < ν < π/2, then by [14, Theorem 13.14], the operators A and Bsatisfy the conditions of Theorem 5.6, so that (A,B) has a stochastic H∞(Sμ−)-functional calculus ϕ �−→ ϕλ,W ;t(A + B) on L2(Rd) and on the sector Sμ− forevery π/2− ν < μ < π/2.

Moreover, the L2(Rd)-valued process

t �−→ eA+Bλ,W ;tu0, t ∈ [0, T ],

218 B. Jefferies

is the solution of equation (5.20) and has paths in

C([0, T ];L2(Rd)) ∩ C((0, T ];H2(Rd)).

If u0 ∈ H2(Rd), then the solution has paths in H2(Rd).

5.3. Further developments

In Lp-spaces with 1 < p < ∞, if the bounds (5.2) and (5.3) are replaced by thesquare function estimates∥∥∥∥∥

(∫ ∞

0

|ψt(T )u|2dtt

) 12

∥∥∥∥∥p

≤ cν‖u‖p, u ∈ Lp, (5.21)

∥∥∥∥∥(∫ ∞

0

|ψt(T′)u|2 dt

t

) 12

∥∥∥∥∥p′

≤ cν‖u‖p′, u ∈ Lp′, (5.22)

then we obtain conditions equivalent to the existence of an H∞-functional calculusfor T [3, Theorem 6.1, Corollary 4.5]. As mentioned in [3, p. 87], the bounds (5.2)and (5.3) may fail for the Laplacian T = Δ on Lp(Rn) for p > 2, where they areassociated with Besov spaces, see for example [13, §1.2]

In order to utilise the bound (5.21) to obtain a stochastic functional calculusin an Lp-space for 1 < p < ∞ and, say, the Laplacian operator A = Δ, we wouldneed to obtain the multilinear estimate

E

∥∥∥∥∫ t

0

∫ sk

0

· · ·∫ s2

0


]dWs1 . . . dWsk

∥∥∥∥2p

≤ C2kp

∥∥∥∥∥(∫ t

0

∫ sk

0

· · ·∫ s2

0

∣∣∣e(t−sk)ABe(sk−sk−1)A · · ·Bes1Au0

∣∣∣2 ds1 . . . dsk

) 12

∥∥∥∥∥2

p

,

analogous to the one-sided Ito inequality used in the proof of Theorem 4.2. Fur-thermore, Lp-spaces for 1 < p < 2 are not of type 2, and so, not of M-type 2 andthe one-sided Ito inequality fails to hold. Here we would hope to apply the theoryof γ-radonifying maps mentioned in Section 3.2 and a multilinear version of theembedding results of [13].

References

[1] Z. Brzezniak, Stochastic partial differential equations in M-type 2 Banach spaces,.Potential Anal. 4 (1995), 1–45.

[2] Z. Brzezniak, J. van Neerven, M.C. Veraar and L. Weis, Ito’s formula in UMD Ba-nach spaces and regularity of solutions of the Zakai equation. J. Differential Equations245 (2008), 30–58

[3] M. Cowling, I. Doust, A. McIntosh and A. Yagi, Banach space operators with abounded H∞ functional calculus. J. Austral. Math. Soc. Ser. A 60 (1996), 51–89.


[4] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Encyclope-dia of Mathematics and its Applications 44, Cambridge University Press, Cambridge,1992.

[5] J. Dettweiler, J. van Neerven and L. Weis, Space-time regularity of solutions of theparabolic stochastic Cauchy problem. Stoch. Anal. Appl. 24 (2006), 843–869.

[6] F. Flandoli, On the semigroup approach to stochastic evolution equations. StochasticAnalysis and Appl. 10 (1992), 181–203.

[7] D.J.H. Garling, Brownian motion and UMD-spaces, in: “Probability and BanachSpaces” (Zaragoza, 1985), 36–49, Lecture Notes in Math. 1221, Springer-Verlag,Berlin, 1986.

[8] H. Heinich, Esperance conditionelle pour les fonctions vectorielles. C.R. Acad. Sci.Paris Ser. A 276 (1973), 935–938.

[9] B. Jefferies, Conditional expectation for operator-valued measures and functions.Bull. Austral. Math. Soc. 30 (1984), 421–429.

[10] , Feynman’s operational calculus and the stochastic functional calculus inHilbert space, in “The AMSI-ANU Workshop on Spectral Theory and HarmonicAnalysis”, Proc. Centre Math. Appl. Austral. Nat. Univ. 44, Austral. Nat. Univ.,Canberra, 2010, 183–210.

[11] B. Jefferies and G.W. Johnson, Feynman’s operational calculi for noncommutingoperators: Definitions and elementary properties. Russ. J. Math. Phys. 8 (2001),153–171.

[12] , Feynman’s operational calculi for noncommuting systems of operators: ten-sors, ordered supports and disentangling an exponential factor. Math. Notes 70(2001), 815–838.

[13] N.J. Kalton, J.M.A.M. van Neerven, M.C. Veraar, and L.W. Weis, Embedding vector-valued Besov spaces into spaces of γ-radonifying operators. Math. Nachr. 281 (2008),238–252.

[14] P. Kunstmann and L. Weis, Lp-regularity for parabolic equations, Fourier multipliertheorems and H∞-functional calculus. Functional analytic methods for evolutionequations, 65–311, Lecture Notes in Math. 1855, Springer, Berlin, 2004.

[15] H.H. Kuo, Gaussian measures in Banach spaces. Lecture Notes in Math. 463,Springer, Berlin, 1975.

[16] S. Kwapien, Decoupling inequalities for polynomial chaos. Ann. Probab. 15 (1987),1062–1071.

[17] S. Kwapien and W. Woyczynski, Random series and stochastic integrals: single andmultiple. Birkhauser Boston, Inc., Boston, MA, 1992.

[18] J. Maas, Malliavin calculus and decoupling inequalities in Banach spaces. J. Math.Anal. Appl. 363 (2010), 383–398.

[19] A. McIntosh, Operators which have an H∞-functional calculus, in: Miniconferenceon Operator Theory and Partial Differential Equations 1986, 212–222. Proc. Centrefor Mathematical Analysis 14, ANU, Canberra, 1986.

[20] J. van Neerven and L. Weis, Stochastic integration of functions with values in aBanach space. Studia Math. 166 (2005), 131–170.

220 B. Jefferies

[21] J. van Neerven, M.C. Veraar and L. Weis, Stochastic evolution equations in UMDBanach spaces. J. Funct. Anal. 255 (2008), 940–993.

[22] , Stochastic maximal Lp-regularity. Ann. Probab. 40 (2012), 788–812.

[23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differen-tial Equations. Springer-Verlag, Applied Mathematical Sciences, Vol. 44, NewYork/Berlin/Heidelberg/Tokyo, 1983.

[24] G. Pisier, Probabilistic methods in the geometry of Banach spaces. Probability andanalysis (Varenna, 1985), 167–241, Lecture Notes in Math. 1206, Springer, Berlin,1986.

[25] J. Rosinski and Z. Suchanecki, On the space of vector-valued functions integrablewith respect to the white noise. Colloq. Math. 43 (1980), 183–201.

[26] G. Samorodnitsky and M. Taqqu, Multiple stable integrals of Banach-valued func-tions. J. Theoret. Probab. 3 (1990), 267–287

[27] H. Schaefer, Topological Vector Spaces, Springer-Verlag, Berlin/Heidelberg/NewYork, 1980.

[28] L. Schwartz, Radon Measures in Arbitrary Topological Spaces and Cylindrical Mea-sures, Tata Inst. of Fundamental Research, Oxford Univ. Press, Bombay, 1973.

[29] A.V. Skorohod, Random Linear Operators, Riedel, 1984.

Brian JefferiesSchool of MathematicsThe University of New South WalesNSW 2052 Australiae-mail: [email protected]


Subideals of Operators – A Survey andIntroduction to Subideal-Traces

Sasmita Patnaik and Gary Weiss

Dedicated to the memory of Mihaly Bakonyi

Abstract. Operator ideals in B(H) are well understood and exploited butideals inside them have only recently been studied starting with the 1983seminal work of Fong and Radjavi and continuing with two recent articles bythe authors of this survey. This article surveys this study embodied in thesethree articles. A subideal is a two-sided ideal of J (for specificity also called aJ-ideal) for J an arbitrary ideal of B(H). In this terminology we alternativelycall J a B(H)-ideal.

This surveys [5], [13] and [14] in which we developed a complete char-acterization of all J-ideals generated by sets of cardinality strictly less thanthe cardinality of the continuum. So a central theme is the impact of gen-erating sets for subideals on their algebraic structure. This characterizationincludes in particular finitely and countably generated J-ideals. It was ob-tained by first generalizing to arbitrary principal J-ideals the 1983 work ofFong–Radjavi who determined which principal K(H)-ideals are also B(H)-ideals. A key property in our investigation turned out to be J-softness of aB(H)-ideal I inside J , that is, IJ = I , a generalization of a recent notionof K(H)-softness of B(H)-ideals introduced by Kaftal–Weiss and earlier ex-ploited for Banach spaces by Mityagin and Pietsch. This study of subidealsand the study of elementary operators with coefficient constraints are closelyrelated. Here we also introduce and study a notion of subideal-traces whereclassical traces (unitarily invariant linear functionals) need not make sense forsubideals that are not B(H)-ideals.

Mathematics Subject Classification (2010). Primary: 47L20, 47B10, 47B07;Secondary: 47B47, 47B37, 47-02, 13C05, 13C12.

Keywords. Ideals, operator ideals, principal ideals, subideals, lattices, traces,subideal-traces.

The first author was partially supported by various The Taft Foundation awards including aCharles Phelps Taft Dissertation Fellowship.

The second author was partially supported by Simons Foundation Collaboration Grant 245014,The Taft Foundation and CIRM.


222

1. Introduction

For general rings, an ideal (all ideals herein are two-sided ideals) is a commutativeadditive subgroup of a ring that is closed under left and right multiplication byelements of the ring. Herein H denotes a separable infinite-dimensional complexHilbert space and B(H) denotes the C∗-algebra of all bounded linear operatorson H . Ideals of B(H), with the latter regarded as a ring, have become ubiqui-tous throughout operator theory since their celebrated characterization by Calkinand Schatten [1], [15], in terms of “characteristic sets” of singular number se-quences s(T ) of the operators T in the ideal. Herein these ideals, alternativelyand for specificity, are called B(H)-ideals as one class among the classes of J-ideals defined below (next paragraph and expanded upon in Definition 2.1). ThisCalkin–Schatten characterization of B(H)-ideals has had and continues to havesubstantial impact in operator theory. As commutative objects in analysis, char-acteristic sets make more accessible the subtler properties of B(H)-ideals, particu-larly illuminating and expanding the knowledge of some of their noncommutativefeatures. Some well-known B(H)-ideals are the ideal of compact operators K(H),the finite rank operators F (H), principal ideals (S) (i.e., singly generated B(H)-ideals), Banach ideals, the Hilbert–Schmidt class C2, the trace class C1, Orliczideals, Marcinkiewicz ideals and Lorentz ideals, to name a few. Definitions andproperties of these ideals among others may be found in [4].

A subideal of operators is an ideal of J , for J an arbitrary B(H)-ideal. (Forspecificity we called these J-ideals.) That is, a subideal is an ideal of a B(H)-ideal.“Subideal” is a name coined by Gary Weiss motivated from the 1983 seminal workof Fong–Radjavi and by the new perspectives on operator ideals from work ofDykema, Figiel, Weiss and Wodzicki [4]. It is clear that every B(H)-ideal is asubideal, but the converse is less clear, i.e., whether or not every subideal is also aB(H)-ideal. Fong–Radjavi constructed the first example of a principal K(H)-idealthat is not a B(H)-ideal (Example 2.4). This shows that the class of subideals isstrictly larger than the class of B(H)-ideals.

The main and most general results in this survey are Theorem 3.5 and Theo-rem 3.7 (Structure Theorem for Subideals (S)J for |S| < c) in which we character-ize, in terms of a new notion called softness, when a subideal generated by strictlyless than c elements is also a B(H)-ideal (c denotes the cardinality of the contin-uum); and then we characterize its algebraic structure. Softness was first noticedby Kaftal and Weiss in [9], [11]–[12] and further exploited in [13]–[14]. Section 4compares B(H)-ideals to subideals via some of their differences and similarities.And Section 5 is new research that begins the investigation of subideal-traces, anattempt at a useful analog to traces on B(H)-ideals which traces are themselvesubiquitous in operator theory.

S. Patnaik and G. Weiss

Subideals of Operators – A Survey 223

2. Preliminaries

Every B(H)-ideal J is linear because for each α ∈ C, α1 ∈ B(H), so then for eachA ∈ J , αA = (α1)A ∈ J . But surprisingly a subideal (i.e., a J-ideal) may not belinear (Section 4-Example 4.1, see also [13, Example 3.5]). A reason this proof failsfor J-ideals when J �= B(H) is that α1 (α �= 0) is never contained in J .

In Subideals of Operators [13] we found three types of principal and finitelygenerated subideals: linear, real-linear and classical subideals (i.e., ideals not as-sumed to be linear inside B(H)-ideals). Indeed both the latter two types are some-times nonlinear. Such differences in types also carry over to non-finitely generatedJ-ideals. The linear K(H)-ideals, meaning traditionally the linear ones, were stud-ied in 1983 by Fong–Radjavi [5]. They found principal linear K(H)-ideals that arenot B(H)-ideals. Herein we take all J-ideals to be linear, but as shown in [13],we expect here also that most of the results and methods apply to the two othertypes of subideals (the real-linear and the sometimes nonlinear classical ones).

Noting the obvious fact that intersections of ideals in any ring are themselvesideals, we begin with the following definition.

Definition 2.1.

(i) The principal B(H)-ideal generated by the single operator S is defined by

(S) :=⋂{I | I is a B(H)-ideal containing S}.

(ii) The principal J-ideal generated by S is defined by

(S)J :=⋂{I | I is a J-ideal containing S}.

(iii) As above for principal J-ideals, likewise for an arbitrary subset S ⊂ J , (S)and (S)J denote respectively, via intersections, the smallest B(H)-ideal andthe smallest J-ideal generated by the set S.

(iv) Since herein all J-ideals are taken to be “linear,” (iii) characterizes all ofthem if you set S = J . But in [13]–[14] where J-ideals are not necessarilydefined as linear, (i)–(iii) define possibly nonlinear J-ideals and are discussedbelow in Section 4.

Definition 2.2. For B(H)-ideals I, J , ideal I is called “J-soft” if IJ = I. (Clearlythis applies only when I ⊂ J.) Equivalently in the language of s-numbers (seeRemark 2.3(i), (ii), (v) below):

For every A ∈ I, sn(A) = O(sn(B)sn(C)) for some B ∈ I, C ∈ J .

(s(A) := 〈sn(A)〉 is the singular number (s-number) sequence of operator A, count-ing multiplicities of course.)

Remark 2.3 (Standard facts and tools for operator ideals).

(i) If I, J are B(H)-ideals, then the traditional ideal product IJ is the B(H)-idealwhich is alternatively described via its characteristic set

Σ(IJ) = {ξ ∈ c∗o | ξ ≤ ηρ for some η ∈ Σ(I) and ρ ∈ Σ(J)}

224 S. Patnaik and G. Weiss

[4, Sections 2.8, 4.3] (see also [9, Section 4]). (See also Historical Backgroundbelow-first paragraph.) This product operation on the lattice of B(H)-ideals isboth associative and commutative.

(ii) If I and J are B(H)-ideals for which A ∈ IJ , then A = XY for some X ∈I, Y ∈ J [4, Lemma 6.3].

(iii) For T ∈ B(H), A ∈ (T ) if and only if s(A) = O(Dm(s(T ))) for some m ∈ N.Dmξ is the m-fold sequence ampliation recalled just below in Historical Back-ground.

Moreover, for B(H)-ideals I, as is well known from the polar decomposition,the inclusions A ∈ I, A∗ ∈ I, |A| ∈ I and diag s(A) ∈ I are equivalent.

(iv) The lattice of B(H)-ideals forms a commutative semiring with multiplicativeidentity B(H). That is, this lattice is commutative and associative under idealaddition and multiplication (see [4, Section 2.8]) and it is distributive. Distribu-tivity with multiplier K(H) is stated without proof in [9, Lemma 5.6 – precedingcomments].

One important feature of principal ideals in a general ring R is that they arebuilding blocks for all ideals I that contain them in that:

I =⋃

r1,...,rn∈I, n∈N

(r1) + · · ·+ (rn).

Note also (r) = r +Rr + rR +∑

finite sum RrR, and if R is unital, this reduces to(r) = Rr + rR +

∑finite sum RrR.

When R = J is a B(H)-ideal,∑

finite sumRrR = RrR [4, Lemma 6.3], inwhich case (r) collapses to (r) = r +Rr + rR +RrR.

(v) When T =

n∑i=1

AiTBi with each Ai or Bi ∈ J , one has the important s-number

relation: s(T ) = O(Dm(s(T ))s(C)) for some C ∈ J (since then T ∈ (T )J , see [9,Section 1, p. 6] and Remark 2.3(i)).

Historical Background. Calkin–Schatten completely characterizedB(H)-ideals viathe lattice preserving isomorphism between B(H)-ideals and characteristic setsΣ ⊆ c∗0 where c∗0 denotes the cone of nonnegative sequences decreasing to zero;characteristic sets Σ are those subsets of c∗0 that are additive, hereditary (solid) andampliation invariant (invariant under each m-fold ampliation Dmξ := 〈ξ1, . . . , ξ1,ξ2, . . . , ξ2, · · · 〉 with each entry ξi repeated m times); the characteristic set Σ(I) :={η ∈ c∗0 | diag η ∈ I}, so Σ(K(H)) = c∗0.

In 1983 Fong–Radjavi [5] investigated principal K(H)-ideals. They foundprincipal K(H)-ideals that are not B(H)-ideals (Example 2.4 below) by deter-mining necessary and sufficient conditions for a principal K(H)-ideal to be also aB(H)-ideal [5, Theorem 2]. And in doing so, at least for the authors of this paper,they initiated the study of subideals. The main results of Fong–Radjavi [5] aresummarized in the following theorem.


Theorem ([5, Theorems 1–2]). For T a compact operator of infinite rank,

P := (T ∗T )12 , I the ideal in K(H) generated by T , and P the ideal of K(H)

generated by P , the following are equivalent.

(i) I is an ideal in B(H).(ii) P is an ideal in B(H).(iii) P is a Lie ideal in B(H).(iv) T = A1TB1 + · · ·+AkTBk for some k, Ai ∈ K(H), Bi ∈ B(H).(v) T = A1TB1 + · · ·+AkTBk for some k, Ai, Bi ∈ K(H).(vi) For some integer k > 1, snk(P ) = o(sn(P )) as n→∞.

Fong–Radjavi proved this via the positive case employing Lie ideal condition(iii), but our approach below avoids considering separately the positive case andany Lie ideal considerations. Notably also, conditions (iv)–(v) above indicate therelevance of elementary operators with coefficient constraints.

Example 2.4. Condition (vi) of the above theorem shows that if the singular num-ber sequence of the operator P is given by s(P ) =

⟨12n

⟩, then the principal K(H)-

ideal generated by P is a B(H)-ideal. But if s(P ) =⟨1n

⟩, then the principal

K(H)-ideal generated by P is not a B(H)-ideal.

Indeed,1

2nk12n

= 12n(k−1) → 0 but

1nk1n

= 1k � 0 as n→∞.

3. Subideals of operators

Motivated by the Calkin–Schatten characterization and the seminal work of Fong–Radjavi, a natural question to ask is:

What can be said about subideals? Can they be characterized in some way?

A conventional approach to attack the characterization problem for J-ideals is tobegin at the elementary level as did Fong–Radjavi, albeit they did not considercharacterizations except implicitly for principal K(H)-ideals in one of their proofs.So we first investigate principal J-ideals, then finitely generated J-ideals and thenJ-ideals I = (S)J generated by sets S of higher cardinalities including the count-able case. We fully generalize Fong–Radjavi’s result [5, Theorem 2] from principalK(H)-ideals to arbitrary principal J-ideals and then to finitely generated J-ideals.The reason to consider the finitely generated case separate from the principal caseis that, unlike B(H)-ideals where every finitely generated B(H)-ideal is always aprincipal B(H)-ideal, a finitely generated J-ideal need not be a principal J-ideal(see Section 4 – Example 4.2 for the case J = K(H)). Consequently, we character-ize all J-ideals generated by sets of cardinality strictly less than the cardinality ofthe continuum, including finitely and countably generated J-ideals. A key propertyin this characterization turned out to be J-softness of a B(H)-ideal I inside J , thatis, IJ = I (Definition 2.2) a generalization of a recent notion of K(H)-softness ofB(H)-ideals introduced by Kaftal–Weiss [9] and earlier exploited for Banach andHilbert spaces by Mityagin and Pietsch.


We first begin with the following algebraic description of the principal J-idealgenerated by S ∈ J (see Remark 2.3(iv)).

Proposition 3.1. For S ∈ J , an algebraic description of principal J-ideal (S)J isgiven by

(S)J =

{αS +AS + SB +

m∑i=1

AiSBi | A, B, Ai, Bi ∈ J, α ∈ C, m ∈ N}

That is, (S)J = CS + JS + SJ + J(S)J .

The following theorem generalizes Fong–Radjavi’s result from principalK(H)-ideals to principal J-ideals by determining necessary and sufficient con-ditions for a principal J-ideal to be also a B(H)-ideal. Here is where J-softnessfirst played a prominent role.

For compact operators S, T , the product s(S)s(T ) denotes the pointwiseproduct of their s-number sequences.

Theorem 3.2. For S ∈ J and (S)J , the principal J-ideal generated by S, thefollowing are equivalent.

(i) (S)J is a B(H)-ideal.(ii) The principal B(H)-ideal (S) is J-soft, i.e., (S) = J(S);

(equivalently, (S) = (S)J).

(iii) S = AS + SB +

m∑i=1

AiSBi for some A, B, Ai, Bi ∈ J, m ∈ N.

(iv) s(S) = O(Dk(s(S))s(T )) for some T ∈ J and k ∈ N.

Proof of (i) ⇒ (ii) only. This is the main part of the proof so we provide here anoutline. For every unitary map φ : H → H ⊕H , S �→ φSφ−1 preserves s-numbersequences and hence also ideals via Calkin–Schatten’s representation. Since (S)Jis a B(H)-ideal containing S, φ−1(S⊕ 0)φ, φ−1(0⊕S)φ ∈ (S)J since they possessthe same s-numbers as S. Then by Proposition 3.1 for principal J-ideal (S)J ,

φ−1(S ⊕ 0)φ = αS +X and φ−1(0⊕ S)φ = βS + Y

for some X,Y ∈ JS + SJ + J(S)J, α, β ∈ C.If α = 0 or β = 0, then φ−1(S ⊕ 0)φ or φ−1(0⊕ S)φ ∈ J(S). Then, in either

case, S ∈ J(S), hence (S) ⊆ J(S) and since the other inclusion is automatic,one has (S) = J(S). If α, β �= 0, multiplying the first equation by −β and thesecond equation by α and adding obtains φ−1(−βS⊕αS)φ = −βX +αY ∈ J(S).Multiplying −βS⊕αS in B(H⊕H) by a suitable diagonal projection one obtainsφ−1(S ⊕ 0)φ ∈ J(S). Hence, also S ∈ J(S), again equivalent to (ii). �

Remark 3.3. Using basic linear algebra techniques, we extended Theorem 3.2 fromprincipal J-ideals to finitely generated J-ideals by solving a large system of lin-ear equations which we then project into a finite-dimensional quotient space [14,Theorem 4.5].


The techniques for finitely generated subideals do not work for countablygenerated subideals because the latter case involves an intractable infinite systemof equations, so a more sophisticated approach was needed. Based on the Hameldimension of a related quotient space (Proposition 3.4 next), a necessary and suffi-cient softness condition is found for a subideal with a generating set of cardinalitystrictly less than c, to be also a B(H)-ideal. In particular, this softness conditionapplies to all countably generated subideals (Theorem 3.5, see also [14, Theorem4.1]). We then use this condition to characterize the structure of these subideals(Theorem 3.7, see also [14, Theorem 4.4]). To investigate this in [14], we beganwith the following proposition.

Proposition 3.4 ([14, Proposition 3.1]). For the J-ideal (S)J generated by a set Sand defining

(S)0J := span{SJ + JS}+ J(S)J,

the Hamel dimension of the quotient space (S)J/(S)0J is at most the cardinality of

the generating set S.

The main softness theorem for when a J-ideal is also a B(H)-ideal [14]:

Theorem 3.5 ([14, Theorem 4.1]). A J-ideal (S)J generated by a set S of cardinalitystrictly less than c is a B(H)-ideal if and only if the B(H)-ideal (S) is J-soft.

Sketch of proof. Here we sketch only the proof of the first implication, that is, that(S)J is a B(H)-ideal implies (S) is J-soft. The reverse implication is somewhatroutine. The algebraic structure of (S)J is given by (S)J = span {S} + (S)0J andso the quotient space (S)J/(S)

0J = span {[Sα]} where Sα ranges over S. Hence the

Hamel dimension of (S)J/(S)0J is strictly less than c. And by minimality (S)J = (S),

since (S)J is also a B(H)-ideal.

The assumption that (S)J � (S) provides an operator in their differencewhich we use to construct an imbedding of �p into (S)J/(S)

0J . But the Hamel

dimension of �p is c [7, Lemma 3.4] and the Hamel dimension of (S)J/(S)0J is

strictly less than c, a contradiction. Therefore, the condition (S)J is a B(H)-idealimplies that (S)J = (S), that is, (S) is J-soft. �

Remark 3.6. Theorem 3.5 on the equivalence of a J-ideal (S)J being a B(H)-ideal and (S), the B(H)-ideal it generates, being J-soft motivates the question onwhether this is always true independent of its various classes of generators. Theanswer is no from the following example. And Theorem 3.5 yields new informationabout the possible cardinality of any class of its generators.

The K(H)-ideal (diag⟨1n

⟩) is also a principal B(H)-ideal but is not K(H)-

soft [14, Section 4, Example 4.5]. Thus I being a B(H)-ideal is not equivalent toJ-softness of the B(H)-ideal (I), for I a J-ideal and (I) the B(H)-ideal generatedby I. Moreover, by Theorem 3.5, (diag

⟨1n

⟩) which is also a K(H)-ideal, cannot

be generated in K(H) by less than c generators.


As a consequence of Theorem 3.5 we obtain a characterization of all J-idealsgenerated by sets of cardinality strictly less than the cardinality of the continuum.These are the countably generated J-ideals when assuming the continuum hy-pothesis, and otherwise these include more J-ideals than the countably generatedones.

Theorem 3.7 (Structure Theorem for (S)J when |S| < c). The algebraic structureof the J-ideal (S)J generated by a set S of cardinality strictly less than c is given by

(S)J = span{S+ JS+ SJ}+ J(S)J,

J(S)J is a B(H)-ideal, span{JS+ SJ}+ J(S)J is a J-ideal, and

J(S)J ⊂ span{JS+ SJ}+ J(S)J ⊂ (S)J

This inclusion collapses to

J(S)J = (S)J

if and only if

(S) is J-soft (i.e., (S)J = (S)).

4. Comparison of subideals to B(H)-ideals

As mentioned in Preliminaries Section 2, a subideal may not be linear. This ledthe authors of this paper to introduce three kinds of J-ideals, namely, linear, real-linear and classical J-ideals ([13, Definition 2.1])(the latter two are nonlinear).The term “classical” is meant in the sense of abstract rings, for instance, idealswhere scalar multiplication may not make sense. The classical principal J-idealgenerated by S is defined by 〈S〉J :=

⋂{I | I is a classical J-ideal containing S}.From Remark 2.3(iv) one deduces that

〈S〉J =

{nS +AS + SB +

m∑i=1

AiSBi | A, B, Ai, Bi ∈ J, n ∈ Z, m ∈ N}.

Example 4.1 (A concrete nonlinear principal ideal is: 〈diag 〈1/n〉〉K(H)). Indeed,

if it were linear, then the principal B(H)-ideal (diag⟨1n

⟩) would be K(H)-soft,

which is not the case. (Combine Example 2.4 and Theorem 3.2.)

The explicit description of the principal J-ideal generated by S given inProposition 3.1 implies that every principal J-ideal contains J(S)J . It is wellknown that every proper B(H)-ideal contains F (H), the B(H)-ideal of all finiterank operators [6, Chapter III, Section 1, Theorem 1.1]. Likewise one sees herethat every nonzero principal J-ideal contains F (H) (since S �= 0 implies (S)J ⊃J(S)J �= {0} and so the B(H)-ideal J(S)J ⊃ F (H)) and hence so also for everynonzero J-ideal because each is algebraically spanned by its principal ones. Theintersection of all B(H)-ideals properly containing F (H) is precisely F (H) [12,Corollary 3.8(ii)], and since every B(H)-ideal is a J-ideal, it is clear then that theintersection of all J-ideals properly containing F (H) is also precisely F (H).


Some striking differences between J-ideals and B(H)-ideals are describednext for the case J = K(H) in Examples 4.2–4.5. Every finitely generated B(H)-ideal is always a principal B(H)-ideal because, as is straightforward to see, theB(H)-ideal generated by S = {S1, . . . , Sn} ⊂ B(H), namely (S), is precisely theprincipal ideal (|S1|+ · · ·+ |Sn|) where |S| := (S∗S)1/2. But finitely generated J-ideals (classical, linear or real-linear) may not be principal as seen in the followingexample.

Example 4.2 (A doubly generated J-ideal of any of the three types that is notprincipal). For J = K(H),

S1 = diag

(1, 0,

1

2, 0,

1

3, · · ·

)and S2 = diag

(0, 1, 0,

1

2, 0,

1

3, · · ·

),

({S1, S2})K(H) is not a principal linear K(H)-ideal, and likewise for the classical

and real-linear cases 〈{S1, S2}〉J and ({S1, S2})RJ [13, Section 4, Example 4.1].

For T ∈ B(H), (T ) = (|T |), but this need not be true for principal linearK(H)-ideals (Example 4.3). Moreover, all B(H)-ideals are selfadjoint, but thisis not necessarily true for principal linear K(H)-ideals (Example 4.4) and un-like B(H)-ideals, K(H)-ideals need not necessarily commute under ideal product(Example 4.5).

Example 4.3. If J = K(H) and operator T = diag⟨in

n

⟩, then (T )K(H) �= (|T |)K(H).

In fact, (|T |)K(H) � (T )K(H) and (T )K(H) � (|T |)K(H) [13, Section 5, Exam-ple 5.1].

Example 4.4 (K(H)-ideal that is not closed under the adjoint operation).

T ∗ /∈ (T )K(H) where T = diag⟨in

n

⟩, [13, Section 5, Example 5.2].

Example 4.5 (K(H)-ideals that do not commute). For J = K(H) and with re-spect to the standard basis take S to be the diagonal matrix

S := diag(1, 0, 1/2, 0, 1/3, 0, . . .)

and T to be the weighted shift with this same weight sequence.Then (S)K(H)(T )K(H) �= (T )K(H)(S)K(H), [14, Section 5, Example 5.4].

5. Subideal-Traces

Subideals I that are not B(H)-ideals need not be invariant under unitary equiv-alence, i.e., UIU∗ � I for some unitary operator U (Examples 5.1–5.2 below).Therefore, the definition of trace on a B(H)-ideal, that is, a unitarily invariantlinear functional, need not make sense on a subideal. Motivated by our work in[2] on unitary operators of the form U = 1 + A for A ∈ K(H) we observe thatsubideals I are invariant under these unitaries (i.e., UIU∗ ⊂ I). This led theauthors of this paper to introduce the notion of a subideal-trace as defined belowin Definition 5.3 (see also Remark 5.9).


Example 5.1 (A K(H)-ideal that is not invariant under unitary equivalence). ForJ = K(H) and a unitary map φ : H → H ⊕ H , consider S = φ−1(D ⊕ 0)φfor D = diag

⟨1n

⟩. Then (S)K(H) the principal K(H)-ideal generated by S is not

invariant under unitary equivalence. We prove this by constructing one unitaryoperator U for which USU∗ /∈ (S)K(H). Indeed, assume (S)K(H) is invariant underunitary equivalence. We then have the following contradiction. Since

φ−1

(0 11 0

)φ is a unitary operator in B(H),

it follows that

φ−1

(0 11 0

)φ S φ−1

(0 11 0

)φ = φ−1

(0 00 D

)φ ∈ (S)K(H)

Using the algebraic structure of (S)K(H) (Proposition 3.1) one obtains,

φ−1

(0 00 D

)φ = αS +X,

where X ∈ K(H)S + SK(H) +K(H)(S)K(H) ⊂ (diag ⟨ 1n

⟩)K(H) (since s(S) =

s(D), (S) =(diag

⟨1n

⟩)). That is,

φ−1

(−αD 00 D

)φ ∈ (diag

⟨1

n

⟩)K(H).

This implies that D ∈ (diag⟨1n

⟩)K(H), a contradiction to the non-softness of

(diag⟨1n

⟩) [13, Example 3.3]. Therefore, (S)K(H) is not invariant under unitary

equivalence.

Example 5.2 (K(H)-ideal that is invariant under unitary equivalence). Varga [16]constructed a concrete example of a K(H)-ideal generated by the unitary orbit ofa positive compact operator that is not a B(H)-ideal, namely, (U(A))K(H) where

0 ≤ A ∈ K(H) and U(A) = {UAU∗ |U∗ = U−1}. Using Remark 2.3 (iv) for anideal written as the union of finite sums of its principal ideals, and Proposition 3.1giving the algebraic structure of the principal K(H)-ideal (UAU∗)K(H) generatedby UAU∗: for each T ∈ (UAU∗)K(H) and V a unitary operator in B(H), fromProposition 3.1 one has

V TV ∗ = V (αUAU∗ +BUAU∗ + UAU∗C +A′XB′)V ∗

(where B,C,A′, B′ ∈ K(H), X ∈ (UAU∗))

= αV UAU∗V ∗ + V BV ∗V UAU∗V ∗

+ V UAU∗V ∗V CV ∗ + V A′V ∗V XV ∗V B′V ∗

(V XV ∗ ∈ (V UAU∗V ∗) since X ∈ (UAU∗))

∈ (V UAU∗V ∗)K(H) ⊂ (U(A))K(H) (since V U is unitary).

Therefore the K(H)-ideal (U(A))K(H) is invariant under unitary equivalence.


Denote by U(H) the full group of unitary operators in B(H). Recall theessential feature of traces: their unitary invariance, that is, τ is a trace on aB(H)-ideal I when it is a linear functional for which τ(UTU∗) = τ(T ) for allT ∈ I, U ∈ U(H). And essential for this is that AdU preserves I, that is, for everyX ∈ I and U ∈ U(H), AdU (X) := UXU∗ ∈ I. But for J-ideals I, AdU maynot preserve I (Example 5.1 above). However some adjustments can be made topreserve much of the trace notion.

Definition 5.3. For a J-ideal I and the subgroup of unitary operators

UJ(H) := {1+A ∈ U(H)|A ∈ J},a linear functional

τ : I → C

is called a subideal-trace if τ(X) = τ(UXU∗) for every X ∈ I, U ∈ UJ(H).In other words, τ is called a subideal-trace if τ is AdUJ (H)-invariant, that is, ifτ(X) = τ(AdU (X)) for U ∈ UJ (H) and X ∈ I.Remark 5.4. In particular, if J = B(H) (so UB(H)(H) = U(H)), then I is aB(H)-ideal and hence AdU preserves I for U ∈ U(H) and Definition 5.3 becomesthe standard definition of a trace on a B(H)-ideal.

Example 5.5 (A simple example of a subideal-trace). Consider (S)J , a principallinear J-ideal generated by S ∈ J that is not a B(H)-ideal, and recall Proposition3.1 on the structure of its elements. Define the map τ : (S)J → C as

τ

(αS +AS + SB +

m∑k=1

AkSBk

):= α,

where A,B,Ak, Bk ∈ J, α ∈ C,m ∈ N. By our methods developed earlier, it iselementary to show that τ is a well-defined linear functional on (S)J when (S)J isnot a B(H)-ideal. Indeed, if αS+X = βS+ Y for X,Y ∈ SJ +JS+ J(S)J , then(α − β)S ∈ SJ + JS + J(S)J . Since (S)J is not a B(H)-ideal, α = β (otherwiseS ∈ J(S) which by Theorem 3.2 implies (S)J is a B(H)-ideal). Therefore τ(αS +X) = τ(βS + Y ), hence τ is a well-defined map. It is elementary to show that τ isa linear map. And since

(1+A)

(αS+AS+SB+

m∑k=1

AkSBk

)(1+A∗) = αS+X for X ∈ SJ+JS+J(S)J,

it follows that τ is AdUJ (H)-invariant. Hence τ is a subideal-trace on (S)J .

The commutator space of a B(H)-ideal I, [I, B(H)], is the linear span ofsingle commutators [A,B] for A ∈ I, B ∈ B(H). Since UXU∗ −X = [UX,U∗] ∈[I, B(H)] for every X ∈ I and every unitary operator U ∈ U(H), and since uni-tary operators span B(H), unitarily invariant linear functionals on I are preciselythe linear functionals on I that vanish on the commutator space [I, B(H)] [11,Section 2].


Because every operator is the linear combination of four unitary operators,the well-known commutator space [I, B(H)] is also the linear span of the singlecommutators [A,U ] for A ∈ I, U ∈ U(H). That is, [I,U(H)] = [I, B(H)]. Observ-ing that UB(H)(H) = U(H), we make the following analog.

Definition 5.6. The UJ(H)-commutator space of J-ideal I is defined as

[I,UJ(H)] := linear span{[X,U ] |X ∈ I, U ∈ UJ(H)}.Notice that if I is a B(H)-ideal, then the UJ (H)-commutator space of I is

precisely [I, B(H)], the commutator space of I.In the following proposition we obtain a necessary and sufficient condition

for a linear functional on a subideal to be a subideal-trace. This is an analog ofthe trace case just described.

Proposition 5.7. For a J-ideal I, a linear functional τ : I → C is a subideal-traceif and only if τ vanishes on the UJ(H)-commutator space of I, that is, τ vanisheson [I,UJ(H)].

Proof. Suppose τ is a subideal-trace. It suffices to show that τ vanishes on singlecommutators [X,U ] for X ∈ I and U ∈ UJ (H). For X ∈ I and 1 + B ∈ U(H)where B ∈ J , X(1+B) = X +XB ∈ I. Since τ is AdUJ (H)-invariant,

τ(X(1+ B)) = τ((1+B)X(1+B)(1+B∗))

= τ((1+B)X)),

i.e.,

τ([X, (1+B)]) = 0.

Therefore τ([X,U ]) = 0 for every U ∈ UJ(H).Next we prove the reverse implication, that is, if τ vanishes on the

UJ(H)-commutator space of I, [I,UJ(H)], then τ is a subideal-trace. That is,for U ∈ UJ (H), τ(X) = τ(UXU∗).

Since τ vanishes on [I,UJ (H)], in particular, τ([X, (1 + B)]) = 0 implyingτ(BX) = τ(XB) for all X ∈ I and (1+B) ∈ UJ(H). Since U = 1+B is a unitaryoperator, (1+B)(1+B∗) = 1 hence B +B∗ +BB∗ = 0.

τ((1 +B∗)X(1+B))− τ(X) = τ(XB) + τ(B∗X) + τ(B∗XB)

= τ(BX) + τ(B∗X) + τ(BB∗X)

(since B∗X ∈ I)= τ((B +B∗ +BB∗)X) = τ(0) = 0

Therefore linear functional τ is AdUJ (H)-invariant, and so by Definition 5.3, τ is asubideal-trace on I. �Corollary 5.8. The set of all subideal-traces on a J-ideal I can be identified withthe elements of the linear dual of the quotient space I

[I,UJ(H)] .

Indeed, for a given subideal-trace τ on a subideal I, define a functional fτ :I

[I,UJ(H)] → C as fτ ([X ]) := τ(X) where [X ] is the coset of the element X ∈ I.


Since [X ] = [Y ] implies X − Y ∈ [I,UJ(H)] and τ a subideal-trace, τ(X − Y ) = 0which implies that fτ is a well-defined linear functional on the quotient space. Onthe other hand, given a linear functional f on the quotient space I

[I,UJ(H)] , define

a function τ : I → C as τ(X) := f([X ]). Since f is a linear functional, τ is alsoa linear functional. And for every element Y ∈ [I,UJ(H)], f([Y ]) = 0 implyingτ(Y ) = 0. Hence τ vanishes on [I,UJ (H)]. Therefore by Proposition 5.7, τ is asubideal-trace on I.Remark 5.9. A subideal I may be invariant under a larger class than UJ(H)but not invariant under the full group of unitary operators U(H). For instance,U = λ(1 + B) for |λ| = 1 and (1 + B) ∈ I. But there may be more less obviousunitary operators under which I is invariant (Example 5.10 below). This leads usto suggest the following alternative definition of a subideal-trace (Definition 5.11below). However we will not explore it further here.

Example 5.10 (A K(H)-ideal invariant under a larger class of unitaries, but notinvariant under the full group U(H)). Using the principal K(H)-ideal (S)K(H)

and the unitary map φ of Example 5.1, the unitary operator U := φ−1(1⊕(−1))φ ∈U(H) \ UK(H)(H). That U /∈ UK(H)(H) is a simple computation. Then (S)K(H)

is invariant under AdU because USU∗ = S (an easy verification combining thedefinition of U here with the definition of S in Example 5.1), but (S)K(H) is notinvariant under AdU for U ∈ U(H) which again follows from Example 5.1.

Definition 5.11. For a J-ideal I and UI(H) := {U ∈ U(H)|UXU∗ ∈ I for X ∈ I},a linear functional

τ : I → C

is called a UI(H)-subideal-trace if τ(X) = τ(UXU∗) for every X ∈ I andU ∈ UI(H), that is, τ is AdUI(H)-invariant.

The following inclusion holds for a subideal I:AdUI(H)-invar. subideal-traces of I ⊂ AdUJ (H)-invar. subideal-traces of IThe next natural question is whether or not these inclusions are proper. In

particular, do Definition 5.3 and Definition 5.11 define different classes of func-tionals on a subideal that is not a B(H)-ideal? When I is a B(H)-ideal, Remark5.4 tells us that they are the same class.

References

[1] Calkin, J.W., Two-sided ideals and congruences in the ring of bounded operators inHilbert space, Ann. of Math. 42 (2)(1941), 839–873.

[2] Beltita, D., Patnaik, S., and Weiss, G., On Cartan subalgebras of operator ideals, inpreparation.

[3] Dixmier, J., Existence de traces non normales, C. R. Acad. Sci. Paris Ser. A-B 262(1966), A1107–A1108.


[4] Dykema, K., Figiel, T., Weiss, G., and Wodzicki, M., The commutator structure ofoperator ideals, Adv. Math., 185 (1) (2004), 1–79.

[5] Fong, C.K. and Radjavi, H., On ideals and Lie Ideals of Compact Operators, Math.Ann. 262, 23–28 (1983).

[6] Gohberg, C.I. and Krein, M.G., Introduction to the theory of nonselfadjoint operators,Transl. Amer. Math. Soc. 18, Providence, RI (1969).

[7] Halbeisen, Lorenz and Hungerbuhler, Norbert, The cardinality of Hamel bases ofBanach spaces, East-West J. Math., (2000) 153–159.

[8] Kaftal, V. and Weiss, G., Traces, ideals, and arithmetic means, Proc. Nat. Acad.Sci. U.S.A. 99(2002), 7356-7360.

[9] Kaftal, V. and Weiss, G., Soft ideals and arithmetic mean ideals, Integral equationsand Operator Theory 58 (2007), 363–405.

[10] Kaftal, V. and Weiss, G., A survey on the interplay between arithmetic mean ideals,traces, lattices of operator ideals, and an infinite Schur–Horn majorization theorem.Hot topics in operator theory, Theta 2008, 101–135.

[11] Kaftal, V. andWeiss, G., Traces on operator ideals and arithmetic means, J. OperatorTheory, 63 Issue 1, Winter 2010, 3–46.

[12] Kaftal, V. and Weiss, G., B(H) lattices, density and arithmetic mean ideals, HoustonJ. Math., 37(1)(2011), 233–283.

[13] Patnaik, S. and Weiss, G., Subideals of Operators, Journal of Operator Theory, 101–122 (2011).

[14] Patnaik, S. andWeiss, G., Subideals of Operators II, Integral Equations and OperatorTheory, Volume 74, Issue 4 (2012), pp. 587–600.

[15] Schatten, R., Norm ideals of completely continuous operators, Ergebnisse der Math-ematik und ihrer Grenzgebiete, Neue Folge, Heft, Vol. 27, Springer, Berlin (1960).

[16] Varga, J., On unitary invariant ideals in the algebra of compact operators, Proc.Amer. Math. Soc., Volume 108, Number 3 (1990).

Sasmita PatnaikDepartment of Mathematics and StatisticsIndian Institute of TechnologyKanpur, Uttar Pradesh, India 208016Telephone: 8127989114e-mail: [email protected]

sasmita [email protected]

Gary WeissUniversity of CincinnatiDepartment of MathematicsCincinnati, OH, 45221-0025, USAe-mail: [email protected]


Multipliers and Lp-operator Semigroups

Werner J. Ricker

Abstract. Deciding whether the generator of certain semigroups of operatorsin Lp(R) are unbounded scalar-type spectral operators can be reduced to de-ciding when eiϕ, for specific unbounded functions ϕ : R → R, is a p-multiplier.We illustrate how van der Corput’s lemma is an effective technique in this re-gard.

Mathematics Subject Classification (2010). 42A45, 47B40, 47D06.

Keywords. Operator semigroup, p-multiplier, van der Corput lemma.

When considering spectral properties of infinitesimal generators of certain semi-groups of operators in Lp-spaces one quickly arrives at questions concerning p-multipliers. For ease of presentation we will consider only the real line R. So, letD = −i d

dx be the closed, densely defined operator of differentiation in Lp withdomain D(D) = {f ∈ Lp : f ∈ AC, f ′ ∈ Lp}, where Lp denotes Lp(R) and AC isthe space of functions on R which are absolutely continuous on bounded intervals.By L(Lp), 1 < p < ∞, we denote the space of bounded linear operators of Lp

into itself. Let M(p) denote the Banach algebra of all Fourier multipliers for Lp

relative to the group R; briefly, p-multipliers. Then each function ψ ∈M(p) spec-

ifies an element ψ(D) of L(Lp) via the formula (ψ(D)f ) = ψf , for f ∈ L2 ∩ Lp,where · denotes the Fourier transform. The notation ψ(D) is consistent with the

fact (for 1 < p ≤ 2) that D(D) = {f ∈ Lp : ξf(ξ) = g(ξ), for some g ∈ Lp}and, for f ∈ D(D), that Df = g where g is the unique element of Lp satisfying

ξf(ξ) = g(ξ), for ξ ∈ R. The multiplier norm |||ψ|||p of ψ ∈ M(p) is defined to bethe operator norm ‖ψ(D)‖ of ψ(D) ∈ L(Lp).

Let ϕ(x) =∑n

j=0 ajxj , for x ∈ R, be any polynomial of even degree with

real coefficients aj , for 0 ≤ j ≤ n. Suppose that an > 0. Then there exists α ∈R such that ϕ(x) ≥ α, for all x ∈ R. By translating, if necessary, it may beassumed that α = 0. Let H+ = {z ∈ C : Re(z) > 0}. For each z ∈ H+ thefunction ϕz(x) = e−zϕ(x), for x ∈ R, is rapidly decreasing and so {ϕz : z ∈H+} ⊆ M(p), for every 1 < p < ∞. The corresponding family of Fourier p-multiplier operators {ϕz(D) : z ∈ H+} is then an analytic semigroup in L(Lp)with infinitesimal generator −ϕ(D). In L2 the operator D is selfadjoint and hence,


236 W.J. Ricker

so is ϕ(D) =∑n

j=0 ajDj , where Dj is defined in the usual way for non-negative

integral powers of an unbounded operator. Accordingly, ϕ(D) =∫Rϕ(λ) dQ(λ)

has an integral representation with respect to the resolution of the identity Q of

D. Of particular interest is the Laplace operator − d2

dx2 corresponding to ϕ(x) = x2.The question arises of whether the case p = 2 carries over to other values of p in(1,∞), that is, whether ϕ(D) =

∫Rϕ(λ) dQ(λ) =

∫∞0 λ dP (λ) for some spectral

measure P defined on the Borel subsets of [0,∞) and with values in L(Lp)? Ifthis were the case, then it would follow that the semigroup of Fourier p-multiplieroperators {ϕz(D) : z ∈ H+} coincides with the operators

∫∞0

e−zϕ(λ) dP (λ), for

z ∈ H+, defined via the usual calculus for scalar-type spectral operators, [1, Ch.XVII], and hence, {ϕz(D) : z ∈ H+} is uniformly bounded in L(Lp). Standardmultiplier convergence theorems would then imply that the “boundary group”ϕit(D) = e−itϕ(D), for t ∈ R, exists in L(Lp) and also consists of Fourier p-multiplier operators. So, the question of whether or not operators of the formϕ(D) are scalar-type spectral operators in Lp is reduced to determining whenfunctions of the type

x �→ eiϕ(x), x ∈ R, (1)

belong toM(p). This formulation has a meaning for all R-valued polynomials ϕ onR (not just those of even degree). So, it is of interest to determine which functions(1), with ϕ : R→ R say a polynomial, belong to M(p)? Results of L. Hormander,[3], are closely related to this question. Of course, various techniques are availablein this situation. Our aim in this note is to show how a well-known result of vander Corput can be effectively used in treating this problem.

van der Corput’s lemma. Let j ≥ 1 be an integer and h ∈ C(j)(a, b) be a R-valuedfunction satisfying |h(j)(w)| ≥ λ > 0, for all w ∈ [a, b], where −∞ < a < b < ∞.In the case of j = 1 it is assumed that h′ is monotone. Then there exists a constantcj > 0 (independent of a, b and h) such that∣∣∣∣∫ b

a

eih(w) dw

∣∣∣∣ ≤ cjλ−1/j .

Remark 1. For j = 1, 2 we refer to [6, p.197]; see [5] for arbitrary j.

As an immediate application we have the following result.

Lemma 1. Let 1 < p < ∞. Let ϕ : R → R be a polynomial of degree at least two.Then w �→ eiϕ(w), for w ∈ R, is not an element of M(p) unless p = 2.

Proof. It suffices to consider 2 < p < ∞. Fix x ∈ R and t ∈ R \ {0} and letj = deg(ϕ) denote the degree of ϕ. If hx,t(w) = xw − ϕ(tw), for w ∈ R, then

h(j)x,t(w) = αtj , for w ∈ R, for some constant α ∈ R. Accordingly, for any bounded

interval [a, b] ⊂ R it follows that∣∣h(j)x,t(w)

∣∣ = |α| |t|j , w ∈ R. (2)

Multipliers and Lp-operator Semigroups 237

The L1-function f = χ[a,b]

satisfies f(v) = iv−1(e−ibv − e−iav), for v �= 0, from

which it is clear (observe f ∈ C0(R)) that f ∈ Lr, for every 1 < r < ∞. Inparticular, there exists u ∈ L2 ∩ Lp such that u = f .

For t ∈ R, define

ut(x) = (2π)−1

∫R

ei[xw−ϕ(tw)] u(w) dw, x ∈ R.

Then the formula ut(x) = 12π

∫ b

a eihx,t(w) dw, van der Corput’s lemma and (2)

imply that ‖ut‖∞ ≤ cj |t|−1, for t ∈ R\{0} and some constant cj > 0. Since p > 2,it follows that

‖ut‖pp ≤ ‖ut‖p−2∞ ‖ut‖22 ≤ dj ‖ut‖22 |t|2−p, t �= 0.

But, ut(·) = u(·)eiϕ(t·) and so Parseval’s formula implies that ‖ut‖22 = ‖u‖22 fromwhich it follows that

‖ut‖pp ≤ dj ‖u‖22 |t|2−p, t �= 0.

Arguing as in the proof of Lemma 1.3 in [2] it follows that if eiϕ(t·) ∈ M(p) forsome t �= 0, then actually eiϕ(t·) ∈M(p) for every t ∈ R and supt∈R |||eiϕ(t·)|||p <∞.

So, there is κp > 0 such that ‖u‖p ≤ κp‖ut‖p and hence, ‖u‖p ≤ rj |t|(2p−1−1), fort �= 0 and for some constant rj > 0. Since 2p−1 < 1 this contradicts u �= 0 (let

t→ 0). Accordingly, eiϕ(t·) /∈M(p) for every t �= 0. �

Remark 2.

(a) Since eiϕ ∈ M(p) whenever ϕ(x) = αx + β with α, β ∈ R (for every 1 < p <∞), Lemma 1 answers the question of when eiψ belongs to M(p) for a givenpolynomial ψ : R→ R.

(b) It is an immediate consequence of Lemma 1 that the infinitesimal generator−ϕ(D) of the analytic semigroup {ϕz(D) : z ∈ H+} is not a scalar-typespectral operator in Lp if p �= 2, where ϕ : R→ R is any polynomial of even(and positive) degree. �

Lemma 1 suggests the question of also determining which functions of thetype (1) are p-multipliers for ϕ a R-valued rational function on R. Given any R-valued rational function f/g on R, where f and g are R-valued polynomials on R,it will henceforth be assumed that all linear and quadratic factors (over the fieldR) common to both f and g have been cancelled.

Theorem 1. Let 1 < p < ∞, with p �= 2, and f/g be a R-valued rational functionon R, where f and g are R-valued polynomials on R.

(i) If g has no real zero, then eif/g ∈M(p) if and only if

deg(f) ≤ 1 + deg(g).

(ii) If g has at least one real zero, then eif/g /∈ M(p).

This result, which incorporates Lemma 1, will be proved in a series of steps.

238 W.J. Ricker

Lemma 2. Let 1 < p <∞, with p �= 2, and f/g be a R-valued rational function onR, where f and g are R-valued polynomials on R, and suppose that g has no realzero. Then eif/g ∈M(p) if and only if

deg(f) ≤ 1 + deg(g).

Proof. We begin with the following

Claim. If deg(f) < deg(g), then eif/g ∈ M(p).The stated condition implies that f/g is of bounded variation on R (it is

bounded and piecewise monotonic) and hence, by Steckin’s theorem, f/g ∈ M(p)

for every 1 < p < ∞. Since M(p) is a commutative Banach algebra and z �→ eiz

is an entire function in C, it follows that eif/g =∑∞

n=0(if/g)n/n! determines an

element of M(p); the series converges absolutely with respect to the norm ||| · |||p in

M(p) because∞∑n=0

|||(if/g)n|||pn!

≤∞∑n=0

|||if/g|||npn!

= e|||if/g|||p <∞.

Of course, eif/g so defined is the function x �→ eif(x)/g(x), for x ∈ R. This estab-lishes the claim.

Suppose now that deg(f) ≤ 1 + deg(g). Then there exist constants α, β ∈ Rand a real polynomial r with deg(r) < deg(g) such that

f(x)

g(x)= α+ βx+

r(x)

g(x), x ∈ R.

By the above claim (for the pair r, g in place of f, g) we have that eir/g ∈ M(p).Since also ei(α+βx) ∈M(p), it follows that eif/g ∈ M(p).

Assume now that deg(f) > 1+deg(g). Then there exist real polynomials q, swith deg(s) < deg(g) and deg(q) ≥ 2 such that

f(x)

g(x)= q(x) +

s(x)

g(x), x ∈ R.

Since eis/g ∈ M(p) (via the above claim for the pair s, g in place of f, g), it wouldfollow that eiq ∈ M(p) if it were the case that eif/g ∈ M(p). But, this wouldcontradict Lemma 1 as deg(q) ≥ 2. Accordingly, we must have that eif/g /∈ M(p).This proves that necessarily deg(f) ≤ 1 + deg(g) whenever eif/g ∈ M(p). �

Lemma 3. Let 1 < p < ∞, with p �= 2, and ϕ(x) = α(βx − γ)−n, for eachx ∈ R \ {γ/β}, where α, β, γ are real numbers and n is a positive integer. Theneiϕ /∈M(p).

Proof. It suffices to consider 2 < p <∞. Since ϕ = ψ ◦ ρ, where ρ(x) = βx− γ isaffine and ψ(w) = αw−n, it suffices to show that eiψ /∈ M(p), [2, Theorem 1.3].

Fix x ∈ R and t ∈ R \ {0}. Define

hx,t(w) = xw − ψ(tw) = xw − αt−nw−n, w ∈ [1/2, 1],


in which case h(2)x,t(w) = −αn(n+ 1)t−nw−(n+2), for w ∈ [1/2, 1]. Hence,∣∣h(2)

x,t(w)∣∣ ≥ |α|n(n+ 1)|t|−n, w ∈ [1/2, 1]. (3)

Let u ∈ L2 ∩ Lp satisfy u = χ[1/2,1]

. Define, for each t ∈ R, the function

ut(x) =1

2π

∫R

ei(xw−ψ(tw)) u(w) dw =

∫ 1

1/2

eihx,t(w) dw, x ∈ R.

Via van der Corput’s lemma (with j = 2) and (3) it follows that ‖ut‖∞ ≤ c|t|n/2for some constant c. Assume eiψ(t·) ∈ M(p) for some t �= 0. Arguing as in theproof of Lemma 1 we have ‖ut‖p−2

∞ ≤ c∗|t|n(p−2)/2 and hence, for some d > 0, that

‖u‖p ≤ d|t|n(1−2p−1). Arguing again as in the proof of Lemma 1 and using the fact

that limt→0+ |t|n(1−2p−1) = 0 (for p > 2) gives the desired contradiction. �

A slightly more general result is the following one.

Lemma 4. Let 1 < p <∞, with p �= 2, and

ϕ(x) =

n∑j=1

aj(x− α)−j , x ∈ R \ {α},

where aj, for 1 ≤ j ≤ n, and α are real numbers with an �= 0 and n is a positive

integer. Then eiϕ /∈M(p).

Proof. It suffices to consider 2 < p < ∞ and (by Lemma 3) the case n ≥ 2.Furthermore, as ϕ = ψ ◦ ρ, where ρ(x) = x− α is affine and ψ(w) =

∑nj=1 ajw

−j ,

for w �= 0, it suffices to show eiψ /∈ M(p). Multiplying by −1, if necessary, it maybe assumed that an > 0.

Fix x ∈ R and t ∈ R \ {0}. Define

hx,t(w) = xw − ψ(tw) = xw −n∑

j=1

aj t−jw−j , w �= 0,

in which case its second derivative

h(2)x,t(w) = −w−2−nt−n

n∑j=1

j(j + 1)aj tn−jwn−j , w �= 0.

Since lims→0+∑n

j=1 j(j + 1)ajsn−j = 0 there exists 0 < δ < 1 such that∣∣∣∣ n−1∑

j=1

j(j + 1)ajsn−j

∣∣∣∣ < 1

2n(n+ 1)an, |s| < δ.

240 W.J. Ricker

Accordingly, for all t and w satisfying 0 < |tw| < δ we have∣∣h(2)x,t(w)

∣∣ = |w|−2−n|t|−n

∣∣∣∣ n(n+ 1)an −(−

n−1∑j=1

j(j + 1)aj(tw)n−j

)∣∣∣∣≥ |w|−2−n|t|−n

∣∣∣∣ n(n+ 1)an −∣∣∣∣ n−1∑j=1

j(j + 1)aj(tw)n−j

∣∣∣∣ ∣∣∣∣≥ 1

2|w|−2−n|t|−n n(n+ 1)an.

Choose any γ ∈ (0, δ). Then for every 0 < t < 1 and w ∈ [γ, δ] it is the case that|tw| < δ and |w|−2−n > 1, from which it follows that∣∣h(2)

x,t(w)∣∣ ≥ 1

2n(n+ 1)ant

−n, w ∈ [γ, δ], (4)

for every 0 < t < 1. Assume that eiψ(s ·) ∈ M(p) for some s �= 0. If u ∈ L2 ∩ Lp

satisfies u = χ[γ,δ]

, then it is possible to argue as in the proof of Lemma 3 (with

(4) replacing (2) there) that ‖u‖p ≤ ctn(1−2p−1) for 0 < t < 1 and some c > 0.

Since tn(1−2p−1) → 0 as t→ 0+ this gives the desired contradiction. �

Lemma 5. Let 1 < p <∞, with p �= 2, and define

ϕ(x) :=

k∑r=1

n(r)∑j=1

a(r)j (x− αr)

−j , x ∈ R \ {αr}kr=1,

where, for 1 ≤ r ≤ k, the a(r)j , 1 ≤ j ≤ n(r), are real numbers with a

(r)n(r) �= 0, the

αr are distinct real numbers and the n(r) are positive integers. Then eiϕ /∈M(p).

Proof. Applying an affine transformation, if needed, we may assume that α1 = 0

and a(1)n(1) > 0. For ease of reading set n := n(1). Assume 2 < p < ∞. Write

ϕ = ϕ1+ϕ2, with ϕ1(w) =∑n

j=1 a(1)j w−j and ϕ2(w) =

∑kr=2

∑n(r)j=1 a

(r)j (w−αr)

−j .

Fix elements x ∈ R and t ∈ R \ {0}. If hx,t(w) = xw − ϕ(tw), then h(2)x,t(w) =

−ϕ(2)1 (tw) − ϕ

(2)2 (tw) and hence,∣∣h(2)

x,t(w)∣∣ ≥ ∣∣∣ ∣∣ϕ(2)

1 (tw)∣∣− ∣∣ϕ(2)

2 (tw)∣∣ ∣∣∣, (5)

where the second derivatives of ϕ1 and ϕ2 are taken with respect to w. An exami-nation of the proof of Lemma 4 shows that there exist γ and δ with 0 < γ < δ < 1and ∣∣ϕ(2)

1 (tw)∣∣ ≥ 1

2(n+ 1)na(1)n t−n, w ∈ [γ, δ], (6)

for all 0 < t < 1. It is clear that δ can be chosen so that [0, δ] does not containany of the other singular points αr, for 2 ≤ r ≤ k, of ϕ. Then

μ := sup{∣∣ϕ(2)

2 (tw)∣∣; w ∈ [γ, δ], t ∈ [0, 1]

}<∞


and there exists s ∈ (0, 1) such that 12 (n+ 1)na

(1)n > μsn. It follows from (5) and

(6) that ∣∣h(2)x,t(w)

∣∣ ≥ 1

2(n+ 1)na(1)n t−n − μ, w ∈ [γ, δ],

for all t ∈ (0, s). Then van der Corput’s lemma implies that there is a constantc > 0 such that∣∣∣∣ ∫ δ

γ

eihx,t(w) dw

∣∣∣∣ ≤ c((n+ 1)na(1)n 2−1t−n − μ

)−1/2,

for all t ∈ (0, s). Since the right-side of this inequality tends to zero as t → 0+ itcan be argued as in the proof of Lemma 4 that eiϕ /∈ M(p). �

An examination of the proof of Lemma 5 shows that whenever ϕ(x) = ψ(x)+∑nj=1 aj(x − α)−j , where the α, aj, for 1 ≤ j ≤ n are real numbers (and an �= 0)

with n a positive integer, ψ is a C(2)-function in (u, v) for some u and v (satisfyingu < α and α+ 1 < v) and

μ := sup{∣∣∂2ψ(t(w + α))/∂w2

∣∣ : t ∈ [0, 1], w − α ∈ [ρ1, ρ2]}<∞

for some interval [ρ1, ρ2] ⊆ (α− u, v − α− 1) for which it is known that∣∣∣∣∂2

( n∑j=1

ajt−jw−j

)/∂w2

∣∣∣∣ ≥ ct−n, w ∈ [ρ1, ρ2],

for all 0 < t < 1, then eiϕ /∈ M(p) for every p �= 2. This applies, in particular, tothe case of

ψ(x) = h(x) +

k∑r=1

n(r)∑j=1

a(r)j (x− αr)

−j ,

where h is a polynomial, the a(r)j are real numbers and the αr, for 1 ≤ r ≤ k, are

distinct real numbers with α /∈ {α1, . . . , αk}.The proof of Theorem 1 follows from these remarks, Lemmas 1, 2 and 5 and

the partial fraction decomposition of rational functions.We conclude with a few remarks. As noted previously, if ϕ : R→ R is already

an element of M(p), then so is eiϕ. For ϕ within the class of polynomials it wasobserved that eiϕ ∈M(p), for 1 < p <∞ with p �= 2, if an only if ϕ(x) = αx+β forsome α, β ∈ R. There are, of course, other unbounded (non-polynomial) functionsϕ for which eiϕ ∈ M(p), e.g., for ϕ(x) = |x| and ϕ(x) = ln|x| (in which caseeiϕ(x) = |x|i); see [4, p.96]. For certain ϕ it is again possible to apply van derCorput’s lemma to show that eiϕ /∈ M(p) unless p = 2. For example, this is thecase if ϕ(x) = |x|α, for some α ∈ R \ {0, 1}. Indeed, if x ∈ R and t ∈ R \ {0} arefixed and

hx,t(w) = xw − ϕ(tw) = xw − |t|α|w|α, w �= 0,

then (for some c > 0) we have |h(2)x,t(w)| ≥ cα|α− 1|.|t|α, for w ∈ [1/2, 1], whenever

α > 0 (with α �= 1) and, for some c∗ > 0, that |h(2)x,t(w)| ≥ c∗|α|(1 + |α|)|t|−α,

242 W.J. Ricker

for w ∈ [1/2, 1], whenever α < 0. Using these estimates and the van der Corputlemma it can be argued as before that ei|x|

α

/∈ M(p), for every 1 < p < ∞ with

p �= 2. The same is true of eiex

; in this case |h(2)x,t(w)| ≥ t2, for all w ∈ R. However,

these are all ad hoc cases. It is clear that the arguments given in this note areof limited use when considering the question of when eiϕ ∈ M(p) for arbitrarymeasurable functions ϕ : R→ R.

Similar questions as considered in this note also arise in Rn; see [3], forexample, and the references therein.

References

[1] Dunford N. and Schwartz J.T., Linear Operators III: Spectral Operators, Wiley-Interscience, New York, 1971.

[2] Hormander L., Estimates for translation invariant operators in Lp spaces. ActaMath. 104 (1960), 93–140.

[3] Ricker W.J., Non-spectrality of generators of some classical analytic semigroups.Indag. Math. (New Ser.) 1 (1990), 95–103.

[4] Stein E.M., Singular Integrals and Differentiability Properties of Functions, Prince-ton Math. Series No. 30, Princeton University Press, Princeton, 1970.

[5] Wainger S., Averages and singular integrals over lower dimensional sets. Ann. Math.Studies 112 (1986), 357–421.

[6] Zygmund A., Trigonometric Series I (2nd ed.), Cambridge University Press, Cam-bridge, 1988.

Werner J. RickerMath.-Geogr. FakultatKatholische Universitat Eichstatt-IngolstadtD-85072 Eichstatt, Germanye-mail: [email protected]


Taylor Approximations of Operator Functions

Anna Skripka

Abstract. This survey on approximations of perturbed operator functions ad-dresses recent advances and some of the successful methods.

Mathematics Subject Classification (2010). Primary 47A55, 47B10.

Keywords. Perturbation theory, Taylor approximation.

1. Introduction

An active mathematical investigation of perturbed operator functions started in asearly as 1950’s, following a series of physics papers by I.M. Lifshits on the change ofthe free energy of a crystal due to appearance of a small defect. The latter researchin physics gave birth to the Lifshits–Krein spectral shift function [32, 27, 28],which has become a fundamental object in perturbation problems of mathematicalphysics. Subsequent attempts to include more general perturbations than those in[27, 28] have resulted in consideration of higher-order Taylor approximations ofperturbed operator functions and introduction of Koplienko’s higher-order spectralshift functions [26, 37, 18, 43, 47, 45].

Approximation of operator functions also arises in problems of noncommuta-tive geometry involving spectral flow (see, e.g., [8]) and spectral action functional(see, e.g., [16]). This investigation was initially carried out independently of thestudy of the spectral shift functions. However, a recent unified approach to theLifshits–Krein spectral shift function and the spectral flow allowed to establishthat these two objects essentially coincide [5]. Higher-order Taylor formulas havebeen derived for spectral actions in [58], with restrictions on the operators relaxedin [56] by applying more universal perturbation theory techniques.

The proof of existence of the first-order (Lifshits–Krein) spectral shift func-tion, which is due to M.G. Krein, relied on the theory of analytic functions andwas of a different nature than the proofs of the other mentioned results on the

Research supported in part by NSF grant DMS-1249186.


244 A. Skripka

approximations of operator functions. An important object in higher-order Tay-lor approximations is the Gateaux derivative of an operator function. When theinitial operator and the perturbation do not commute, the Gateaux derivative isa complex object, whose complexity increases with the order of differentiation.Treatment of such derivatives and subsequent derivation of Taylor approximationswas based on a delicate noncommutative analysis, which had been developing forsome 60 years.

To proceed to a detailed discussion of the aforementioned and further resultsand methods, we need to fix some notation. We work with a pair of operatorsdefined in a separable Hilbert space H, denoting the initial operator H0 and itsperturbation V . The perturbation is always a bounded operator and, moreover,some summability restrictions are imposed either on V or H0. In some instances,H0 is allowed to be unbounded, and we will consider only closed densely definedunbounded operators. For sufficiently nice scalar functions f , we consider the op-erator functions f(H0) and f(H0 + V ) given by the functional calculus. We areinterested in some scalar characteristics associated with perturbations that arecalculated using traces (a canonical trace Tr, a Dixmier trace Trω, a normal traceon a semi-finite von Neumann algebra τ , and, more generally, any trace τI on anormed ideal I continuous with respect to the ideal norm).

We consider the remainders of the Taylor approximations

Rn,H0,V (f) = f(H0 + V )−n−1∑k=0

1

k!

dk

dtk

∣∣∣∣t=0

f(H0 + tV ),

where n ∈ N and the Gateaux derivatives ddt

∣∣t=0

f(H0 + tV ) are evaluated in theuniform operator topology. If the nth order Gateaux derivative is continuous on[0, 1], then we have the integral representation for the remainder

Rn,H0,V (f) =1

(n− 1)!

∫ 1

0

(1− t)n−1 dn

dtnf(H0 + tV ) dt, (1.1)

which can be proved by applying functionals in the dual space (B(H))∗ of the alge-bra of bounded linear operators on H and reducing the problem to the scalar case.The questions we are interested in consist in establishing more specific propertiesof the remainders Rn,H0,V (f).

2. Schatten class perturbations

In this section, we discuss Taylor approximations in the classical setting of pertur-bations belonging to the Schatten–von Neumann ideals of compact operators

Sα ={A ∈ B(H) : ‖A‖α :=

(Tr(|A|α)) 1

α <∞}, α ∈ [1,∞)

(see, e.g., [52]). The operator functions under consideration come from either poly-nomials P or the functions with nice Fourier transforms

Wn ={f : f (j), f (j) ∈ L1(R), j = 0, . . . , n

}.

Taylor Approximations of Operator Functions 245

The class Wn includes such widely used sets of functions as Cn+1c (R) and the

rational functions in C0(R), which we denote by R.

2.1. Spectral shift functions

As a joint finding of many investigations, we have the following representation forthe Taylor remainders corresponding to self-adjoint perturbations of self-adjointoperators.

Theorem 2.1. If H0 = H∗0 and V = V ∗ ∈ Sn, n ∈ N, then there exists a unique

real-valued function ηn = ηn,H0,V ∈ L1(R) and a constant cn > 0 such that

‖ηn‖1 ≤ cn‖V ‖nnand

Tr

(f(H0 + V )−

n−1∑k=0

1

k!

dk

dtk

∣∣∣∣t=0

f(H0 + tV )

)=

∫R

f (n)(t) ηn(t) dt, (2.1)

for f ∈ Wn.

The cases n = 1, n = 2, and n ≥ 3 are due to [27] (see also [29]), [26], and[43], respectively. The formula (2.1) has been extended fromWn to the Besov classBn

∞1(R) in [38], [39], and [3], respectively. Differentiability of operator functions inthe setting most applicable to Theorem 2.1 is discussed in [40] and [6]. The resultsof [6, 40] can also be used to justify that the trace on the left-hand side of (2.1) iswell defined.

The function ηn provided by Theorem 2.1 is called the nth order spectralshift function associated with the pair of operators (H0, H0 + V ). The name to η1was given by M.G. Krein and can be understood from the formula

η1(λ) = Tr(EH0((−∞, λ))

) − Tr(EH0+V ((−∞, λ))

)holding for H0 and V finite matrices, where EH denotes the spectral measure ofH . A number of remarkable connections of the first-order spectral shift functionto other objects of mathematical physics can be found in the brief survey [9].More detailed discussion of the first-order spectral shift function can be foundin [12, 52, 59] and of the second-order one in [23]. When a perturbation V is inthe Hilbert–Schmidt class S2, the higher-order spectral shift functions ηn can beexpressed via the lower-order ones (see [26] in case n = 2 and [18, 54] in casen ≥ 3). The former are more sensitive to the displacement of the spectrum underperturbation, as demonstrated in [53, 55].

The question of validity of

Tr(f(H0 + V )− f(H0)

)=

∫Ω

f ′(t) η1(t) dt, (2.2)

was also investigated for non-self-adjoint operators H0 and H0 + V . Here the setΩ ⊂ C is determined by H0 and V . The trace formula (2.2) with Ω = T (the unitcircle) was proved in [28] for unitary operators H0 and H0 + V such that V ∈ S1.The case of arbitrary bounded operators H0 and H0 + V differing by V ∈ S1 is

246 A. Skripka

naturally harder than the case of self-adjoint operators. If H0 and H0 + V arecontractions, then (2.2) holds with Ω = T for every f analytic on a disc centeredat zero of radius r > 1 [31]. Attempts to extend (2.2) to more general functionsf resulted in consideration of only selected pairs of contractions and brought tomodification of (2.2) with passage to a more general type of integration. Therelevant discussion (also for dissipative operators H0 and H0 + V ) can be foundin [1, 2, 30, 34, 35, 36, 49, 50, 51].

The higher-order version of (2.2) for pairs of bounded operators has thefollowing formulation.

Theorem 2.2. Let H0 and H0+V be contractions and assume that V ∈ Sn, n ≥ 2.Then, there exists a function ηn = ηn,H0,V in L1(T) such that

Tr

(f(H0 + V )−

n−1∑k=0

1

k!

dk

dtk

∣∣∣∣t=0

f(H0 + tV )

)=

∫T

f (n)(z) ηn(z) dz, (2.3)

for f ∈ P. Furthermore, there exists a constant cn > 0 such that a function ηnsatisfying (2.3) can be chosen so that

‖ηn‖1 ≤ cn‖V ‖nn. (2.4)

The case n = 2 for H0 and H0+V unitaries, where the derivative is evaluatedalong a multipicative path of unitaries instead of the path of contractions t �→H0+ tV , is due to [37] (with later extension of the class of functions f in [39]) andfor arbitrary pairs of contractions H0 and H0+V joined by the path t �→ H0+ tVis due to [47]. The case n ≥ 3 is established in [45]. The spectral shift functionηn satisfying Theorem 2.2 is determined uniquely only up to an analytic term(that is, the equivalence class of ηn in the quotient space L1(T)/H1(T) is uniquelydetermined). Theorem 2.2 can be extended to analytic functions f .

2.2. Proof strategy

The proofs of Theorems 2.1 and 2.2 are very subtle and technically involved, so wewill give only a flavor of some basic ideas. For simplicity we assume that ‖H0‖ ≤ 1,‖H0 + V ‖ ≤ 1, V ∈ Sn, and f ∈ P . Then our goal is the formula

Tr(Rn,H0,V (f)

)=

∫T

f (n)(z)νn(dz) dz, (2.5)

where νn is a finite measure, with total variation bounded by

‖νn‖ ≤ cn‖V ‖nn. (2.6)

From the integral representation for the remainder (1.1), we derive

Tr(Rn,H0,V (f)

)=

1

(n− 1)!

∫ 1

0

(1− t)n−1 Tr

(dn

dsn

∣∣∣∣s=t

f(H0 + sV )

)dt.

Thus, if we prove

supt∈[0,1]

∣∣∣∣ 1n! Tr(

dn

dsn

∣∣∣∣s=t

f(H0 + sV )

)∣∣∣∣ ≤ cn‖V ‖nn ·∥∥f (n)

∥∥∞, (2.7)


then application of the Hahn–Banach theorem and the Riesz representation theo-rem for the dual space of C(T) implies existence of a measure νn satisfying (2.5)and (2.6).

For n = 1, we have

Tr

(d

ds

∣∣∣∣s=t

f(H0 + sV )

)= Tr

(f ′(H0 + tV )V

),

which in case of f a polynomial follows from the straightforward calculation ofthe derivative and some combinatorics. Applying the Holder and von Neumanninequalities then implies (2.7) with n = 1 and c1 = 1. This reasoning does notallow to establish the absolute continuity of ν1 (which was established in [27, 48]),but it can be generalized to apply to the higher-order case. If, in addition, wetake H0 and V to be self-adjoint, then application on the spectral theory allowsto derive an explicit formula for ν1, as it was done in [10].

Apart from the case of commuting H0 and V ∈ S2, we do not have the conve-

nient equality Tr(

d2

ds2

∣∣s=t

f(H0 + sV ))= Tr

(f ′′(H0 + tV )V 2

). However, since the

set function A1 ×A2 �→ Tr(EH0+tV (A1)V EH0+tV (A2)V

), where A1, A2 are Borel

subsets of R, uniquely extends to a measure on R2 with total variation ‖V ‖22, wehave

Tr

(d2

dt2

∣∣∣∣s=t

f(H0 + sV )

)=

∫R2

(f ′)[1](λ1, λ2)Tr(EH0+tV (dλ1)V EH0+tV (dλ2)V

)(see, e.g., [55, Theorem 3.12]), which along with the estimate for the divided dif-ference ‖(f ′)[1]‖∞ ≤ ‖f ′′‖∞ implies (2.7) with n = 2 and c2 = 1

2 .

When n ≥ 3, the set functionA1 × · · · × An �→ Tr

(EH0+tV (A1)V . . . EH0+tV (An)V

)can fail to extend to a

measure of finite variation on Rn (see [18, Section 4]). This is one of the reasonssuggesting that the case n ≥ 3 requires much more delicate (noncommutative)analysis of operator derivatives than the case n < 3.

Pioneering estimates for norms of nth order operator derivatives are attrib-uted to Yu.L. Daleckii and S.G. Krein [17]. In [17], H0 = H∗

0 and V = V ∗ ∈ B(H),a scalar function f belongs to C2n(R), and the estimates depend on the size of thespectrum of the operator H0. Development of the Birman–Solomyak double oper-ator integration (see, e.g., [11]) and subsequent multiple operator integration (see[40] and also [6]) resulted in significant improvement of the estimates for operatorderivatives. It follows from [40] that for H0 = H∗

0 and V = V ∗ ∈ Sn,

supt∈[0,1]

∣∣∣∣Tr( dn

dsn

∣∣∣∣s=t

f(H0 + sV )

)∣∣∣∣ ≤ cn ‖f‖Bn∞1(R)

· ‖V ‖nn ,

where f ∈ Bn∞1(R); however, the norm ‖f‖Bn

∞1(R)is greater than the norm

‖f (n)‖L∞(R). The powerful estimates (2.7) are established in the following the-orems.

248 A. Skripka

Theorem 2.3. ([45]) If ‖H0‖ ≤ 1, ‖H0 + V ‖ ≤ 1, and n ∈ N, then there exists aconstant cn > 0 such that for every f ∈ P the following estimates hold.

(i) If β > n and V ∈ Sβ, then

supt∈[0,1]

∥∥∥∥ dn

dsn

∣∣∣∣s=t

f(H0 + sV )

∥∥∥∥βn

≤ cn‖V ‖nβ ·∥∥f (n)

∥∥L∞(T)

.

(ii) If V ∈ Sn, then

supt∈[0,1]

∣∣∣∣Tr( dn

dsn

∣∣∣∣s=t

f(H0 + sV )

)∣∣∣∣ ≤ cn‖V ‖nn ·∥∥f (n)

∥∥L∞(T)

.

Theorem 2.4. ([43]) If H0 = H∗0 , V = V ∗, and n ∈ N, then there exists a constant

cn > 0 such that for every f ∈ Wn the following estimates hold.

(i) If β > n and V ∈ Sβ, then

supt∈[0,1]

∥∥∥∥ dn

dsn

∣∣∣∣s=t

f(H0 + sV )

∥∥∥∥βn

≤ cn‖V ‖nβ ·∥∥f (n)

∥∥L∞(R)

.

(ii) If V ∈ Sn, then

supt∈[0,1]

∣∣∣∣Tr( dn

dsn

∣∣∣∣s=t

f(H0 + sV )

)∣∣∣∣ ≤ cn‖V ‖nn ·∥∥f (n)

∥∥L∞(R)

.

The proofs of Theorems 2.3 and 2.4 (and also analogous estimates for poly-linear transformations more general than operator derivatives) include a subtlesynthesis of advanced techniques from harmonic, functional, complex analysis andnoncommutative Lp spaces as well as development of a novel approach to multi-ple operator integration. The principal two cases here are the ones of self-adjointsand unitaries, while the case of contractions reduces to the case of unitaries byapplying the Sz.-Nagy–Foias dilation theory [57].

2.3. Operator Lipschitz functions

Derivation of the estimates of Theorems 2.3 and 2.4 was preceded by resolution ofKrein’s conjecture on whether every Lipschitz function on R is operator Lipschitz.Detailed discussion of the problem, including references to partial results, canbe found in [41, 46]; here we only state the concluding result and mention somegeneralizations.

Theorem 2.5. ([46]) Let f be a Lipschitz function on R. Then, for every α ∈ (1,∞),there is a constant cα > 0 such that

‖f(B)− f(A)‖α ≤ cα‖B −A‖α · ‖f‖Lip,

for all A = A∗, B = B∗, defined in H with B −A ∈ Sα.

The best constant cα ∼ α2

α−1 is obtained in [15]. It is known that not every

Lipschitz function is operator Lipschitz in S1 and in B(H) (i.e., when α ∈ {1,∞})[20, 21, 22]. Operator Lipschitzness of functions of normal operators and of func-tions of several variables is discussed in [4, 24].


3. Some natural generalizations

If a perturbation V is not compact and no additional restriction on H0 is im-posed, then the canonical trace Tr of Rn,H0,V (f) is not defined. Depending on theproblem, one can consider another trace that is defined on Rn,H0,V (f) for rathergeneral H0, V , and f , or impose extra restrictions on H0, f , and/or V to ensureRn,H0,V (f) ∈ S1.

3.1. Compact resolvents and similar conditions

Perturbations that arise in the study of differential operators are multiplicationsby functions defined on Rd, which are not compact operators. In this case, thecondition V ∈ Sn gets replaced by a restriction on the resolvent of the initialoperator H0.

If H0 equals the negative Laplacian −Δ and the operator V act as multipli-cation by a real-valued function in L1(R3) ∩ L∞(R3), then

(H0 − zI)−1 − (H0 + V − zI)−1 ∈ S1, z ∈ C \ R (3.1)

(see, e.g., [12]). Due to the invariance principle for the first-order spectral shiftfunction (see, e.g., [12]), the problem for a pair of self-adjoint operators (H0, V )satisfying (3.1) reduces to the problem for a pair of unitaries with difference inS1, and (2.1) with n = 1 holds for f ∈ C∞

c (R) ∪R, as established in [28]. In thiscase, η1 is an element of L1

(R, 1

1+t2 dt). Existence of the first-order spectral shift

function under more general resolvent conditions is discussed in [25, 60].

If H0 = −Δ and V is a multiplication by a real-valued function in L2(R3) ∩L∞(R3), then instead of the condition (3.1), we have

(I +H20 )

−1/4V ∈ S2 (3.2)

(see, e.g., [44]). It is established in [26] that for a pair of self-adjoint operators(H0, V ) satisfying (3.2), there exists η2 ∈ L1

(R, 1

1+t2 dt)such that the trace formula

(2.1) with n = 2 holds for f ∈ R. A modified trace formula is obtained in [44] fora pair (H0, V ) satisfying (I +H2

0 )−1/2V ∈ S2. The proofs are based on multiple

operator integration techniques developed to partly compensate for the lack of theinvariance principle under the assumption (3.2).

In perturbation problems of noncommutative geometry, typical assumptionson the operators are that the resolvent of H0 is compact and V ∈ B(H). Thefollowing result is obtained in [56], relaxing assumptions onH0 and V made in [58].

Theorem 3.1. Let H0 = H∗0 be defined in H and have compact resolvent and

let V = V ∗ ∈ B(H). Let {μk}∞k=1 be a sequence of eigenvalues of H0 countingmultiplicity and let {ψk}∞k=1 be an orthonormal basis of the respective eigenvectors.

250 A. Skripka

Then, for each function f ∈ Cn+1c (R), with n ∈ N,

Tr(f(H0 + V )

)− Tr(f(H0)

)=

n−1∑p=1

1

p

∑i1,...,ip

(f ′)[p−1](μi1 , . . . , μip) 〈V ψi1 , ψi2〉 · · ·⟨V ψip , ψi1

⟩+Tr

(RH0,f,n(V )

),

whereTr(RH0,f,n(V )

)= O(‖V ‖n).

Moreover, the trace formula (2.1) with f ∈ C3c (R) is established in [5] for

n = 1 (this is also discussed in the next subsection) and, under the additionalassumption (I + H2

0 )−1/2 ∈ S2, in [56] for n = 2. The respective spectral shift

functions η1 and η2 are locally integrable.Taylor asymptotic expansions and spectral distributions have also been con-

sidered in the study of pseudodifferential operators (see, e.g., [13]).

3.2. Operators in a semifinite von Neumann algebra

Let M be a semifinite von Neumann algebra of bounded linear operators definedon H and let τ be a semifinite normal faithful trace τ on M. (The definitionscan be found in, e.g., [33].) Note that (B(H),Tr) is one of examples of (M, τ).Let H0 be either an element of M or an unbounded closed densely defined self-adjoint operator affiliated with M (that is, all the spectral projections of H0 areelements of M). The perturbation V is taken to be a bounded element of thenoncommutative Lp-space associated with (M, τ), that is,

V ∈ Ln ={A ∈ M : ‖A‖n := τ(|A|n) 1

n <∞}, n ∈ N.

Theorem 3.2. If H0 = H∗0 is affiliated with M and V = V ∗ ∈ Ln, n ∈ N, then

there exists a unique real-valued function ηn = ηn,H0,V ∈ L1(R) and a constantcn > 0 such that

‖ηn‖1 ≤ cn‖V ‖nnand

τ

(f(H0 + V )−

n−1∑k=0

1

k!

dk

dtk

∣∣∣∣t=0

f(H0 + tV )

)=

∫R

f (n)(t) ηn(t) dt, (3.3)

for f ∈ Wn.

The case n = 1 was established first for a bounded operator H0 in [14] andthen for an unbounded operator in [7]. The case n = 2 is due to [18, 54] andn ≥ 3 is due to [43]. The strategy of the proof is as described in Subsection 2.2;this strategy can be implemented because noncommutative Lp-spaces have muchin common with Schatten ideals (see, e.g., [42]).

The first-order spectral shift function for a pair of τ -Fredholm operatorsdiffering by a τ -compact perturbation is known to coincide with the spectral flow[5, Theorem 3.18]. It is also established in [5] that (3.3) with n = 1 holds for


H0 having τ -compact resolvent. (In the case (M, τ) = (B(H),Tr), a τ -compactoperator is merely a compact operator.)

Theorem 3.3. ([5]) If H0 = H∗0 is affiliated with M and has a τ-compact resolvent

and if V = V ∗ ∈M, then, for f ∈ C3c ((a, b)),

τ(f(H0 + V )

)= τ(f(H0)

)+

∫R

f ′(λ)τ(EH0((a, λ]) − EH0+V ((a, λ])

)dλ.

Analogs of (3.3) with n = 1 and n = 2 for pairs of arbitrary (non-self-adjoint)operators in M differing by a perturbation V ∈ Ln are obtained in [19]. As to thecase n ≥ 3, the results of Theorem 2.4 can be extended to pairs of operators in Mby applying dilation of contractions in M to unitary operators in semi-finite vonNeumann algebras constructed in [19].

3.3. General traces

The canonical trace Tr is widely used, but it is not the most “typical” trace. Thedistinctive feature of Tr is that it is normal, i.e, has the property of monotonicity.A continuous trace on a normed ideal of compact operators in B(H) other than S1

has a singular component, which vanishes on finite rank operators. Detailed discus-sion of traces and applications of singular traces to classical and noncommutativegeometry can be found in [33].

Let M be a semifinite (von Neumann) factor and I a normed ideal of Mwith norm ‖ · ‖I . (The definitions can be found, e.g., in [19, 33].) Let τI be atrace on I bounded with respect to the ideal norm ‖ · ‖I . Examples of (I, τI)include (S1,Tr), (L1, τ), where τ is the normal faithful semifinite trace onM, and(L(1,∞),Trω), where L(1,∞) denotes the dual Macaev ideal and Trω the Dixmiertrace on it corresponding to a generalized limit ω on �∞(N).

The following results are obtained in [19].

Hypotheses 3.4. Consider a set Ω, a closed, densely defined operator H0 affiliatedto M, an operator V ∈ I and a space F of functions that satisfy one of thefollowing assertions.

(i) Ω = conv(σ(H0) ∪ σ(H0 + V )

), H0 = H∗

0 ∈M, V = V ∗, F = C3(R);(ii) Ω = R, H0 and H0 + V are maximal dissipative operators (that is, closed,

densely defined operators whose quadratic forms have nonnegative imaginaryparts), and

F = span{λ �→ (z − λ)−k : k ∈ N, Im(z) < 0

};

(iii) Ω = T, ‖H0‖ ≤ 1, ‖H0 + V ‖ ≤ 1, and F is the set of all functions that areanalytic on discs centered at 0 and of radius strictly larger than 1.

Theorem 3.5. Let Ω, H0, V and F satisfy Hypotheses 3.4. Then, there exists a(countably additive, complex) measure ν1 = ν1,H0,V on Ω such that

‖ν1‖ ≤ min{τI(|Re(V )|)+ τI

(|Im(V )|), ‖τI‖I∗ · ‖V ‖I}

252 A. Skripka

and

τI(f(H0 + V )− f(H0)

)=

∫Ω

f ′(λ) ν1(dλ),

for all f ∈ F . If Hypotheses 3.4(i) are satisfied, then the measure ν1 is real andunique.

When I = S1, the measure ν1 is absolutely continuous, but when I is thedual Macaev ideal (with the Dixmier trace), the measure ν1 can be of any type[19, Theorem 4.4]. Moreover, we do not have an explicit formula for ν1 in case of ageneral trace τI . Derivation of an explicit formula for ν1 in case I = S1, H0 = H∗

0 ,and V = V ∗ relies on the fact that Tr

(EH0 (·)V

)is a (countably-additive) measure,

while the set function Trω(EH0(·)V

)can fail to be countably-additive (see [19,

Section 3]).As another consequence of singularity of Trω (and, more generally, of every

trace satisfying τI(I2) = {0}), we have the following linearization formula.

Theorem 3.6. Assume Hypotheses 3.4 and assume τI(I2) = {0}. Then,τI(f(H0 + V )− f(H0)

)= τI

(f ′(H0)V

).

Below we consider perturbations in the normed ideal I1/2 ={A ∈ M :

|A|2 ∈ I} and impose an additional natural assumption ‖AB‖I ≤ ‖A‖I1/2‖B‖I1/2,

which, in particular, holds for the ideals S1, L1, and L(1,∞).

Hypotheses 3.7. Consider a set Ω, a closed, densely defined operator H0 affiliatedwith M, V ∈ I1/2 and a set F of functions that satisfy one of the followingassertions:

(i) Ω = R, H0 and H0 + V are maximal dissipative operators, and

F = span{λ �→ (z − λ)−k : k ∈ N, Im(z) < 0

};

(ii) Ω = T, ‖H0‖ ≤ 1, ‖H0 + V ‖ ≤ 1, and F is the set of all functions that areanalytic on discs centered at 0 and of radius strictly larger than 1.

Theorem 3.8. Let Ω, H0, V and F satisfy Hypotheses 3.7. Then, there exists a(countably additive, complex) measure ν2 = ν2,H0,V on Ω such that

‖ν2‖ ≤ 1

2τI(|V |2)

and

τI

(f(H0 + V )− f(H0)− d

dt

∣∣∣∣t=0

f(H0 + tV )

)=

∫Ω

f ′′(λ) ν2(dλ),

for every f ∈ F .

Theorem 3.9. Suppose τI(I3/2) = {0}. Either assume Hypotheses 3.7 or else takeH0 = H∗

0 ∈M, V = V ∗ ∈ I1/2, and F = C4(R). Then, for every f ∈ F ,

τI

(f(H0 + V )− f(H0)− d

dt

∣∣∣∣t=0

f(H0 + tV )

)=

1

2τI

(d2

dt2

∣∣∣∣t=0

f(H0 + tV )

).


The major components in the proofs of Theorems 3.5 and 3.8 are analogs ofthe estimates (2.7), which hold due to the continuity of τI with respect to ‖ · ‖I .However, presence of a singular component in the trace τI requires more carefultreatment of the operator derivatives than in the case of the normal trace Tr.

References

[1] V.M. Adamjan, H. Neidhardt, On the summability of the spectral shift function forpair of contractions and dissipative operators, J. Operator Theory 24 (1990), no. 1,187–205.

[2] V.M. Adamjan, B.S. Pavlov, Trace formula for dissipative operators, VestnikLeningrad. Univ. Mat. Mekh. Astronom. 1979, no. 2, 5–9, 118 (Russian).

[3] A.B. Aleksandrov, V.V. Peller, Trace formulae for perturbations of class Sm, J.Spectral Theory, 1 (2011), no. 1, 1–26.

[4] A.B. Aleksandrov, V.V. Peller, D. Potapov, F.A. Sukochev, Functions of normaloperators under perturbations, Adv. Math. 226 (2011), no. 6, 5216–5251.

[5] N.A. Azamov, A.L. Carey, F.A. Sukochev, The spectral shift function and spectralflow, Comm. Math. Phys. 276 (2007), no. 1, 51–91.

[6] N.A. Azamov, A.L. Carey, P.G. Dodds, F.A. Sukochev, Operator integrals, spectralshift, and spectral flow, Canad. J. Math. 61 (2009), no. 2, 241–263.

[7] N.A. Azamov, P.G. Dodds, F.A. Sukochev, The Krein spectral shift function insemifinite von Neumann algebras, Integral Equations Operaor Theory 55 (2006),347–362.

[8] M.-T. Benameur, A.L. Carey, J. Phillips, A. Rennie, F.A. Sukochev, K.P. Woj-ciechowski, An analytic approach to spectral flow in von Neumann algebras. Analysis,geometry and topology of elliptic operators, 297–352, World Sci. Publ., Hackensack,NJ, 2006.

[9] M. Sh. Birman, A.B. Pushnitski, Spectral shift function, amazing and multifaceted.Dedicated to the memory of Mark Grigorievich Krein (1907–1989), Integral Equa-tions Operator Theory 30 (1998), no. 2, 191–199.

[10] M.Sh. Birman, M.Z. Solomyak, Remarks on the spectral shift function, ZapiskiNauchn. Semin. LOMI 27 (1972), 33–46 (Russian). Translation: J. Soviet Math.3 (1975), 408–419.

[11] M.Sh. Birman, M. Solomyak, Double operator integrals in a Hilbert space, IntegralEquations Operator Theory 47 (2003), no. 2, 131–168.

[12] M. Sh. Birman, D.R. Yafaev, The spectral shift function. The papers of M.G. Kreinand their further development, Algebra i Analiz 4 (1992), no. 5, 1–44 (Russian).Translation: St. Petersburg Math. J. 4 (1993), no. 5, 833–870.

[13] J.-M. Bouclet, Trace formulae for relatively Hilbert-Schmidt perturbations, Asymp-tot. Anal. 32 (2002), 257–291.

[14] R.W. Carey, J.D. Pincus, Mosaics, principal functions, and mean motion in vonNeumann algebras, Acta Math. 138 (1977), 153–218.

[15] M. Caspers, S. Montgomery-Smith, D. Potapov, F. Sukochev, The best constants foroperator Lipschitz functions on Schatten classes, preprint.

254 A. Skripka

[16] A.H. Chamseddine, A. Connes, The spectral action principle, Comm. Math. Phys.186 (1997), 731–750.

[17] Yu.L. Daleckii, S.G. Krein, Integration and differentiation of functions of Hermitianoperators and applications to the theory of perturbations, (Russian) Vorone. Gos.Univ. Trudy Sem. Funkcional. Anal. 1956 (1956), no. 1, 81–105.

[18] K. Dykema, A. Skripka, Higher order spectral shift, J. Funct. Anal. 257 (2009), 1092–1132.

[19] K. Dykema, A. Skripka, Perturbation formulas for traces on normed ideals, Comm.Math. Phys. 325 (2014), no 3, 1107–1138.

[20] Y.B. Farforovskaya, An estimate of the nearness of the spectral decompositions ofself-adjoint operators in the Kantorovic–Rubinstein metric, Vestnik Leningrad Univ.22 (1967), no. 19, 155–156.

[21] Y.B. Farforovskaya, The connection of the Kantorovic–Rubinstein metric for spectralresolutions of self-adjoint operators with functions of operators, Vestnik LeningradUniv. 23 (1968), no. 19, 94–97.

[22] Y.B. Farforovskaya, An example of a Lipschitz function of self-adjoint operators withnon-nuclear difference under a nuclear perturbation, Zap. Nauchn. Sem. Leningrad.Otdel. Mat. Inst. Steklov. (LOMI) 30 (1972), 146–153.

[23] F. Gesztesy, A. Pushnitski, B. Simon, On the Koplienko spectral shift function, I.Basics, Zh. Mat. Fiz. Anal. Geom. 4 (2008), no. 1, 63–107.

[24] E. Kissin, D. Potapov, V. Shulman, F. Sukochev, Operator smoothness in Schattennorms for functions of several variables: Lipschitz conditions, differentiability andunbounded derivations, Proc. London Math. Soc. 105 (2012), no. 4, 661–702.

[25] L.S. Koplienko, Local conditions for the existence of the function of spectral shift.Investigations on linear operators and the theory of functions, VIII. Zap. Nauchn.Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 73 (1977), 102–117 (Russian).Translation: J. Soviet Math. 34 (1986), no. 6, 2080–2090.

[26] L.S. Koplienko, Trace formula for perturbations of nonnuclear type, Sibirsk. Mat.Zh. 25 (1984), 62–71 (Russian). Translation: Siberian Math. J. 25 (1984), 735–743.

[27] M.G. Krein, On a trace formula in perturbation theory, Matem. Sbornik 33 (1953),597–626 (Russian).

[28] M.G. Krein, On the perturbation determinant and the trace formula for unitary andself-adjoint operators, Dokl. Akad. Nauk SSSR 144 (1962), 268–271 (Russian). Trans-lation: Soviet Math. Dokl. 3 (1962), 707–710.

[29] M.G. Krein, Some new studies in the theory of perturbations of self-adjoint operators,First Math. Summer School, Part I, 1964, pp. 103–187, Izdat. “Naukova Dumka”,Kiev (Russian). Translation: Topics in differential and integral equations and oper-ator theory, Birkhauser Verlag, Basel, 1983, pp. 107–172.

[30] M.G. Krein, Perturbation determinants and a trace formula for some classes of pairsof operators, J. Operator Theory 17 (1987), no. 1, 129–187 (Russian).

[31] H. Langer, Eine Erweiterung der Spurformel der Storungstheorie, Math. Nachr. 30,123–135 (1965) (German).

[32] I.M. Lifshits, On a problem of the theory of perturbations connected with quantumstatistics, Uspehi Matem. Nauk (N.S.) 7 (1952), no. 1 (47), 171–180 (Russian).


[33] S. Lord, F. Sukochev, D. Zanin, Singular Traces, de Gruyter Studies in Mathematics,46, Walter de Gruyter & Co., Berlin, 2012.

[34] K.A. Makarov, A. Skripka, M. Zinchenko, On perturbation determinant for antidis-sipative operators, preprint.

[35] H. Neidhardt, Scattering matrix and spectral shift of the nuclear dissipative scatteringtheory. Operators in indefinite metric spaces, scattering theory and other topics(Bucharest, 1985), 237–250, Oper. Theory Adv. Appl., 24, Birkhauser, Basel, 1987.

[36] H. Neidhardt, Scattering matrix and spectral shift of the nuclear dissipative scatteringtheory. II. J. Operator Theory 19 (1988), no. 1, 43–62.

[37] H. Neidhardt, Spectral shift function and Hilbert–Schmidt perturbation: extensionsof some work of L.S. Koplienko, Math. Nachr. 138 (1988), 7–25.

[38] V.V. Peller, Hankel operators in the perturbation theory of unbounded self-adjointoperators. Analysis and partial differential equations, Lecture Notes in Pure andApplied Mathematics, 122, Dekker, New York, 1990, pp. 529–544.

[39] V.V. Peller, An extension of the Koplienko–Neidhardt trace formulae, J. Funct. Anal.221 (2005), 456–481.

[40] V.V. Peller, Multiple operator integrals and higher operator derivatives, J. Funct.Anal. 223 (2006), 515–544.

[41] V.V. Peller, The behavior of functions of operators under perturbations. A glimpse atHilbert space operators, 287–324, Oper. Theory Adv. Appl., 207, Birkhauser Verlag,Basel, 2010.

[42] G. Pisier, Q. Xu, Noncommutative Lp-spaces. Handbook of the Geometry of Banachspaces, 2, North-Holland, Amsterdam, 2003, pp. 1459–1517.

[43] D. Potapov, A. Skripka, F. Sukochev, Spectral shift function of higher order, Invent.Math., 193 (2013), no. 3, 501–538.

[44] D. Potapov, A. Skripka, F. Sukochev, On Hilbert–Schmidt compatibility, Oper. Ma-trices, 7 (2013), no. 1, 1–34.

[45] D. Potapov, A. Skripka, F. Sukochev, Higher order spectral shift for contractions,Proc. London Math. Soc. 108 (2014), no 3, 327–349.

[46] D. Potapov, F. Sukochev, Operator-Lipschitz functions in Schatten–von Neumannclasses, Acta Math., 207 (2011), 375–389.

[47] D. Potapov, F. Sukochev, Koplienko spectral shift function on the unit circle, Comm.Math. Phys., 309 (2012), 693–702.

[48] D. Potapov, F. Sukochev, D. Zanin, Krein’s trace theorem revisited, J. Spectral The-ory, in press.

[49] A.V. Rybkin, The spectral shift function for a dissipative and a selfadjoint operator,and trace formulas for resonances, Mat. Sb. (N.S.) 125(167) (1984), no. 3, 420–430(Russian).

[50] A.V. Rybkin, The spectral shift function, the characteristic function of a contractionand a generalized integral, Mat. Sb. 185 (1994), no. 10, 91–144 (Russian). Translation:Russian Acad. Sci. Sb. Math. 83 (1995), no. 1, 237–281.

[51] A.V. Rybkin, On A-integrability of the spectral shift function of unitary operatorsarising in the Lax–Phillips scattering theory, Duke Math. J. 83 (1996), no. 3, 683–699.

256 A. Skripka

[52] B. Simon, Trace ideals and their applications. Second edition. Mathematical Surveysand Monographs, 120. American Mathematical Society, Providence, RI, 2005.

[53] A. Skripka, Trace inequalities and spectral shift, Oper. Matrices 3 (2009), no. 2,241–260.

[54] A. Skripka, Higher order spectral shift, II. Unbounded case, Indiana Univ. Math. J.59 (2010), no. 2, 691–706.

[55] A. Skripka, Multiple operator integrals and spectral shift, Illinois J. Math., 55 (2011),no. 1, 305–324.

[56] A. Skripka, Asymptotic expansions for trace functionals, J. Funct. Anal. 266 (2014),no 5, 2845–2866.

[57] B. Sz.-Nagy, C. Foias, Harmonic analysis of operators on Hilbert space. Translatedfrom the French and revised, North-Holland Publishing Co., Amsterdam-London,1970.

[58] W.D. van Suijlekom, Perturbations and operator trace functions, J. Funct. Anal. 260(2011), no. 8, 2483–2496.

[59] D.R. Yafaev, Mathematical scattering theory: general theory, Providence, R.I., AMS,1992.

[60] D.R. Yafaev, The Schrodinger operator: perturbation determinants, the spectral shiftfunction, trace identities, and more, Funktsional. Anal. i Prilozhen. 41 (2007), no. 3,60–83 (Russian). Translation: Funct. Anal. Appl. 41 (2007), no. 3, 217–236.

Anna SkripkaDepartment of Mathematics and StatisticsUniversity of New Mexico400 Yale Blvd NE, MSC01 1115Albuquerque, NM 87131, USAe-mail: [email protected]

Documents

Operator Theory in Harmonic and Non-commutative Analysis: 23rd International Workshop in Operator Theory and its Applications, Sydney, July 2012