Recent Progress on Operator Theory and Approximation in Spaces of Analytic Functions

679

Recent Progress on OperatorTheory and Approximation inSpaces of Analytic Functions

Conference onCompleteness Problems, Carleson Measures,

and Spaces of Analytic FunctionsJune 29–July 3, 2015

Institut Mittag-Leffler, Djursholm, Sweden

Catherine BénéteauAlberto A. Condori

Constanze LiawWilliam T. RossAlan A. Sola

Editors

American Mathematical Society







Editors

679







Editors

American Mathematical SocietyProvidence, Rhode Island

EDITORIAL COMMITTEE

Dennis DeTurck, Managing Editor

Michael Loss Kailash Misra Catherine Yan

2010 Mathematics Subject Classification. Primary 11M41, 30H05, 30H20, 35P10,47A16, 47B32.

Library of Congress Cataloging-in-Publication Data

Names: Beneteau, Catherine, 1960- editor.Title: Recent progress on operator theory and approximation in spaces of analytic functions :

Conference on Completeness Problems, Carleson Measures, and Spaces of Analytic Functions,June 29-July 3, 2015, Institut Mittag-Leffler, Djursholm, Sweden / Catherine Beneteau [andfour others], editors.

Description: Providence, Rhode Island : American Mathematical Society, [2016] | Series: Con-temporary mathematics ; volume 679 | Includes bibliographical references.

Identifiers: LCCN 2016023124 | ISBN 9781470423056 (alk. paper)Subjects: LCSH: Operator theory–Congresses. | Analytic spaces–Congresses. | Analytic functions–

Congresses. | AMS: Number theory – Zeta and L-functions: analytic theory – Other Dirichletseries and zeta functions. msc | Functions of a complex variable – Spaces and algebras ofanalytic functions – Bounded analytic functions. msc | Functions of a complex variable –Spaces and algebras of analytic functions – Bergman spaces, Fock spaces. msc | Partial differ-ential equations – Spectral theory and eigenvalue problems – Completeness of eigenfunctions,eigenfunction expansions. msc | Operator theory – General theory of linear operators – Cyclicvectors, hypercyclic and chaotic operators. msc | Operator theory – Special classes of linearoperators – Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces). msc

Classification: LCC QA329 .R434 2016 | DDC 515/.724–dc23 LC record available at https://lccn.loc.gov/2016023124

Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online)

DOI: http://dx.doi.org/10.1090/conm/679

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Contents

Preface vii

List of Participants ix

Properties of vector-valued submodules on the bidiskKelly Bickel and Constanze Liaw 1

Composition operators on the Hardy–Hilbert space H2 and model spaces KΘ

Abdellatif Bourhim and Javad Mashreghi 13

A survey of some recent results on truncated Toeplitz operatorsIsabelle Chalendar, Emmanuel Fricain, and Dan Timotin 59

Approximating z in the Bergman SpaceMatthew Fleeman and Dmitry Khavinson 79

Real complex functionsStephan Ramon Garcia, Javad Mashreghi,

and William T. Ross 91

Thin Interpolating SequencesPamela Gorkin and Brett D. Wick 129

Kernels of Toeplitz operatorsAndreas Hartmann and Mishko Mitkovski 147

Some open questions in analysis for Dirichlet seriesEero Saksman and Kristian Seip 179

Some problems on optimal approximantsDaniel Seco 193

Some open problems in complex and harmonic analysis: Report on problemsession held during the conference Completeness problems, Carleson measures,and spaces of analytic functions

Catherine Beneteau, Alberto A. Condori, Constanze Liaw,

William T. Ross, and Alan A. Sola 207

v

Preface

This volume contains the proceedings of the conference Completeness problems,Carleson measures, and spaces of analytic functions that took place at the InstitutMittag-Leffler (IML), Sweden, from June 29 to July 3, 2015.

The conference brought together experienced researchers and promising youngmathematicians from many countries to discuss recent progress made in functiontheory, model spaces, completeness problems, and Carleson measures. It consistedof roughly 14 hours of lectures, an open problems session, and discussion periodsthat promoted collaboration among the participants.

In these refereed proceedings, we have included contributed articles that havenot appeared elsewhere, both on cutting-edge research questions as well as longersurvey papers, and a report on the problem session that contains a collection ofattractive open problems in complex and harmonic analysis.

We would like to take the opportunity to thank the IML staff and Director,Ari Laptev, for providing an outstanding venue that allowed the development ofinformal collaborations among the participants. In addition, we are grateful to theNational Science Foundation for their support of the US participants, in particulargraduate students, post-docs, and pre-tenure faculty. We would also like to thankall the authors who submitted articles to this collection. A special thanks goesto Alexandru Aleman, Emmanuel Fricain, Alexei Poltoratski, Rishika Rupam, andKristian Seip, for their presentations at the problem session at the conference aswell as their written contributions to the volume. We also thank the anonymous ref-erees whose prompt and thorough reviews improved the quality of the proceedings.Finally, we thank the participants of the conference whose enthusiastic participationmade the event a great success.

Catherine BeneteauAlberto A. Condori

Constanze LiawWilliam T. Ross

Alan A. Sola

vii

List of Participants

Alexandru AlemanLund University

Catherine BeneteauUniversity of South Florida

Roman BessonovSt. Petersburg State University

Kelly BickelBucknell University

Alberto A. CondoriFlorida Gulf Coast University

Matthew FleemanUniversity of South Florida

Emmanuel FricainUniversite Lille 1

Dale FrymarkBaylor University

Pamela GorkinBucknell University

Andreas HartmannUniversite Bordeaux 1

Hakan HedenmalmRoyal Institute of Technology

Irina HolmesWashington University - St. Louis

Dmitry KhavinsonUniversity of South Florida

Michael T. LaceyGeorgia Institute of Technology

Constanze LiawBaylor University

Javad MashreghiUniversite Laval

Mishko MitkovskiClemson University

Stefanie PetermichlUniversite Paul Sabatier

Alexei PoltoratskiTexas A & M University

William T. RossUniversity of Richmond

Rishika RupamUniversite Lille 1

Eero SaksmanUniversity of Helsinki

Daniel SecoUniversitat de Barcelona

Kristian SeipNorwegian University of Science andTechnology

Alan A. SolaUniversity of South Florida

Brett D. WickWashington University - St. Louis

ix

Contemporary MathematicsVolume 679, 2016http://dx.doi.org/10.1090/conm/679/13668

Properties of vector-valued submodules on the bidisk

Kelly Bickel and Constanze Liaw

Abstract. In previous work, the authors studied the compressed shift oper-

ators Sz1 and Sz2 on two-variable model spaces H2(D2) � θH2(D2), where θis a two-variable scalar inner function. Among other results, the authors usedAgler decompositions to characterize the ranks of the operators [Szj , S

∗zj] in

terms of the degree of rational θ. In this paper, we examine similar questionsfor H2(D2)�ΘH2(D2) when Θ is a matrix-valued inner function. We extendseveral results from our previous work connecting Rank[Szj , S

∗zj] and the de-

gree of Θ to the matrix setting. When results do not clearly generalize, weconjecture what is true and provide supporting examples.

1. Introduction

Both Beurling’s theorem on shift invariant subspaces for the Hardy space onthe disk H2(D) [4] and the model theory of Sz.-Nagy–Foias (see e.g. [19]) were ofindisputable importance to central developments in function and operator theory.In this paper, we are interested in generalizations of this classical Hardy spacetheory to the Hardy space on the bidisk H2(D2). For examples, see e.g. [7,8,18].In analogy with objects important in the one-variable setting, we consider Hilbertsubmodules – namely subspaces of H2(D2) that are invariant under the Toeplitz(or shift) operators Tz1 and Tz2 . Because of their close connections to one-variableresults and the structure of inner functions, we restrict attention to submodulesof Beurling-type, which are submodules of the form θH2(D2), where θ is an innerfunction on the bidisk.

Given a submodule of Beurling-type θH2(D2), one can define the associatedtwo-variable model space Kθ ≡ H2(D2) � θH2(D2). As in the one-variable set-ting, the compressed shift operators on these Kθ spaces possess many interestingproperties. Specifically, define

Sz1 ≡ PθTz1 |Kθand Sz2 ≡ PθTz2 |Kθ

,

where Pθ denotes the projection onto Kθ. Interestingly, the behaviors of the crosscommutators [Sz1 , S

∗z2 ] and [Sz1 , Sz2 ] are closely related to both properties of θ

2010 Mathematics Subject Classification. Primary 47A13, 47A20, 46E22.Key words and phrases. Model spaces, two complex variables, compressed shift, Agler de-

composition, essential normality.The first author’s research was supported in part by National Science Foundation DMS grant

#1448846.

c©2016 American Mathematical Society

1

2 K. BICKEL AND C. LIAW

and the structure of Kθ. See e.g. [10,12,13,21, 22]. However, the properties ofindividual operators Sz1 and Sz2 are not as well-understood.

One interesting result by Guo–Wang in [11] concerns rational inner functions.To state it, we first recall that the degree of a rational function θ is (m1,m2) ifθ = p/q, where the polynomials p and q share no common factors and each mj isthe maximum degree of p and q in zj . Then, Guo–Wang’s theorem in [11] statesthat both [Sz1 , S

∗z1 ] and [Sz2 , S

∗z2 ] are compact if and only if θ is a rational inner

function of degree at most (1, 1). Complementing Guo–Wang’s result, the authors[6, Theorem 1.1] proved:

Theorem 1.1. Let θ be an inner function in H2(D2). Then, the commutator[S∗

z1 , Sz1 ] has rank n if and only if θ is rational inner of degree (1, n) or (0, n).

In this paper, we seek a generalization of Theorem 1.1 to d× d matrix-valuedinner functions Θ. One-variable matrix-valued inner functions appeared in themodel theory of Sz.-Nagy–Foias and ever since, matrix inner functions have beenfrequently studied alongside scalar inner functions in both the one and two-variabletheory. For examples, see [3,5,9,14–17,20]. Nevertheless, many proof techniquesand results become much more complicated in this matrix setting.

Before discussing our results, let us introduce several standard definitions. Atwo-variable d × d matrix-valued function Θ is called inner if the entries of Θ areholomorphic functions and

Θ(τ )Θ(τ )∗ = Θ(τ )∗Θ(τ ) = Id×d for a.e. τ ∈ T2.

The vector-valued Hardy space is given by H2d(D

2) ≡ H2(D2)⊗ Cd, and

KΘ ≡ H2d(D

2)�ΘH2d(D

2)

is the vector-valued model space associated to Θ.We will use decompositions of vector-valued KΘ spaces induced via Agler ker-

nels to study the compressed shift operators Sz1 and Sz2 . To begin, recall thatpositive matrix-valued kernel functions K1,K2 : D2×D2 →Md(C) are called Aglerkernels of Θ if they decompose Θ as follows

I −Θ(z)Θ(w)∗ = (1− z1w1)K2(z, w) + (1− z2w2)K1(z, w), ∀ z, w ∈ D2.

J. Agler proved the existence of Agler kernels in [1]. Subsequent work in [3] gavecanonical constructions of Agler kernels, which were further explored in [5]. Hereis the basic setup. Define Smax

1 to be the maximal Tz1-invariant subspace of KΘ.Then Smax

1 is the set of functions f with zk1f ∈ KΘ for all k ∈ N. Define Smin2 =

KΘ � Smax1 . It is not hard to show that there are matrix-valued kernel functions

(Kmax1 ,Kmin

2 ) such that

Smin2 = H

(Kmin

2 (z, w)

1− z2w2

)and Smax

1 = H(Kmax

1 (z, w)

1− z1w1

),

where H(K) indicates the Hilbert space with reproducing kernel K. Easy com-putations imply that (Kmax

1 ,Kmin2 ) are Agler kernels of Θ. One can similarly de-

fine shift-invariant subspaces Smin1 and Smax

2 of KΘ, which yield Agler kernels(Kmin

1 ,Kmax2 ). See [5] for details.

Our main results concern matrix-valued inner functions whose entries are alsorational functions. We say that a rational inner matrix-valued function is of degree(m1,m2), if mj is the maximum degree of its scalar-valued entries in zj , for j = 1, 2.We also write this as degj Θ = mj for j = 1, 2. It is worth pointing out that if Θ is

VECTOR-VALUED SUBMODULES ON THE BIDISK 3

a matrix-valued rational inner function, then its determinant, denoted detΘ, is ascalar-valued rational inner function.

1.1. Summary of Results. In this paper, we partially extend the resultsof [6] to the matrix setting. As with [6], we first examine the situation whereΘ is a product of one-variable inner functions. In the scalar setting, this studyilluminated the connections between Agler kernels and compressed shift operatorsand provided a roadmap for obtaining more general results. In this matrix-setting,these product inner functions are not as helpful. Indeed, rather than illuminatinggeneral results, this preliminary study illustrates that non-commutativity makeseven seemingly-simple situations very complicated.

Nevertheless, in Section 3, we generalize several parts of Theorem 1.1 to ma-trices. First, we show that if Θ is a rational inner function of a particular degree,then the associated commutator [Sz1 , S

∗z1] will have finite rank. The details are as

follows.

Theorem. 3.2. Assume that Θ = Qp as in (3.1) is a d× d matrix-valued inner

function on the bidisk with degQ ≤ (1, n). Then

Rank[Sz1 , S∗z1 ] ≤ dn.

We also study the other direction of Theorem 1.1. Here, several of the scalararguments completely break down in the matrix setting. Still, we are able toconclude that if the commutator has finite rank, then a certain object associatedto Θ is also finite. Specifically, we conclude the following:

Theorem. 3.3. Assume that Θ is a d× d matrix-valued inner function on thebidisk with Rank[Sz1 , S

∗z1 ] = n. Then

dimH(Kmax1 ) ≤ n.

At first glance, this result appears quite different from Theorem 1.1. To rewriteit in terms of the degree of Θ, we use Theorem 3.1, which links the degree of thedeterminant detΘ with the dimension of such subspaces. Indeed, if Θ is rationalinner, then Theorem 3.3 paired with Theorem 3.1 says that Rank[Sz1 , S

∗z1 ] = n

implies that deg2 detΘ ≤ n. This is much more in line with Theorem 1.1. Forseveral reasons, the results obtained in Theorems 3.2 and 3.3 are unsatisfactory.These points are discussed in detail in Remark 3.4 and lead us to the followingconjecture:

Conjecture. 3.5. Let Θ be a d×d matrix-valued inner function on the bidisk.Then Rank[Sz1 , S

∗z1 ] = n if and only if deg1 Θ ≤ 1 and deg2 detΘ = n.

This conjecture is supported by several nontrivial examples detailed in Section4. It is worth noting that the proof of Guo–Wang’s result also uses formulas specificto scalar-valued rational inner functions. For this reason, it is also not clear howto fully generalize their proofs and results to the matrix setting.

2. Products of One-Variable Inner Functions

In the scalar setting, if θ(z) = φ(z1)ψ(z2) is a product of one-variable inner func-tions, then the properties of Szj and its commutator [Szj , S

∗zj ] are well-understood


for j = 1, 2. Specifically, see [6, Section 2] for results concerning the reducing sub-spaces, essential normality, and spectra of these operators. The obtained resultsrest on the simple decompositions

1− φ(z1)ψ(z2)ψ(w2)φ(w1)

=(1− ψ(z2)ψ(w2)

)+ ψ(z2)

(1− φ(z1)φ(w1)

)ψ(w2)

=(1− φ(z1)φ(w1)

)+ φ(z1)

(1− ψ(z2)ψ(w2)

)φ(w1).

Using these, one can obtain nice formulas for the reproducing kernels of the shift-invariant subspaces Smax

1 ,Smin1 and Smax

2 ,Smin2 of Kθ. Most results follow from

studying Szj and [Szj , S∗zj ] on these well-understood subspaces.

When Θ is a matrix-valued product of one-variable inner functions, this methodno longer works. Indeed, non-commutativity implies that such Θ could be of theform

(2.1) Θ(z) =N∏i=1

Φi(z1)Ψi(z2),

with no apparent simplification. Even in the simplest case, when Θ(z)=Φ(z1)Ψ(z2),finding reproducing kernels for Smax

j and Sminj is complicated. Indeed, non-

commutativity means that, in general,

I − Φ(z1)Ψ(z2)Ψ(w2)∗Φ(w1)

∗

=(I −Ψ(z2)Ψ(w2)∗) + Ψ(z2) (1− Φ(z1)Φ(w1)

∗)Ψ(w2)∗.

Because this factorization fails, we do not know how to obtain reproducing kernelformulas for the spaces Smax

1 and Smin2 . In contrast, for this particular Θ, the

symmetric factorization

I − Φ(z1)Ψ(z2)Ψ(w2)∗Φ(w1)

∗

=(I − Φ(z1)Φ(w1)∗) + Φ(z1) (1−Ψ(z2)Ψ(w2)

∗)Φ(w1)∗

does hold, which, in some cases, makes it possible to write down formulas for thereproducing kernels of Smax

2 and Smin1 . Such formulas can be proved using the

characterizations of Smaxj and Smin

j in [5]. However, for the more general Θ givenin (2.1), it is not clear how to obtain formulas for any of the subspaces. Without thereproducing kernel formulas for Smax

j and Sminj , many of the proofs from [6, Section

2] establishing results about the Szj and [Sz1 , S∗zj ] do not generalize. This motivates

the question:

Open question. Is there a method for determining the reproducing kernelformulas for Smax

j and Sminj when Θ is a d× d matrix-valued inner function of the

form (2.1)?

The previous question may be asking too much. Indeed, it may be possible toestablish certain results, such as the characterization of reducing subspaces, withoutestablishing concrete formulas for the reproducing kernels. This seems especiallypossible since various characterizations of the spaces Smax

j , Sminj were obtained in

[5] for matrix-valued Θ. This leads to the general question:

Open question. Do any of the results about Szj and [Szj , S∗zj ] from [6, Section

2] generalize to case where Θ is a d× d matrix-valued inner function of form (2.1)?


These open questions indicate the complexity of many seemingly-simple prob-lems in the matrix setting.

3. Relationship Between Degree of Θ and Rank of [Szj , S∗zj ]

In [6], the authors proved Theorem 1.1 by exploiting connections between thedegree of Θ and the structure of related subspaces H(Kmax

j ) and H(Kminj ). The

needed connections are detailed in [6, Theorem 3.2]. These connections do gener-alize to matrix-valued inner functions. To state them, recall that if Θ is a rationalinner d× d matrix-valued function, then we can write

(3.1) Θ(z) =1

p(z)Q(z),

where the polynomial p(z) is the least common multiple of the denominators of theentries of Θ after each entry is put in reduced form and Q(z) satisfies

Q(τ )Q(τ )∗ = Q(τ )∗Q(τ ) = |p(τ )|2I for a.e. τ ∈ T2.

Given this representation, we can state the following result, which generalizes [6,Theorem 3.2] to matrix-valued inner functions. The proof is in [5]; the degreebounds appear in [5, Theorem 1.7] and dimension results appear in [5, Theorem1.8].

Theorem 3.1. Let Θ = Qp be a d× d matrix-valued rational inner function on

the bidisk of degQ = (m,n). Then

dimH(Kmax1 ) = dimH(Kmin

1 ) = deg2 detΘ,

dimH(Kmax2 ) = dimH(Kmin

2 ) = deg1 detΘ.

Furthermore, if f is a function in H(Kmax1 ) or H(Kmin

1 ) then f = qp where deg q ≤

(m,n − 1) and if g is a function in H(Kmax2 ) or H(Kmin

2 ) then g = rp , where

deg r ≤ (m− 1, n).

We use this result to obtain the following generalization of one direction ofTheorem 1.1:

Theorem 3.2. Let Θ = Qp as in (3.1) be a d× d matrix-valued rational inner

function on the bidisk with degQ ≤ (1, n), then

Rank[Sz1 , S∗z1 ] ≤ dn.

The proof is similar to that of the corresponding direction in Theorem 1.1. Forthe convenience of the reader, we include some details.

Proof. Let Θ = Qp be a d × d matrix-valued rational inner function with

degQ ≤ (1, n) and let N = deg2 detΘ and M = deg1 detΘ. Notice that N ≤ dnand M ≤ d. Theorem 3.1 with m ≤ 1 informs us that we can find vector-valuedfunctions fi, i = 1, . . . , N with deg fi ≤ (1, n − 1) and gj , j = 1, . . . ,M withdeg gj ≤ (0, n) such that

Kmax1 (z, w) =

N∑i=1

fi(z)fi(w)∗

p(z)p(w)and Kmin

2 (z, w) =M∑j=1

gj(z)gj(w)∗

p(z)p(w).


Without loss of generality, we assume orthogonality and normality (or trivial norms)

of{

fip

}, and likewise for

{gjp

}. Then, since KΘ = Smax

1 ⊕ Smin2 , we can write the

reproducing kernel of KΘ as the sum of the reproducing kernels K1w(z) and K2

w(z)of the spaces Smax

1 and Smin2 as follows

I −Θ(z)Θ(w)∗

(1− z1w1)(1− z2w2)=K1

w(z) +K2w(z)

=

∑Ni=1 fi(z)fi(w)

∗

p(z)p(w)(1− z1w1)+

∑Mj=1 gj(z)gj(w)

∗

p(z)p(w)(1− z2w2).

Now, fix e ∈ Cd and w ∈ D2. Using the structures of K1w(z)e and K2

w(z)e, weestablish formulas for [S∗

z1 , Sz1 ]Kjwe. As the proofs are quite technical and follow

the scalar arguments from [6] closely, we omit the details. Here are the obtainedformulas [

S∗z1 , Sz1

]K1

we = PΘ

(N∑i=1

fi(0, z2)

p(0, z2)

(fi(w)

p(w)

)∗e

)and similarly [

S∗z1 , Sz1

]K2

we = PΘ

(N∑i=1

Tz1fi(z)

p(0, z2)

(Tz1

fip (w)

)∗e

),

where PΘ denotes the projection onto KΘ. Combining these two formulas showsthat

[S∗z1 , Sz1 ]

I −Θ(z)Θ(w)∗

(1− z1w1)(1− z2w2)e =

PΘ

(1

p(0, z2)

N∑i=1

Tz1fi(z)(Tz1

fip (w)

)∗e+ fi(0, z2)

(fi(w)p(w)

)∗e

).

Since deg fi ≤ (1, n− 1), then deg Tz1fi ≤ (0, n− 1) and deg fi(0, z2) ≤ (0, n− 1).Thus, considering all w ∈ D2 and e ∈ Cd, the set of vector-valued functions of theform

1

p(0, z2)

N∑i=1

Tz1fi(z)(Tz1

fip (w)

)∗e+ fi(0, z2)

(fi(w)p(w)

)∗e

can have at most dimension nd. By the definition of KΘ, linear combinations offunctions of the form

I −Θ(z)Θ(w)∗

(1− z1w1)(1− z2w2)e

are dense in KΘ. Thus, we can immediately conclude that

Rank[Sz1 , S∗z1 ] ≤ nd,

as desired. �

We can similarly study the other direction of Theorem 1.1 in the matrix setting.The following result provides a partial generalization.

Theorem 3.3. Assume that Θ is a d× d matrix-valued inner function on thebidisk with Rank[Sz1 , S

∗z1 ] = n. Then

dimH(Kmax1 ) ≤ n.


Proof. First, observe that if f ∈ Smax1 , then z1f ∈ KΘ and so(

Sz1S∗z1 − S∗

z1Sz1

)f = PΘ (z1Tz1f − f) = −PΘ (f(0, z2)) .

Now, assume that Rank[Sz1 , S∗z1 ] = n and by way of contradiction, assume

dimH(Kmax1 ) > n. Then there is some nontrivial f ∈ H(Kmax

1 ) such that f ∈ker[Sz1 , S

∗z1 ]. This means

PΘf(0, z2) = 0

and so, there is some vector-valued h ∈ H2d(D

2) such that we have f(0, z2) =Θ(z)h(z). But then, using basic orthogonality relations,

‖f(0, z2)‖2H2 = 〈f, f(0, z2)〉H2 = 〈f,Θh〉H2 = 0.

Thus, f(0, z2) ≡ 0 and f(z) = z1Tz1f(z). As f ∈ Smax1 , this implies zk1Tz1f(z) ∈ KΘ

for all k ∈ N. Thus, we can conclude that Tz1f ∈ Smax1 and so, f ∈ z1Smax

1 . AsH(Kmax

1 ) = Smax1 �z1Smax

1 , we conclude that f ⊥ f and so f ≡ 0, a contradiction.�

Remark 3.4. Theorems 3.2 and 3.3 are unsatisfactory for two reasons. First,in the scalar setting, Theorem 1.1 shows that if [Sz1 , S

∗z1 ] is finite rank, then θ

is a rational function. An important part of that result involves the fact thatRank[Sz1 , S

∗z1 ] <∞ implies deg1 θ ≤ 1. Unfortunately, the proof of that result relies

on scalar arguments that do not generalize to the matrix setting. Nevertheless, westill conjecture that Rank[Sz1 , S

∗z1 ] < ∞ implies deg1 Θ ≤ 1 and will discuss this

further in the next section.Now assume Θ is a d×d rational inner function with deg1 Q ≤ 1. In the scalar

setting, Theorem 1.1 shows that if deg1 θ ≤ 1, then Rank[Sz1 , S∗z1 ] = n if and only if

deg2 θ = n. Let us consider the matrix analogue of this result encoded in Theorems3.2 and 3.3. It says

If deg2 Q = n, then Rank[Sz1 , S∗z1 ] ≤ dn.

If Rank[Sz1 , S∗z1 ] = N, then deg2 detΘ ≤ N.

For Θ with deg2 detΘ = d · deg2 Q, then these results combine to give:

(3.2) Rank[Sz1 , S∗z1 ] = N if and only if deg2 detΘ = N.

We do not currently have this if and only if condition for all Θ because in general,

deg2 detΘ ≤ d · deg2 Q,

with strict inequality possible. However, we conjecture that (3.2) is actually truefor all Θ. We also think that the condition deg1 Q ≤ 1 can likely be loosened todeg1 Θ ≤ 1.

The conjectures discussed in Remark 3.4 combined with our known resultsyield:

Conjecture 3.5. Let Θ be a d×d matrix-valued inner function on the bidisk.Then Rank[Sz1 , S

∗z1 ] = n if and only if deg1 Θ ≤ 1 and deg2 detΘ = n.


4. Some Examples

In this section, we consider several examples supporting Conjecture 3.5. Wedemonstrate that this conjecture is true for d × d diagonal matrix-valued innerfunctions. We then investigate several 2×2 non-diagonal inner functions and showthat the conjecture holds for them as well.

Example 4.1. Consider the d× d diagonal matrix function

Θ(z) =

⎡⎢⎣θ1(z) . . .

θd(z)

⎤⎥⎦ ,where each θi(z) is a scalar two-variable inner function. Then, KΘ is a direct sumof the Kθi spaces and so,

Rank[Sz1 , S∗z1 ] on KΘ =

d∑i=1

(Rank[Sz1 , S

∗z1 ] on Kθi

).

Thus, by Theorem 1.1, Rank[Sz1 , S∗z1 ] = n on KΘ if and only if each θi is rational

inner with

deg1 θi ≤ 1 and

d∑i=1

deg2 θi = n.

Furthermore, using the structure of rational inner functions, for example, as givenin [2], one can show that

deg2 detΘ = deg2

(d∏

i=1

θi

)=

d∑i=1

deg2 θi.

It follows that Rank[Sz1 , S∗z1 ] = n on KΘ if and only if we have both deg1 Θ =

maxi deg1 θi ≤ 1 and deg2 detΘ = n. Thus, Conjecture 3.5 holds for diagonalmatrix-valued inner functions.

This further implies that Theorem 3.2 is not sharp. Specifically, consider

Θ(z) =

[z1z2 00 1

].

Theorem 3.2 implies that Rank[Sz1 , S∗z1 ] ≤ 2 · 1 = 2. However, as deg det2 Θ = 1,

our earlier arguments show that Rank[Sz1 , S∗z1 ] = 1.

We investigate two examples of inner functions Θ that are not diagonal.

Example 4.2. Consider the matrix-valued function

Θ(z) =1

2

[z1 + z2 z1 − z2z1 − z2 z1 + z2

].

A simple computation shows that Θ is unitary-valued on T2 and hence, is inner.Observe that we can decompose the reproducing kernel of KΘ as follows

I −Θ(z)Θ(w)∗

(1− z1w1)(1− z2w2)=

12

1− z1w1

[−11

] [−1 1

]+

12

1− z2w2

[11

] [1 1

].

This reproducing kernel decomposition induces the following orthogonal decompo-sition

KΘ =

[−11

]H2

1 (D)⊕[11

]H2

2 (D) = H1 ⊕H2,


where H2j (D) denotes the one variable Hardy space with independent variable zj .

We turn our attention to [Sz1 , S∗z1 ]. We first show that this operator is identically

zero on H2. Fix an arbitrary f ∈ H2, so f(z) =

[11

]g(z2), where g ∈ H2(D).

Observe that, as g is a function in z2,

Sz1S∗z1

[11

]g(z2) = 0.

Now we focus on Sz1f = PΘz1f. Fix an arbitrary element

H(z) =

[11

]h(z2) +

[−11

]h(z1) ∈ KΘ,

where h, h ∈ H2(D). A simple computation shows that

〈Sz1f,H〉KΘ= 〈z1f,H〉H2

=

⟨[z1z1

]g(z2),

[11

]h(z2) +

[−11

]h(z1)

⟩H2

=

⟨[g(0)g(0)

],

[−h′(0)

h′(0)

]⟩C2

= 0.

Thus, we can conclude that Sz1f = 0 and hence S∗z1Sz1f ≡ 0. As f ∈ H2 was

arbitrary, this implies [Sz1 , S∗z1 ]|H2

≡ 0.

Let us compute [Sz1 , S∗z1 ] on H1. Fix an arbitrary f ∈ H1, so f =

[−11

]g(z1),

where g ∈ H2(D). It is easy to calculate

[Sz1 , S∗z1 ]f =

(Sz1S

∗z1 − S∗

z1Sz1

) [−11

]g(z1)

=

[−11

](z1Tz1g − g) =

[g(0)−g(0)

].

From this, we can conclude that the image of [Sz1 , S∗z1 ] on H1 is

[−11

]C and so,

Rank[Sz1 , S∗z1 ]|H1

= 1.

Combining this with our result for [Sz1 , S∗z1 ]|H2

implies that

Rank[Sz1 , S∗z1 ] = 1.

We observe that deg1 Θ = 1 and as detΘ = 2z1z2, we have deg2 detΘ = 1. Thus,Conjecture 3.5 says Rank[Sz1 , S

∗z1 ] = 1, which agrees with our computed result.

In our last example we consider what happens when deg1 Θ > 1. In the scalarsetting, this always causes Rank[Sz1 , S

∗z1 ] =∞. Conjecture 3.5 claims that the same

holds true for matrix-valued inner functions. To test this conjecture, let us considerthe following example:

Example 4.3. Consider the matrix-valued function

Θ(z) =1

2

[z1(z1 + z2) z1(z1 − z2)z1 − z2 z1 + z2

].


A simple computation shows that Θ is unitary-valued on T2 and hence, is inner.Observe that we can decompose the reproducing kernel of KΘ as follows

I −Θ(z)Θ(w)∗

(1− z1w1)(1− z2w2)=

12

1− z1w1

[−z11

] [−w1 1

]+

12

1− z2w2

[z11

] [w1 1

]+

1

1− z2w2

[10

] [1 0

].

This reproducing kernel decomposition induces the following orthogonal decompo-sition

KΘ =

[−z11

]H2

1 (D)⊕[z11

]H2

2 (D)⊕[10

]H2

2 (D) = H1 ⊕H2 ⊕H3,

where H2j (D) denotes the one variable Hardy space with independent variable zj .

Consider [Sz1 , S∗z1 ]. As deg1 Θ = 2, Conjecture 3.5 indicates that the rank of this

operator should be infinite. To see why this is true, we consider [Sz1 , S∗z1 ]|H3

. Fix

an arbitrary f ∈ H3. Then f(z) =

[10

]g(z2) for some g ∈ H2(D). It is immediate

that

Sz1S∗z1

[10

]g(z2) = 0.

Now observe that

z1f(z) =

[z10

]g(z2) =

[z11

]g(z2)

2+

[z1−1

]g(0)

2+

[z1−1

]z2Tz2g(z2)

2.

The first two terms come from H2 and H1 respectively. Simple computations showthat the [

z1−1

]z2Tz2g(z2)

2⊥ H1 ⊕H2 ⊕H3 = KΘ.

Thus, we can compute:

Sz1f = PΘ

[z10

]g(z2) =

[z11

]g(z2)

2+

[z1−1

]g(0)

2.

Finally, we can conclude that

[Sz1 , S∗z1 ]f = −S∗

z1

([z11

]g(z2)

2+

[z1−1

]g(0)

2

)= −

[10

](g(z2) + g(0)

2

).

From this, it is clear that Rank[Sz1 , S∗z1 ] =∞.

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Department of Mathematics, Bucknell University, One Dent Drive, Lewisburg,

Pennsylvania 17837

E-mail address: [email protected]

CASPER and Department of Mathematics, Baylor University, One Bear Place

#97328, Waco, Texas 76798

E-mail address: Constanze [email protected]


Composition operators on the Hardy–Hilbert space H2 andmodel spaces KΘ

Abdellatif Bourhim and Javad Mashreghi

Abstract. We study the composition operators on the Hardy–Hilbert spaceH2 and the model spaces KΘ. We just address the question of ‘boundedness’and leave the ‘compactness’ for future studies. In the classical case of H2,several proofs of boundedness are provided.

1. Introduction

Let Hol(D) denote the family of all analytic functions on D, and let ϕ : D→ D

be an analytic self-map of D. Then the composition mapping Cϕ is defined by

Cϕ : Hol(D) −→ Hol(D)f �−→ f ◦ ϕ.

One of the major questions in the theory of composition operators is to find Xand Y , two Banach spaces of analytic functions which as a set sit inside Hol(D),such that the restricted mapping Cϕ : X → Y is well-defined and bounded. Inthis setting, and mostly for X = Y , this question has been extensively studiedfor the following function spaces: Hardy spaces [15, 42, 44, 66], weighted Hardyspaces [75], Hardy–Orlicz spaces [14, 16, 42, 43, 52, 64, 67, 77, 78, 80], Dirichletseries [4,5,29,36,41,62,65], Bergman space [20,21,38,45,51,74], Besov spaces[40, 53, 70], Bergman–Orlicz spaces [16, 17, 45, 63], Bloch or Bloch-type spaces[18,19,35,53,61,81–83], Wiener–Dirichlet algebra [9], Dirichlet space [28,46–49,54], weighted Dirichlet spaces [13, 76, 79], several complex variable versions ofthese spaces [6–8]. We would like to add two important facts. First, there areseveral other function spaces that were not mentioned in the above list. Second,the literature is abundant and the above list is far from being complete. The shortlist just reflects the authors’ interests.

In this note, we first consider X = Y = H2. This is precisely the classical sub-ordination principle of Littlewood that is expressed in the language of compositionoperators. There are several proofs of this important result. In Section 3, severaldifferent proofs are provided. These proofs are based on different, but equivalent,ways to define the norm on the Hardy–Hilbert space H2. This preliminary subjectis discussed in Section 2.

2010 Mathematics Subject Classification. Primary 30D50; Secondary 47B33.Key words and phrases. Composition operator, Hardy space, Model space.This work was supported by NSERC (Canada).


13

14 ABDELLATIF BOURHIM AND JAVAD MASHREGHI

Then we study X = Ku, Y = Kv, where u and v are inner functions. Thispart contains some new and some recent published results. Let us explain thefoundation. The mapping

S : H2 −→ H2

f �−→ zf

is called the forward shift operator. The closed invariant subspaces of S werecompletely characterized by A. Beurling in his seminal work [10]. If Θ is an innerfunction, i.e. bounded on D and unimodular almost everywhere on T, then ΘH2

is closed and invariant under S. More importantly, Beurling showed that, apartfrom the trivial case {0}, there is no more! That is why subspaces ΘH2 are calledBeurling subspaces of H2. Using a standard Hilbert space technique, we know thatthe invariant subspaces of the backward shift operator S∗ are precisely (ΘH2)⊥.These subspaces, i.e.,

KΘ := (ΘH2)⊥ = H2 �ΘH2,

are important in their own right, and due to their numerous applications in mathe-matical analysis and in operator theory, in particular in the theory of contractions,they are referred to as model spaces. The monograph [22] and the upcoming text[34] contains more detailed information about model spaces and their operators.

Studying composition operators on model spaces is a new topic, and yet therestill are several open questions about them. In [3, 50], the authors studied thecompactness and membership in Schatten classes of the mapping Cϕ : KΘ −→ H2.In [57], a complete characterization of ϕ’s for which Cϕ leaves KΘ invariant, whenΘ is a finite Blaschke product, is given. The papers [11,32,58] are devoted to acomprehensive study of Cϕ when ϕ is an inner function. In this situation, we arefaced with the functional equation

(1.1) ψ(ϕ(z)

)× ω(z) = ψ(z), (z ∈ D),

where all three functions ψ, ϕ and ω are inner. Using an iteration technique andappealing to the structure of inner function, (1.1) simplifies to

(1.2) ψ(ϕ(z)

)= λψ(z), (λ ∈ T, z ∈ D),

where ϕ has its fixed (Denjoy–Wolff) point on T. This equation is a special case ofthe celebrated Schroder equation which has a long and rich history. As a matter offact, its first treatment dates back to 1884 when Konigs [39] classified the eigen-values λ for mappings ϕ with a fixed points inside D. See [69, Chapter 6] for moredetail on Konigs’ solution. Despite the vast literature on Schroder’s equation, notmuch is known when the Denjoy–Wolff point of ϕ is on T and it calls for furtherinvestigation. Our goal is to analyze the solution of (1.2) and explore its relationto composition operators on model spaces.

2. The Hardy–Hilbert space H2

There are various ways to define the Hardy–Hilbert space H2. Each way hasits own merits and better fit for various applications. In the following, we discussfour approaches to define a norm on H2. For detailed information on Hardy spaces,see [56].

COMPOSITION OPERATORS 15

2.1. Taylor or Fourier coefficients. The most straightforward method todefine H2 is to say that it is a copy of the sequence space 2 inside Hol(D). Moreprecisely, each function f ∈ Hol(D) has the Taylor expansion

(2.1) f(z) =

∞∑n=0

anzn, (z ∈ D).

We use this representation to define

‖f‖2 :=

( ∞∑n=0

|an|2) 1

2

.

Then the Hardy–Hilbert space H2 is the collection of all analytic functions f on D

for which ‖f‖2 <∞.Assume that, according to the above definition, a function f ∈ H2 is given.

Then a classical theorem of Fatou says that, for almost all eiθ ∈ T, the boundaryvalues

f∗(eiθ) = limr→1−

f(reiθ)

exist. Moreover, f∗ ∈ L2(T) and the Fourier coefficients of f∗ are given by

f∗(n) =

{an if n ≥ 0,0 if n < 0.

The Parseval identity also says that

‖f∗‖L2(T) =

( ∞∑n=−∞

|f∗(n)|2) 1

2

=

( ∞∑n=0

|an|2) 1

2

= ‖f‖2.

Reciprocally, if a function g ∈ L2(T) is given such that g(n) = 0, n ≤ −1, then wecan define the analytic function

f(z) =

∞∑n=0

g(n)zn, (z ∈ D).

Then, by the first definition, f ∈ H2 and again Parseval’s identity says that ‖f‖2 =‖g‖L2(T). Moreover, the uniqueness theorem for Fourier coefficients also ensuresthat g = f∗ almost everywhere on T.

Therefore, we can also consider H2 as the subspace of L2(T) consisting ofelements whose negative spectrum is vanishing. When we adopt this point of view,to emphasize that our elements live on T, we denote the space by H2(T).

2.2. Integral means. In the preceding definition, if we replace 2 by p, we donot obtain the Hardy space Hp. In other words, Hp is not isomorphically isometricto the sequence space p if p = 2. To overcome this obstacle, we consider theintegral means

m(f, r, 2) =

(1

2π

∫ 2π

0

|f(reiθ)|2 dθ) 1

2

, (0 ≤ r < 1).


If we use the Taylor expansion (2.1) and plug it into the above formula, by Parseval’sidentity, we obtain

m(f, r, 2) =

( ∞∑n=0

|an|2r2n) 1

2

, (0 ≤ r < 1).

Hence, m(f, r, 2) is an increasing function of r and the monotone convergence the-orem (discrete version) says that

sup0≤r<1

m(f, r, 2) = limr→1−

m(f, r, 2) =

( ∞∑n=0

|an|2) 1

2

.

Hence, another way to define the H2-norm is via integral means

‖f‖2 := sup0≤r<1

(1

2π

∫ 2π

0

|f(reiθ)|2 dθ) 1

2

.

The advantage of this approach is that we can consider the integral meansm(f, r, p), where p is a parameter in (0,∞). Hardy [37] showed that logm(f, r, p)is an increasing convex function of log r. This paper is considered as the startingpoint of the theory of Hardy spaces.

2.3. Harmonic majorants. If there is a harmonic function U such that

|f(z)|2 ≤ U(z), (z ∈ D),

then the mean value theorem for harmonic functions [56, page 58] ensures that

1

2π

∫ 2π

0

|f(reiθ)|2 dθ ≤ 1

2π

∫ 2π

0

U(reiθ) dθ = U(0), (0 ≤ r < 1).

Hence, if |f |2 has a harmonic majorant U , certainly f ∈ H2 and ‖f‖22 ≤ U(0).Reciprocally, if f ∈ H2, then we have the Poisson integral representation

f(z) =1

2π

∫ 2π

0

1− |z|2|eiθ − z|2 f(e

iθ) dθ, (z ∈ D).

Hence, by the Cauchy–Schwarz inequality,

(2.2) |f(z)|2 ≤ 1

2π

∫ 2π

0

1− |z|2|eiθ − z|2 |f(e

iθ)|2 dθ, (z ∈ D).

This inequality means that the function

U(z) =1

2π

∫ 2π

0

1− |z|2|eiθ − z|2 |f(e

iθ)|2 dθ, (z ∈ D),

serves as a harmonic majorant for |f |2. Moreover, for this particular choice ofmajorant, we have U(0) = ‖f‖22. Therefore, we can equally define the Hardy spaceH2 as the set of analytic functions f such that |f |2 has a harmonic majorant and‖f‖22 is the infimum of the values of these majorants at the origin. This methodcan also be exploited to define Hp spaces for other values of p.


2.4. Weighted Dirichlet integral. The Dirichlet integral of f is

D(f) =∫D

|f ′(z)|2 dA(z),

where dA(z) = dxdy/π is the normalized Lebesgue measure on D. The restrictionD(f) <∞ leads to an interesting family known as the Dirichlet space [27]. At thesame token, one wonders if the norm of H2 can be obtained via a double integral.The affirmative answer was obtained by Payley–Littlewood as a weighted Dirichletintegral. They showed that

(2.3) ‖f − f(0)‖22 = 2

∫D

|f ′(z)|2 log1

|z| dA(z).

In fact, again by Parseval’s formula,

2

∫D

|f ′(z)|2 log1

|z| dA(z) =2

π

∫ 1

0

∫ 2π

0

|f ′(reiθ)|2 log

(1

r

)rdrdθ

=2

π

∫ 1

0

∫ 2π

0

∣∣∣∣∣∞∑

n=1

annrn−1ei(n−1)θ

∣∣∣∣∣2

log

(1

r

)rdrdθ

= 4

∫ 1

0

∞∑n=1

|an|2n2r2(n−1) log

(1

r

)rdr

= 4

∞∑n=1

|an|2n2

(∫ 1

0

r2n−1 log

(1

r

)dr

)

=

∞∑n=1

|an|2 = ‖f − a0‖22.

This identity has a generalization to other Hp spaces [55].Since log 1/|z| ∼ (1−|z|) as |z| → 1, how much do we lose if we replace log 1/|z|

by 1− |z| in the above formula? We certainly do not obtain the H2-norm, but onthe other hand we do not lose a lot either. As a matter of fact, we have∫

D

|f ′(z)|2 (1− |z|2) dA(z) =1

π

∫ 1

0

∫ 2π

0

|f ′(reiθ)|2 (1− r2) rdrdθ

=1

π

∫ 1

0

∫ 2π

0

∣∣∣∣∣∞∑n=1

annrn−1ei(n−1)θ

∣∣∣∣∣2

(1− r2) rdrdθ

= 2

∫ 1

0

∞∑n=1

|an|2n2r2(n−1)(1− r2) rdr

=

∞∑n=1

n

n+ 1|an|2.

Hence,

(2.4)1

2‖f − f(0)‖22 ≤

∫D

|f ′(z)|2 (1− |z|2) dA(z) ≤ ‖f − f(0)‖22.

This estimate is helpful and easier to use in certain applications.


3. The Littlewood subordination principle

The Littlewood subordination principle says that if f ∈ H2 and ϕ : D −→ D isanalytic with ϕ(0) = 0, then f ◦ ϕ ∈ H2 and, moreover, ‖f ◦ ϕ‖2 ≤ ‖f‖2. In thelanguage of composition operators, this classical result is stated as follows.

Theorem 3.1. Let f ∈ H2, and let ϕ : D −→ D be analytic with ϕ(0) = 0.Then the composition mapping

Cϕ : H2 −→ H2

f �−→ f ◦ ϕ

is a well-defined contraction on H2. In fact, ‖Cϕ‖ = 1.

One may also naturally wonder what happens if ϕ(0) = 0 in Theorem 3.1. Inthis case, the operator remains bounded, but no precise formula for its norm isavailable.

Theorem 3.2. Let f ∈ H2, and let ϕ : D −→ D be analytic. Then thecomposition mapping Cϕ : H2 → H2 is well-defined and

1 ≤ ‖Cϕ‖ ≤(1 + |ϕ(0)|1− |ϕ(0)|

)1/2

.

Using various definitions of the H2-norm, which were presented in Section 2,we provide several proofs of Theorem 3.1, or its generalized version Theorem 3.2.We also show how to pass from the special case to the general version. An excellentsource for such studies is [69].

3.1. A proof of Theorem 3.1 via Taylor coefficients and integralmeans. Since Cϕ1 = 1, we certainly have ‖Cϕ‖ ≥ 1. The reverse inequality isour main task. We need two mappings. The first is the multiplication operator

Mϕ : H2 −→ H2

f �−→ ϕf,

where ϕ ∈ H∞ is well-defined and in fact

(3.1) ‖Mϕ‖ = ‖ϕ‖∞,

see [12,31]. However, at this stage we just need the easily verified estimate

(3.2) ‖Mϕ‖ ≤ ‖ϕ‖∞.

The second mapping is

T : H2 −→ H2

f �−→ f(z)−f(0)z .

This mapping is in fact the adjoint of the forward unilateral shift on H2. However,we do not need these facts at this point. We will get back to this mapping inSection 9, where is it denoted by S∗. At the present stage, we just need somesimply established properties of T which are obtained below.

Using the Taylor expansion (2.1), the mapping T can be rewritten as

(Tf)(z) =

∞∑n=0

an+1zn, (z ∈ D).


Hence, by induction,

(T kf)(z) =

∞∑n=0

an+kzn, (z ∈ D),

which implies

(3.3) (T kf)(0) = ak, (k ≥ 0).

The formula that defines T can be rewritten as

f(z) = f(0) + z (Tf)(z), (z ∈ D).

Since ϕ maps D into itself, we can replace z by ϕ(z) in the above identity to get

f(ϕ(z)) = f(0) + ϕ(z) (Tf)(ϕ(z)), (z ∈ D).

In the language of operator theory, this identity is written as

(3.4) Cϕf = f(0) +MϕCϕTf.

One may question the validity of this identity since we do not know yet if Cϕ mapsH2 into itself. If this fact fails, the combination MϕCϕT is meaningless. But, wewill apply (3.4) when f is a polynomial and in this case there is clearly no obstacle.Now we consider two cases.

Case I: f is a polynomial. At this point, we use the assumption ϕ(0) = 0. Thiscondition implies that

(MϕCϕTf)(0) = 0,

and thus in the Hardy–Hilbert spaceH2 this function is orthogonal to the constants.Hence, by (3.2) and (3.4),

‖Cϕf‖22 = |f(0)|2 + ‖MϕCϕTf‖22≤ |f(0)|2 + ‖CϕTf‖22.

According to (3.3), replacing f by T kf in this estimate gives

(3.5) ‖CϕTkf‖22 ≤ |ak|2 + ‖CϕT

k+1f‖22, (k ≥ 0).

We add up these inequalities for 0 ≤ k ≤ n, where n = deg(f). Since Tn+1f ≡ 0,we get

(3.6) ‖Cϕf‖22 ≤n∑

k=0

|ak|2 = ‖f‖22.

Hence, the theorem holds for analytic polynomials.

Case II: An arbitrary f ∈ H2. Let

fn(z) =

n∑k=0

akzk, (n ≥ 1).

The sequence of polynomials (fn)n≥1 converges uniformly on compact subsets of Dto f . Fix r ∈ (0, 1). Then, by (3.6),

1

2π

∫ 2π

0

|(fn ◦ ϕ)(reiθ)|2 dθ ≤ ‖fn ◦ ϕ‖22 ≤ ‖fn‖22 ≤ ‖f‖22.


Since {ϕ(reiθ) : θ ∈ [0, 2π]} is a compact subset of D, upon letting n→∞, we get

1

2π

∫ 2π

0

|(f ◦ ϕ)(reiθ)|2 dθ ≤ ‖f‖22.

Since this estimate holds uniformly with respect to r, we conclude that f ◦ϕ ∈ H2

and ‖f ◦ ϕ‖2 ≤ ‖f‖2.

3.2. A proof of Theorem 3.1 via Weighted Dirichlet integral. Thisproof is slightly more sophisticated. However, it pays its debts when one studiesthe compactness of Cϕ. The major tools in this proof are an interesting changeof variable formula [2] and Littlewood’s inequality for the Nevanlinna countingfunction.

Given an analytic function ϕ : D −→ D, its Nevanlinna counting function isdefined by

Nϕ(w) =

⎧⎪⎪⎨⎪⎪⎩∑

z∈ϕ−1(w)

log1

|z| if w ∈ ϕ(D),

0 if w ∈ ϕ(D).

The function z �−→ ϕ(z)−w is a bounded and thus, by F. Riesz theorem [56, page166], its zeros satisfy the Blaschke condition, i.e.∑

z∈ϕ−1(w)

(1− |z|) <∞.

But, as |z| → 1, the quantities 1− |z| and log 1/|z| are asymptotic. Hence, the sumin the definition of Nϕ is also convergent. In fact, with some careful estimates, wecan obtain an upper bound for Nϕ.

We remind the reader that, for each fixed w ∈ D,

(3.7) τw(z) =w − z

1− wz, (z ∈ D),

is an automorphism of D such that τw ◦ τw = id. Later on, we will also need therotations

ρλ(z) = λ z, (λ ∈ T, z ∈ D).

Lemma 3.3 (Littlewood’s inequality). Let ϕ : D −→ D be an analytic function.Then, for each w ∈ D \ {ϕ(0)}, we have

Nϕ(w) ≤ log

∣∣∣∣1− wϕ(0)

w − ϕ(0)

∣∣∣∣ .In particular, if ϕ(0) = 0, we have

Nϕ(w) ≤ log1

|w| .

Proof. First note that, by (3.7), the right side of the inequality is precisely− log |τw(p)| at the point p = ϕ(0). Hence, it is a positive quantity.

If w ∈ ϕ(D), then, by definition, Nϕ(w) = 0 and the inequality trivially holds.Now, assume that w ∈ ϕ(D). The Jensen formula says that∑

|zn|≤r

logr

|zn|=

1

2π

∫ 2π

0

log |f(reiθ)| dθ + log1

|f(0)| ,


where f is an analytic function on D, and zn’s represent the zeros of f , countingmultiplicities. In particular, if f is bounded by 1, we deduce∑

|zn|≤r

logr

|zn|≤ log

1

|f(0)| .

Then, appealing to the monotone convergence theorem (discrete version), we letr → 1− to obtain ∑

zn∈D

log1

|zn|≤ log

1

|f(0)| .

This inequality can be rewritten as

(3.8)∑

z∈f−1(0)

log1

|z| ≤ log1

|f(0)| .

This is precisely the inequality we are looking for, in disguised form of course. Letus clarify the situation.

Take f = τw ◦ϕ. Then f satisfies the required conditions of the last paragraphand f−1(0) = ϕ−1(w). Hence,∑

z∈f−1(0)

log1

|z| =∑

z∈ϕ−1(w)

log1

|z| = Nϕ(w).

Moreover,1

|f(0)| =∣∣∣∣1− wϕ(0)

w − ϕ(0)

∣∣∣∣ .It remains to apply (3.8) and get the result. �

Lemma 3.4 (Change of variable). Let f be a nonnegative measurable functionon D, and let ϕ : D −→ D be an analytic function. Then∫

D

f(ϕ(z)

)|ϕ′(z)|2 log

1

|z| dA(z) =

∫D

f(w)Nϕ(w) dA(w).

Proof. If ϕ is a univalent function from D onto ϕ(D), then the ordinary changeof variable formula w = ϕ(z) implies∫

D

f(ϕ(z)

)|ϕ′(z)|2 log

1

|z| dA(z) =

∫ϕ(D)

f(w) log1

|ϕ−1(w)| dA(w)

=

∫ϕ(D)

f(w)Nϕ(w) dA(w)

=

∫D

f(w)Nϕ(w) dA(w).

The general formula is obtained via the same principle if we carefully put aside thepoints that cause trouble. Let us examine the details.

The set of critical points of ϕ, i.e.,

Z := {z ∈ D : ϕ′(z) = 0}is at most countable. Moreover, on the open set D \ Z, the function ϕ is locallyunivalent. Hence, we can divide D\Z into a union of polar rectangles Rn such thatthe interior of these rectangles is disjoint and ϕn := ϕ|Rn is univalent. We apply


the ordinary change of variable formula on the interior of each such rectangle. Thisgives∫

R◦n

f(ϕ(z)

)|ϕ′(z)|2 log

1

|z| dA(z) =

∫ϕ(R◦

n)

f(w) log1

|ϕ−1n (w)|

dA(w)

=

∫D

f(w)χn(w) log1

|ϕ−1n (w)|

dA(w),

where χn is the characteristic function of ϕ(R◦n). We add up all these identities.

Now, on one hand,∑n≥1

∫R◦

n

f(ϕ(z)

)|ϕ′(z)|2 log

1

|z| dA(z) =

∫D

f(ϕ(z)

)|ϕ′(z)|2 log

1

|z| dA(z),

because the set which is missed for integration on the left side has 2-dimensionalLebesgue measure zero. On the other hand,

∑n≥1

∫D

f(w)χn(w) log1

|ϕ−1n (w)|

dA(w)=

∫D

f(w)

⎛⎝∑n≥1

χn(w) log1

|ϕ−1n (w)|

⎞⎠ dA(w),

and

(3.9)∑n≥1

χn(w) log1

|ϕ−1n (w)|

= Nϕ(w)

if w ∈ ∪n≥1ϕ(R◦n) \ E, where E is the image of boundaries of rectangles and also

the critical points under ϕ. Therefore, again the set over which (3.9) does not holdis of 2-dimensional Lebesgue measure zero. Hence, the result follows. �

Now, we combine Lemmas 3.3 and 3.4 to obtain a proof of Theorem 3.1.

Proof. By the Paley–Littlewood identity (2.3)

‖f‖22 = |f(0)|2 + 2

∫D

|f ′(z)|2 log1

|z| dA(z).

Replace f by f ◦ ϕ, and use the assumption ϕ(0) = 0, to get

‖f ◦ ϕ‖22 = |f(0)|2 + 2

∫D

|(f ◦ ϕ)′(z)|2 log1

|z| dA(z)

= |f(0)|2 + 2

∫D

|f ′(ϕ(z))|2 |ϕ′(z)|2 log1

|z| dA(z).

By Lemma 3.4, we obtain

‖f ◦ ϕ‖22 = |f(0)|2 + 2

∫D

f(w)Nϕ(w) dA(w).

Lemma 3.3 then implies that

‖f ◦ ϕ‖22 ≤ |f(0)|2 + 2

∫D

f(w) log1

|w| dA(w).

We use the Paley–Littlewood identity one more time to deduce ‖f ◦ϕ‖22 ≤ ‖f‖22. �


3.3. A proof of Theorem 3.2 via integral means. The following lemmais actually a special case of Theorem 3.2. However, this special case combined withTheorem 3.1 lead us to a proof of the general case.

Lemma 3.5. For each w ∈ D, the composition mapping Cτw is bounded on H2

with

‖Cτw‖ =(1 + |w|1− |w|

)1/2

.

Proof. Fix any f ∈ H2. Then we use the change of variable formula

eis = τw(eit)⇔ eit = τw(e

is) and dt =1− |w|2|1− weis|2 ds

to obtain

‖Cτwf‖2 = ‖f ◦ τw‖2

=1

2π

∫ 2π

0

|f(τw(eit))|2 dt

=1

2π

∫ 2π

0

|f(eis)|2 1− |w|2|1− weis|2 ds.

Consider the multiplication operator Mψ with symbol

ψ(z) =(1− |w|2)1/2

1− wz, (z ∈ D).

Hence, in short, we may write

‖Cτwf‖2 = ‖Mψf‖2.

This simple observation implies ‖Cτw‖ = ‖Mψ‖. But, by (3.1), we know that

‖Mψ‖ = ‖ψ‖∞ =(1− |w|2)1/2

1− |w| =

(1 + |w|1− |w|

)1/2

.

�

Now we combine Theorem 3.1 and Lemma 3.5 to obtain a proof of Theorem3.2

Proof. Since Cϕ1 = 1, we have ‖Cϕ‖ ≥ 1. To verify the other bound, putw = ϕ(0), and define ψ = τw ◦ ϕ. Then ψ is also a self-map of the disc with theextra property ψ(0) = 0. Thus, by Lemma 3.1, ‖Cψ‖ ≤ 1.

The identity ψ = τw ◦ ϕ is equivalent to ϕ = τw ◦ ψ, and the latter impliesCϕ = Cψ ◦ Cτw . Hence, by Lemma 3.5,

‖Cϕ‖ ≤ ‖Cτw‖ ‖Cψ‖ ≤(1 + |w|1− |w|

)1/2

=

(1 + |ϕ(0)|1− |ϕ(0)|

)1/2

.

�


3.4. A proof of Theorem 3.2 via Harmonic majorants. Fix f ∈ H2.Let U be the harmonic function

U(z) =1

2π

∫ 2π

0

1− |z|2|eiθ − z|2 |f(e

iθ)|2 dθ, (z ∈ D).

Note that U(0) = ‖f‖22 and, by (2.2),

|f(z)|2 ≤ U(z), (z ∈ D).

Replace z by ϕ(z) to get

|(f ◦ ϕ)(z)|2 ≤ (U ◦ ϕ)(z), (z ∈ D).

The main point is that U ◦ ϕ is a harmonic function on D. Hence, if we integrateover the circle reiθ, 0 ≤ θ ≤ 2π, then, by the mean value theorem for harmonicfunctions, we get

1

2π

∫ 2π

0

|(f ◦ ϕ)(reiθ)|2dθ ≤ (U ◦ ϕ)(0) = U(ϕ(0) ).

But, Harnack’s inequality for positive harmonic functions says

U(z) ≤ 1 + |z|1− |z|U(0), (z ∈ D).

Therefore, we deduce

1

2π

∫ 2π

0

|(f ◦ ϕ)(reiθ)|2dθ ≤ 1 + |ϕ(0)|1− |ϕ(0)| ‖f‖

22.

Since this estimate holds for all values of r ∈ [0, 1), we conclude that f ◦ ϕ ∈ H2

and, moreover,

‖f ◦ ϕ‖2 ≤(1 + |ϕ(0)|1− |ϕ(0)|

)1/2

‖f‖2.

This seems like an extremely easy and straightforward proof (and actually it worksfor all values of p ∈ [1,∞)), but, note that heavy tools, e.g. mean value theoremfor harmonic functions and Harnack’s inequality, were exploited in the proof.

4. A Schroder-type equation

In studying the inner functions ϕ for which Cϕ maps Ku into itself, we arefaced with the functional equation

(4.1) ψ(ϕ(z)

)× ω(z) = ψ(z), (z ∈ D),

where all functions ψ, ϕ and ω are inner. A variation of (1.1) is known as Schroder’sequation and has a very long and rich history. A detailed discussion of this topicis available in [23]. We discuss the solutions of (1.1) first, and then apply them tofind inner functions ϕ such that Cϕ maps Ku into itself.

Since ϕ is a self map of D, it has a fixed point in D, i.e., a point p ∈ D suchthat ϕ(p) = p; see [69, page 78]. To proceed, depending on the location of p, weconsider two principal cases: p ∈ D and p ∈ T.


4.1. The fixed point is inside D. For this case, there is a complete charac-terization of the solution of (1.1).

Theorem 4.1. Let ϕ be a non-constant inner function with the fixed pointp ∈ D. Assume that ϕ(z) ≡ z. Then the complete solution of the functionalequation

ψ(ϕ(z)

)× ω(z) = ψ(z), (z ∈ D),

where ω and ψ are inner functions, is as follows. First, ω has to be a unimodularconstant. Second, ϕ is the hyperbolic rotation

ϕ = τp ◦ ρλ ◦ τp ,

where λ := ϕ′(p) ∈ T. Third,

ψ =

⎧⎨⎩ γ (τp)m × ψ1

((τp)

n)

if λn = 1 for some n ≥ 2,

γ (τp)m if λn = 1 for all n ≥ 2,

where ψ1 is an arbitrary inner function, γ ∈ T, m ≥ 0. Finally, the functionalequation has the simplified form

ψ ◦ ϕ = λm ψ .

Proof. According to the Schwarz–Pick theorem,

|ϕ′(z)| ≤ 1− |ϕ(z)|21− |z|2 , (z ∈ D).

Hence, at the fixed point p, we have |ϕ′(p)| ≤ 1. Suppose that ψ has a zero of orderm ≥ 0 at p. Then we can write

ψ(z) =

(p− z

1− p z

)m

ψ0(z),

where ψ0 is an inner function with ψ0(p) = 0. Hence, (1.1) becomes(p− ϕ(z)

1− p ϕ(z)

)m

ψ0

(ϕ(z)

)ω(z) =

(p− z

1− p z

)m

ψ0(z).

Divide by (p−z)m and then let z −→ p to deduce(ϕ′(p)

)mω(p) = 1. This identity

reveals that ϕ′(p) and ω(p) are both unimodular constants. Thus, by the maximumprinciple and Schwarz’s lemma, ϕ must have the form ϕ = τp ◦ ρλ ◦ τp for somearbitrary constant λ ∈ T and ω ≡ λm. Note that λ = ϕ′(p) ∈ T \ {1}. The caseλ = 1 is excluded since it leads to the trivial case ϕ(z) = z. The main feature of ϕis that τp ◦ ϕ = λ τp. Hence, (1.1) simplifies to ψ ◦ ϕ = λm ψ. Since ψ = (τp)

mψ0,we can go further and see that ψ0 must satisfy

ψ0 ◦ ϕ = ψ0 ◦ τp ◦ ρλ ◦ τp = ψ0.

By induction, if we repeatedly compose both sides with τp ◦ ρλ ◦ τp, we obtain

ψ0 ◦ τp ◦ ρλk ◦ τp = ψ0, (k ≥ 1),

which we rewrite as

(4.2) (ψ0 ◦ τp)(λkz) = (ψ0 ◦ τp)(z), (k ≥ 1, z ∈ D).


Now, depending on λ being a root of unity or not, we need to consider two casesto obtain ψ0 and ψ.

Case 1: There is no integer n ≥ 2 such that λn = (ϕ′(p) )n = 1. If λ is not a rootof unity, then (4.2) and the uniqueness theorem for analytic functions force ψ0 ◦ τpto be a unimodular constant, and thus ψ0 ≡ γ, for some γ ∈ T. In this situation,the inner function ψ is given by

ψ = γ (τp)m,

where γ ∈ T, m ≥ 0. A trivial, but important, special case of this category isψ(z) = γzm and ϕ(z) = λz.

Case 2: There is an integer n ≥ 2 such that λn = (ϕ′(p) )n = 1. Take n to bethe smallest such integer. Then, with a proper choice of exponent ≥ 1, we haveλ� = ei2π/n and, by (4.2), ψ0 must satisfy

(ψ0 ◦ τp)(ei2kπ/nz) = (ψ0 ◦ τp)(z), (k ≥ 1, z ∈ D).

This identity means that the Taylor expansion of ψ0 ◦ τp is of the form

(ψ0 ◦ τp)(z) =∞∑k=0

ak zkn, (z ∈ D),

or equivalently, we have

(ψ0 ◦ τp)(z) = ψ1(zn), (z ∈ D),

where ψ1 is any ‘arbitrary’ inner function. Hence, the inner function ψ is given by

ψ = γ (τp)m × ψ1

((τp)

n),

where ψ1 is inner, γ ∈ T, m ≥ 0. Note that if ψ1 is a unimodular constant, thenthe function ψ becomes of the type considered in case 1, even though λ is a root ofunity. �

4.2. The fixed point is on T. This is a complicated case. In this situation,p is called the Denjoy–Wolff point of ϕ [25,71–73]. If ω is a unimodular constant,then the equation (1.1) becomes

(4.3) ψ(ϕ(z)

)= λψ(z), (z ∈ D),

where λ ∈ T. This is by itself an interesting equation. However, a complete char-acterization of its solutions is not available yet. We study some solutions below.However, in all cases that we study, ϕ is an automorphism of the open unit disc.In fact, we have not been able to construct, or even to find in the literature, anon-trivial solution of (1.2) when ϕ is not an automorphism of D.

Conjecture 1: The functional equation (1.2) has a non-trivial solution if and onlyif ϕ is an automorphism of D.

Open Problem 1: Characterize the solutions of (1.2).

Note that Theorem 4.1 says that the conjecture is true if the fixed point isinside D. Hence, one should really treat the conjecture when the fixed points sitson T. In studying the solutions of (1.2), the most important case to discus is λ = 1.


In fact, if ψ1 is any solution of (1.2) and ψ2 is another solution for the case λ = 1,then ψ = ψ1ψ2 is again a solution of (1.2). This simple observation is important,in particular, when we study the atomic solutions of (1.2).

We can reduce the case where ω is non-constant to the previous case. Writeϕ[0](z) = z and ϕ[n] = ϕ ◦ · · · ◦ ϕ, n ≥ 1.

Theorem 4.2. Let ω be a non-constant inner function. Let ϕ be an innerfunction with its Denjoy–Wolff point on the unit circle T. Assume that there isa inner function ψ such that the functional equation (1.1) is fulfilled. Then theproduct

ψ1(z) =∞∏

n=0

ω(ϕ[n](z)

)is uniformly convergent on compact subsets of D. It represent an inner functionwhich is a divisor of ψ. Moreover, ψ2 := ψ/ψ1 satisfies the functional equationψ2 ◦ ϕ = ψ2.

Proof. If ω is not a unimodular constant, then by replacing z by ϕ(z) in (1.1)and using induction, we see that

ψ(ϕ[N+1](z)

)×

N∏n=0

ω(ϕ[n](z)

)= ψ(z), (N ≥ 1).

Hence, considering the fact that∏N

n=0 ω(ϕ[n](z)

)is a chain of divisors of ψ, we

deduce that the product

ψ1(z) =

∞∏n=0

ω(ϕ[n](z)

)is a well-defined non-constant inner function. Moreover, ψ1 is a divisor of ψ whichitself satisfies (1.1). Since, for each z ∈ D, the sequence ϕ[n](z), n ≥ 1, tendsnon-tangentially to p and the above product is convergent, ω must have the non-tangential limit 1 at p, i.e.

ω(p) := ∠ limz→p

ω(z) = 1.

By putting ψ = ψ1ψ2 in (1.1) and simplifying the equation, we see that ψ2 mustfulfil ψ2 ◦ ϕ = ψ2. �

The converse of Theorem 4.2 is almost a triviality.

Theorem 4.3. Let ω be a non-constant inner function. Let ϕ be an innerfunction with its Denjoy–Wolff point on the unit circle T. Assume that the product

(4.4) ψ(z) =

∞∏n=0

ω(ϕ[n](z)

)is uniformly convergent on compact subsets of D. Then ψ is an inner functionwhich satisfies the functional equation (1.1).

Even though Theorem 4.3 and the paragraph before it indicate that we even-tually get back to the simpler question 1.2, there is still one important issue thatmust be addressed.


Open Problem 2: Under what conditions on ω and ϕ is the product in (4.4)convergent?

Let us provide a partial answer.

Theorem 4.4 ([58]). Suppose that ϕ and ω are inner functions with the fol-lowing properties:

(i) p, the Denjoy–Wolff point of ϕ, is on T;(ii) ϕ′(p) < 1;(iii) for all z ∈ D, ∣∣∣∣1− ω(z)

p− z

∣∣∣∣2 ≤ C

1− |z|2 ,

where C is a constant.

Then

ψ(z) =∞∏

n=0

ω(ϕ[n](z)

)is a well-defined non-constant inner function that satisfies the equation (1.1).

Proof. If f : D −→ D has finite angular derivative in the sense of Caratheodoryat α ∈ T, then Julia’s inequality says

|f(α)− f(z)|21− |f(z)|2 ≤ |f ′(α)| |α− z|2

1− |z|2 , (z ∈ D).

We apply this inequality to ϕ. Since ϕ(p) = p, by induction, we obtain

|p− ϕ[n](z)|21− |ϕ[n](z)|2 ≤ |ϕ

′(p)|n |p− z|21− |z|2 , (z ∈ D).

Hence, replacing z by ϕ[n](z) in (iii) and exploiting the inequality before, we obtain

|1− ω(ϕ[n](z))|2 ≤(C|p− z|21− |z|2

)|ϕ′(p)|n, (z ∈ D).

The assumption |ϕ′(p)| < 1 now ensures the convergence of the product in thedefinition of ψ. �

Let us discuss a bit about conditions (ii) and (iii). We always have ϕ′(p) ∈(0, 1]. Hence, we essentially exclude the case ϕ′(p) = 1. The condition (iii) impliesω(p) = 1. On the other hand, we saw that for the convergence of product thiscondition is also necessary. If ω : D −→ D has finite angular derivative in the senseof Caratheodory at p ∈ T, then Julia’s inequality says

|ω(p)− ω(z)|21− |ω(z)|2 ≤ |ω′(p)| |p− z|2

1− |z|2 , (z ∈ D).

Rewrite this as ∣∣∣∣1− ω(z)

p− z

∣∣∣∣2 ≤ |ω′(p)| 1− |ω(z)|2

1− |z|2 , (z ∈ D).

Hence, (iii) holds with C = |ω′(p)|.


As a special case of Theorem 4.4, we can take ω = p ϕ[n0], n0 ≥ 0. Hence, weobtain

ψ(z) =

∞∏n=n0

p ϕ[n](z)

which is a non-constant inner function satisfying the equation

ψ(ϕ(z)

)× pϕ[n0](z) = ψ(z).

5. The group (D, ∗)The contents of this section are borrowed from [32]. In a rather surprising way,

the open unit disc D becomes a group. The law of composition is defined by

(5.1) α ∗ β :=β(1− β) + α(1− β)

(1− β) + αβ(1− β), (α, β ∈ D).

This algebraic structure is rewarding and has numerous interesting properties. Thisstrange law of composition comes from the composition of some judiciously chosenautomorphisms of the open unit disc.

5.1. An isomorphism. Let

(5.2) ϕα(z) :=1− α

1− α

z − α

1− α z,

where α is a parameter running through D. A simple computation shows ϕα(1) = 1and the other fixed point of ϕα is

(5.3) κα := − α(1− α)

α(1− α).

Moreover, if f is any automorphism of D such that f(1) = 1, then f = ϕα where αis the unique zero of f . This observation is important and is used in the proof offollowing Lemma.

Lemma 5.1 ([32]). (D, ∗) is a (non-abelian) group. The identity element is 0,and the inverse of α under the operation ∗ is −α 1−α

1−α .

Proof. To reveal the mystery behind the complicated law ∗, consider thecollection

G = {ϕα : α ∈ D}.Then the set G precisely consists of all automorphisms of the open unit disc witha fixed point at 1. Since the automorphisms ϕα ◦ ϕβ and ϕ−1

α fix the point 1, wededuce that ϕα◦ϕβ ∈ G and ϕ−1

α ∈ G. Hence, equipped with the law of compositionof functions, G is a group. Since G is a group, given α, β ∈ D, there is a uniqueγ ∈ D such that ϕα ◦ ϕβ = ϕγ . Let us proceed and find an explicit formula for γ.By definition,

ϕγ(z) = (ϕα ◦ ϕβ)(z)

=1− α

1− α

1−β1−β

z−β1−β z

− α

1− α 1−β1−β

z−β1−β z

=1− α

1− α

(1− β)(z − β)− α(1− β)(1− β z)

(1− β)(1− β z)− α(1− β)(z − β)

=1− α

1− α

((1− β) + αβ(1− β)

)z −

(β(1− β) + α(1− β)

)

((1− β) + αβ(1− β)

)−

(β(1− β) + α(1− β)

)z.


Looking at the zero of the last quotient, which is an automorphism of D whichfixes the point 1, we deduce that γ = α ∗ β, where the latter is given by (5.1). Inother words, given α, β ∈ D, the element α ∗ β ∈ D is such that

(5.4) ϕα ◦ ϕβ = ϕα∗β , (α, β ∈ D),

and this formula provides an isomorphism between the groups (D, ∗) and (G, ◦).We constructed the group (D, ∗) such that it is an isomorphic copy of (G, ◦).

Hence, since ϕ0 = id, the point 0 is the identity element of (D, ∗). Using (5.1), it isalso easy to directly verify that

α ∗ 0 = 0 ∗ α = α, (α ∈ D).

Similarly, the expression

ϕ−1α (z) =

1− α

1− α

z + α 1−α1−α

1 + α 1−α1−α z

= ϕ−α 1−α1−α

(z),

gives the formula for the inverse of α, something that can also be directly verifiedvia (5.1), i.e.,

α ∗(−α 1− α

1− α

)=

(−α 1− α

1− α

)∗ α = 0, (α ∈ D).

�

In our calculation, we will also need the quantity

(5.5) Aα := ϕ′α(1) =

1− |α|2|1− α|2 ,

i.e., Aα is the angular derivative (in the sense of Caratheodory) of ϕα at the fixedpoint 1. Moreover, note that

Aα = 1 ⇐⇒ κα = 1.

Finally, given z0 ∈ D, we define the unimodular constant γα,z0 by

γα,z0 :=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ϕz0(κα) if Aα < 1,

1 if Aα = 1,

ϕz0(κα) if Aα > 1.

5.2. Subgroups Dκ. Since the fixed points of ϕα are 1 and κα, the study of(D, ∗) naturally bifurcates into two cases: κα = 1 and κα = 1. Note that as α runsthrough D, the fixed point κ runs through all of T.

For a fixed κ ∈ T, define

(5.6) Dκ := {α ∈ D : κα = κ} = {α ∈ D : α+ κα = (1 + κ)|α|2}.

The last expression shows that the points of Dκ are part of the circle which is insideD and passing through the points 1, 0 and κ. Also note that for κ = −1, we havethe degenerate case

D−1 = {α ∈ D : α(1− α) = α(1− α)} = (−1, 1).


One special case is of particular interest. If κ = 1, then

D1 = {α ∈ D : α(1− α) = −α(1− α)}= {α ∈ D : α+ α = 2|α|2}= {x+ iy : (x− 1/2)2 + y2 = 1/4} \ {1}.(5.7)

Hence, D1 is precisely the circle of radius 1 inside D which is tangent to point1, of course without counting the boundary point 1. At the points of D1, theautomorphism ϕα has just one fixed point, i.e. the point 1, and this makes itstreatment slightly different from other members of G. The subgroup D1 is also theborder line for the values of Aα. On D1, we precisely have Aα = 1, while inside itAα > 1, and in between D and D1 we have Aα < 1.

Corollary 5.2 ([32]). Let κ ∈ T. Then Dκ is an abelian subgroup of D. More-over, on Dκ, the law of composition simplifies to

α ∗ β =α+ β − (1 + κ)αβ

1− καβ, (α, β ∈ Dκ).

Proof. If κ is the fixed point of ϕα and ϕβ , then it also stays fixed under

ϕα ◦ ϕ−1β = ϕα∗β−1 . Hence, for each α, β ∈ Dκ, we have α ∗ β−1 ∈ Dκ. Clearly


0 ∈ Dκ. Thus, Dκ is a subgroup of D. (The notation β−1 represents the inverse ofβ under the law ∗. It should not be confused with the reciprocal of β as a complexnumber. Later on we will adopt another notation to avoid this issue.)

To obtain a simpler formula for ∗ in Dκ, note that by (5.1) and (5.6), we have

α ∗ β =β(1− β) + α(1− β)

(1− β) + αβ(1− β)

=−κβ(1− β) + α(1− β)

(1− β)− ακβ(1− β)

=(1− β)(−κβ + α)

(1− β)(1− καβ)

=α+ β − (1 + κ)αβ

1− καβ, (α, β ∈ Dκ).

This formula also reveals that Dκ is abelian. �

5.3. Iteration. To avoid the confusion with the law of multiplication in thecomplex plane, for n ≥ 1, we write

αn := α ∗ α ∗ · · · ∗ α, (n times, n ≥ 1),

and, appealing to the formula for the inverse of α in D given in Lemma 5.1, wedefine

α−n :=

(−α 1− α

1− α

)n

, (n ≥ 1).

Since 0 is the identity element in D, we put α0 := 0. Note that the inverse ofα is now written as α−1. Hence, each α ∈ D gives birth to a two-sided sequence(αn)n∈Z, and with this notation, out of (5.4) we have the crucial identity

(5.8) ϕ[n]α = ϕαn

, (n ∈ Z).

This observation immediately implies

(5.9) ϕαm◦ ϕαn

= ϕαm+n, (m,n ∈ Z).

This identity will be used frequently. Lemma 5.1 and (5.2) also reveal that

(5.10) ϕα(0) = −α1− α

1− α= α−1, (α ∈ D).

To obtain another useful formula, note that ϕβ ◦ ϕα−1and ϕ

ϕα(β)both belong

to G and vanish at ϕα(β). Hence, by uniqueness, ϕβ ◦ ϕα−1= ϕ

ϕα(β). Therefore,

we have

(5.11) ϕβ ◦ ϕα−n= ϕϕαn (β)

, (α, β ∈ D, n ∈ Z).

The importance of this formula will be revealed below. Using an interesting tech-nique of complex analysis, we now obtain an explicit formula for αn.

Lemma 5.3 ([32]). Let α ∈ D. Then we have

αn =

⎧⎪⎪⎪⎨⎪⎪⎪⎩κα(1− An

α)

1− καAnα

if κα = 1,

nα

1 + (n− 1)αif κα = 1,

(n ∈ Z).


In particular, except for the identity element 0, no other element of D is of finiteorder.

Proof. Direct verification of the above formula is feasible, but it is not apleasant task. We present another more interesting approach. Given κ ∈ T, define

φκ(z) :=

⎧⎪⎪⎨⎪⎪⎩z − κ

z − 1if κ = 1,

z

z − 1if κ = 1.

This function satisfies φκ ◦ φκ = id. Now, we need to consider two cases.

Case I, κα = 1: We have

(φκα◦ ϕα ◦ φκα

)(z) =z

Aα,

and thus we deduce that

(φκα◦ ϕ[n]

α ◦ φκα)(z) =

z

Anα

, (n ∈ Z).

Therefore,

(5.12) ϕ[n]α (z) = φκα

(φκα

(z)

Anα

), (n ∈ Z),

which simplifies to

(5.13) ϕ[n]α (z) =

(1− καAnα)z − κα(1−An

α)

(1−Anα)z + (An

α − κα), (n ∈ Z).

Now, according to (5.8), ϕ[n]α = ϕαn

, and by considering the zero of ϕ[n]α , we obtain

the required above formula.

Case II, κα = 1: The proof has the same spirit, except that we use φ1. In this case,we have

(φ1 ◦ ϕα ◦ φ1)(z) = z +α

1− α,

and thus

(φ1 ◦ ϕ[n]α ◦ φ1)(z) = z +

nα

1− α, (n ∈ Z).

Therefore,

(5.14) ϕ[n]α (z) = φ1

(φ1(z) +

nα

1− α

), (n ∈ Z),

which simplifies to

(5.15) ϕ[n]α (z) =

(1− α+ nα)z − nα

nαz + 1− α− nα, (n ∈ Z).

The result now follows. �


6. An equivalence relation on D

The operation

� : (D, ∗)× D −→ D

(α, z) �−→ ϕα(z)

defines a group action on the set D. The essential condition α � (β � z) = (α ∗β) � zwhich is required in the definition of a group action is precisely a reformulation of(5.4). Since (ϕw−1

◦ ϕz)(z) = w this action is transitive and thus it creates justone orbit on D. Hence, we restrict ourselves to some subgroups of (D, ∗) to obtaina richer class of equivalence classes. In particular, fixing α ∈ Dκ, the iterates of α,i.e., (αn)n∈Z form a subgroup in Dκ which is exploited below.

At the same token, the operation

� : (D, ∗)× T −→ T

(α, z) �−→ ϕα(z)

defines a group action on the set T.

6.1. Orbits on D. The orbits, or equivalent classes, created by the subgroup(αn)n∈Z are as follows. Two points z1 and z2 are in the same orbit, and we writez1 ∼α z2, if and only if there is an integer n ∈ Z such that

ϕ[n]α (z1) = ϕαn

(z1) = z2.

Since ϕα is an automorphism, it maps D and T respectively to themselves bijectively.Hence, the equivalence class generated by a z ∈ D is entirely in D. A similarstatement holds for the points of T. More information on the equivalence classesis gathered below. Since α = 0 corresponds to the identity mapping on D, thefollowing result becomes trivial in this situation. Hence, we assume that α = 0.

Theorem 6.1 ([32]). Let α ∈ D, α = 0. Then the following assertions hold.

(i) The equivalence class generated by z0 ∈ D is precisely(ϕαn

(z0))n∈Z

, which

consists of distinct points of D.(ii) The equivalence class generated by 0 is the sequence

(αn

)n∈Z

.

(iii) If Aα = 1, then

limn→±∞

ϕαn(z0) = 1.

(iv) If Aα > 1, then

limn→+∞

ϕαn(z0) = κα while lim

n→−∞ϕαn

(z0) = 1.

(v) If Aα < 1, then

limn→+∞

ϕαn(z0) = 1 while lim

n→−∞ϕαn

(z0) = κα.

Proof. (i): That the equivalence class generated by z0 ∈ D is precisely(ϕαn

(z0))n∈Z

is rather trivial. This fact says that the equivalence class gener-ated by z0 consists of the past, present and future of z0 under the transformationϕα. See formulas (5.13) and (5.15). For any α ∈ D, the automorphism ϕα has nofixed point inside D. Hence, the class

(ϕαn

(z0))n∈Z

consists of distinct points.

(ii): To find the equivalence class of 0, apply (5.2) to get

ϕαn(0) = −αn

1− αn

1− αn, (n ∈ Z).


But, by Lemma 5.1 and (5.9),

−αn1− αn

1− αn= inverse of αn in (D, ∗) = α−n, (n ∈ Z).

Thus, by part (i), (ϕαn

(0))n∈Z

=(α−n

)n∈Z

=(αn

)n∈Z

.

(iii): If Aα = 1, then we rewrite (5.15) as

ϕ[n]α (z) =

(1− α)z + nα(z − 1)

(1− α) + nα(z − 1).

This representation shows that

limn→±∞

ϕαn(z0) = 1.

Note that Aα = 1 happens precisely on D1.(iv)− (v): If Aα = 1, then we rewrite (5.13) as

ϕ[n]α (z) =

−Anακα(z − 1) + (z − κα)

−Anα(z − 1) + (z − κα)

.

Now, there are two possibilities. If Aα > 1, which corresponds to the points αinside the disc surrounded by D1, then

limn→+∞

ϕαn(z0) = κα, while lim

n→−∞ϕαn

(z0) = 1.

But, if Aα < 1, which corresponds to the points α ∈ D, but outside the discsurrounded by D1, then

limn→+∞

ϕαn(z0) = 1, while lim

n→−∞ϕαn

(z0) = κα.

�

6.2. Orbits on T. The following result has the same flavor as Theorem 6.1,except that it characterizes the orbits of a point on the unit circle T.

Theorem 6.2 ([11]). Let α ∈ D, α = 0. Then the following assertions hold.

(i) The equivalence class generated by ζ0 ∈ T is precisely(ϕαn

(ζ0))n∈Z

.

(ii) The equivalence classes generated by the fixed points 1 and κα are respectivelythe singletons {1} and {κα}.

(iii) For any other ζ0 ∈ T \ {1, κα}, the corresponding equivalence class,(ζn)n∈Z

=(ϕαn

(ζ0))n∈Z

,

consists of distinct points of T. Moreover, if Aα = 1,

ζn =ζ0 + nβ

1 + nβ, where β =

α(ζ0 − 1)

(1− α),

while, for Aα = 1, we have

ζn =1− καηA

nα

1− ηAnα

, where η =(ζ0 − 1)

(ζ0 − κα).

(iv) For each ζ0 ∈ T \ {1, κα}, if Aα = 1 then

limn→±∞

ϕαn(ζ0) = 1.


(v) For each ζ0 ∈ T \ {1, κα}, if Aα > 1 then

limn→+∞

ϕαn(ζ0) = κα while lim

n→−∞ϕαn

(ζ0) = 1.

(vi) For each ζ0 ∈ T \ {1, κα}, if Aα < 1 then

limn→+∞

ϕαn(ζ0) = 1 while lim

n→−∞ϕαn

(ζ0) = κα.

Proof. (i): Trivial.

(ii): The fixed points of ϕα are 1 and κα. Hence,(ϕ[n]α (1)

)n∈Z

= {1} and(ϕ[n]α (κα)

)n∈Z

= {κα}.

(iii): The fixed points of ϕ[n]α , n ∈ Z \ {0} are also precisely 1 and κα. Hence,

if ζ0 ∈ {1, κα}, then

ϕ[n]α (ζ0) = ϕ[n]

α (ζ0), (m = n, m, n ∈ Z).

If Aα = 1, then, by (5.8),

ϕ[n]α (z) =

(1− α)z + nα(z − 1)

(1− α) + nα(z − 1).

Hence, by (ii),

ζn = ϕ[n]α (ζ0) =

ζ0 + nα(ζ0−1)(1−α)

1 + nα(ζ0−1)(1−α)

.

But, if Aα = 1, then

ϕ[n]α (z) =

−Anακα(z − 1) + (z − κα)

−Anα(z − 1) + (z − κα)

.

Hence, again by (ii),

ζn = ϕ[n]α (ζ0) =

1−Anακα

(ζ0−1)(ζ0−κα)

1−Anα

(ζ0−1)(ζ0−κα)

.

(iv)− (vi): These follow from (iii). �

6.3. Geometric interpretation. We can also provide a geometric interpre-tation of the equivalence classes. Chapter 3 of [59] contains a comprehensive studyof the geometric behavior of Mobius transformations. A very short glimpse of thisvisual interpretation is provided below.

Theorem 6.1 shows that the points(ϕαn

(z0))n∈Z

reside on some curves passingthrough 1, κα and z0, and tend to the frontiers 1 and κα as n→ ±∞.

Parabolic case, κα = 1: The relation (5.14) reveals that the equivalence class(ϕαn

(z0))n∈Z

is on the image of the line

t �−→ φ1(z0) +α

1− αt, (t ∈ R),

under the mapping φ1. Since φ1(∞) = 1 and φ1(φ1(z0)) = z0, the image is acircle passing through the points 1 and z0. Different values of φ1(z0) correspond todifferent parallel lines. Hence, their images are circles which are tangent at 1. One


particular circle corresponds to the line passing through φ1(z0) = 1/2. In this case,we have

φ1

(1

2+

α

1− αt

)=

t+ 1−α2α

t+ 1−α2α

∈ T.

Hence, the image of this last line is the unit circle T. In other words, the iteratesof boundary points stay on T and (except for 1), they form a two-sided sequencewhich converges to 1 from both sides.

Hyperbolic case, κα = 1: By (5.12), we see that the equivalence class(ϕαn

(z0))n∈Z

is on the image of

t �−→ φκα(z0)

Atα

, (t ∈ R).

Since Aα ∈ (0,∞) \ {1}, the image is a line passing through 0 and φκα(z0). Since

φκα(∞) = 1, φκα

(0) = κα, and φκα(φκα

(z0)) = z0, the image is a circle passingthrough the points 1, κα and z0.

Of course, depending on the location of α, the arrows in the above two picturesmay be reversed. The important fact to recall is that as |n| increases the points oforbits get close to one of the fixed points on the boundary.


7. Minimal Blaschke products

In this section, we take the first step in finding the solutions of the equationψ◦ϕα = λψ by showing that each equivalence class of ∼α in D produces a Blaschkeproduct which is a minimal solution of the equation. Therefore, having the freedomto choose α ∈ D and any of the equivalence classes it generates, the following resultprovides a vast variety of solutions of the functional equation.

Theorem 7.1 ([32]). Fix α ∈ D, α = 0. Let z0 ∈ D, and let (zn)n∈Z ⊂ D bethe corresponding equivalence class generated by ∼α. Then (zn)n∈Z is a two-sidedinfinite Blaschke sequence and the corresponding Blaschke product

Bα,z0 =

∞∏n=−∞

bzn

satisfies the functional equation

Bα,z0 ◦ ϕα = γα,z0 Bα,z0 .

Moreover, no proper divisor ψ of Bα,z0 satisfies any functional equation of the formψ ◦ ϕα = λψ, λ ∈ T.

Proof. According to Theorem 6.1(i), without loss of generality, we can assume

zn = ϕαn(z0), (n ∈ Z).

Hence,

1− |zn|2 = 1− |ϕαn(z0)|2

=(1− |αn|2) (1− |z0|2)

|1− αn z0|2

≤ 1 + |z0|1− |z0|

(1− |αn|2).

Therefore, to deal with (1− |αn|2), in light of Lemma 5.3, we consider two cases.Parabolic case, κα = 1: Using (5.7), we have

1− |zn|2 ≤ 1 + |z0|1− |z0|

(1−∣∣∣∣ nα

1 + (n− 1)α

∣∣∣∣2)

=1 + |z0|1− |z0|

1 + (n− 1)(α+ α)− (2n− 1)|α|2|1 + (n− 1)α|2

≤ 1 + |z0|1− |z0|

1− |α|2|1 + (n− 1)α|2 = O(1/n2), (n −→ ±∞).

Hence, (zn)n∈Z is a two-sided Blaschke sequence.

Hyperbolic case, κα = 1: We have

1− |zn|2 ≤ 1 + |z0|1− |z0|

(1−∣∣∣∣κα(1−An

α)

1− καAnα

∣∣∣∣2)

=1 + |z0|1− |z0|

(2− κα − κα)Anα

|1− καAnα|2

= O(q|n|), (n −→ ±∞),

where q := min{Aα, 1/Aα} < 1. Hence, again (zn)n∈Z is a two-sided Blaschkesequence (indeed, with a geometric rate of convergence).


To show thatBα,z0 =

∏n∈Z

bzn

satisfies the functional equation Bα,z0 ◦ ϕα = Bα,z0 , we rewrite Bα,z0 in the form

Bα,z0 =∏n∈Z

γnϕzn ,

where γn are appropriate constants such that bzn = γnϕzn , i.e.,

γn = −|zn|zn· 1− zn1− wn

, (n ∈ Z).

Now, by (5.9) and (5.11),

(7.1) ϕzn ◦ ϕα = ϕz0 ◦ ϕα−n◦ ϕα = ϕz0 ◦ ϕα−n+1

= ϕzn−1.

Therefore,

Bα,z0 ◦ ϕα =∏n∈Z

γnϕzn ◦ ϕα =∏n∈Z

γnϕzn−1=

(∏n∈Z

γnγn−1

)Bα,z0 .

In the first place, even thought it can be directly verified, the above calculationshows that this last product has to be convergent. Secondly, we have∏

n∈Z

γnγn−1

= limN→+∞

N∏n=−N+1

γnγn−1

= limN→+∞

γNγ−N

=limN→+∞ γNlimN→−∞ γN

.

Using Theorem 6.1, we can compute both limits. In fact, the formula

zn = ϕαn(z0) =

1− αn

1− αn

z0 − αn

1− αnz0, (n ∈ Z),

implies1− zn1− wn

=1− z01− z0

1− αn

1− αn

1− αnz01− αnz0

, (n ∈ Z).

Hence,

γn =1− z01− z0

1− αnz0|1− αnz0|

|αn − z0|αn − z0

, (n ∈ Z),

and thusαn → 1 =⇒ γn → 1

while

αn → κα =⇒ γn →1− z01− z0

1− καz0κα − z0

.

Therefore, by Theorem 6.1,

∏n∈Z

γnγn−1

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1−z01−z0

1−καz0κα−z0

if Aα > 1,

1 if Aα = 1,

1−z01−z0

κα−z01−καz0

if Aα < 1.

In fact, the above calculation shows the motivation for the definition of γα,z0 . It isdefined such that

∏n∈Z

γn

γn−1= γα,z0 . Thus, Bα,z0 satisfies the functional equation

Bα,z0 ◦ ϕα = γα,z0 Bα,z0 .Finally, the identity (7.1) reveals that no proper divisor of Bα,z0 satisfies a

functional equation of the form ψ ◦ ϕα = λψ. �


By Theorem 6.1, the equivalence class generated by 0 is(αn

)n∈Z

and, in thiscase, α−n = αn. Hence, the corresponding minimal Blaschke product is

B(z) = z+∞∏n=1

(αn − z) (αn − z)

(1− αn z) (1− αn z).

By Theorem 7.1, this is the minimal Blaschke product which satisfies the equationB ◦ ϕα = B and, moreover, B(0) = 0.

8. Composition of singular inner functions

Based on a special case of the canonical factorization theorem [56], given apositive singular Borel measure σ on T, then

(8.1) ψσ(z) = exp

{−∫T

ζ + z

ζ − zdσ(ζ)

}, (z ∈ D),

is a singular inner function on D. Reciprocally, for each singular inner function ψ,there is a unique positive singular Borel measure σ on T such that ψ = ψσ.

If ϕ and ψ are inner functions, then ψ ◦ϕ is also inner. This nontrivial fact is aconsequence of the Lindelof theorem [60,68]. Hence, in particular, if ψ is a singularinner function then both ψ ◦ϕα and ϕα ◦ψ are inner functions. A classical theoremof Frostman ensures that the composition ϕα ◦ ψ is a Blaschke product for all αexcept possibly for a small (zero logarithmic capacity) range of values of α ∈ D

[33]. Despite this fact, the other combination, i.e. ψ ◦ϕα, is always a singular innerfunction. We proceed to explore further this singular inner function.

Given a Borel measure σ, and α ∈ D, define the weighted push-forward measure

(8.2) σα(E) =

∫ϕα(E)

|ϕ−1α (ζ)− α|21− |α|2 dσ(ζ)

for each Borel set E ⊂ T. Clearly, if σ is positive and singular, then so is σα. Notethat ϕ−1

α = ϕα−1, where α−1 = −α 1−α

1−α , and we can directly verify that

ϕ−1α (ζ)− α = ϕα−1

(ζ)− α =(1− |α|2)ζ1− α−1 ζ

.

Thus|ϕα−1

(ζ)− α|21− |α|2 =

1− |α−1|2|ζ − α−1|2

.

This formulation reveals that σα can also be written as

(8.3) σα(E) =

∫ϕα(E)

1− |α−1|2|ζ − α−1|2

dσ(ζ).

The reason for defining σα is clarified in the following result.

Theorem 8.1 ([11]). Let α ∈ D, and let ψσ be a singular inner function on D

corresponding to the positive singular measure σ. Then

ψσ ◦ ϕα = η ψσα,

where η is a unimodular constant.


Proof. According to (8.1), composing ψσ with ϕα gives

(8.4) (ψσ ◦ ϕα)(z) = exp

{−∫T

ζ + ϕα(z)

ζ − ϕα(z)dσ(ζ)

}, (z ∈ D).

Now, consider the mapping ϕ−1α : T −→ T and the corresponding (ordinary) push-

forward measure

μ(E) = σ(ϕα(E) ), (E is a Borel subset of T).

This measure has the crucial property

(8.5)

∫T

f dμ =

∫T

f ◦ ϕ−1α dσ,

which holds at least for bounded Borel measurable functions f on T. Hence, by(8.4) and (8.5), we have

(ψσ ◦ ϕα)(z) = exp

{−∫T

ϕα(ζ) + ϕα(z)

ϕα(ζ)− ϕα(z)dμ(ζ)

}, (z ∈ D).

However, a simple computation shows

ϕα(ζ) + ϕα(z)

ϕα(ζ)− ϕα(z)=

1−α1−α

ζ−α1−α ζ + 1−α

1−αz−α1−α z

1−α1−α

ζ−α1−α ζ −

1−α1−α

z−α1−α z

=(1 + |α|2)(ζ + z)− 2(α+ αζz)

(1− |α|2)(ζ − z)

=|ζ − α|21− |α|2

ζ + z

ζ − z+ 2i

�(αζ)1− |α|2 .(8.6)

Therefore,

(ψσ ◦ ϕα)(z) = η exp

{−∫T

ζ + z

ζ − z

|ζ − α|21− |α|2 dμ(ζ)

}, (z ∈ D),

where

(8.7) η = exp

{− 2i

1− |α|2∫T

�(αζ) dμ(ζ)}.

Clearly η ∈ T, and ψσ ◦ ϕα = η ψν , where

dν(ζ) =|ζ − α|21− |α|2 dμ(ζ).

But, by (8.5), for each Borel set E ⊂ T,

ν(E) =

∫E

|ζ − α|21− |α|2 dμ(ζ)

=

∫T

1E(ζ)|ζ − α|21− |α|2 dμ(ζ)

=

∫T

1E(ϕ−1α (ζ))

|ϕ−1α (ζ)− α|21− |α|2 dσ(ζ)

=

∫ϕα(E)

|ϕ−1α (ζ)− α|21− |α|2 dσ(ζ)

= σα(E).

�


Given two positive singular measures σ1 and σ2, it is well-known that ψσ1is

a divisor of ψσ2(in the family of inner functions, or equivalently in the algebra

of H∞ functions) if and only if σ1 ≤ σ2. Hence, in the light of Theorem 8.1,we immediately deduce the following result which has essential applications in thetheory of composition operators on model spaces.

Corollary 8.2. Let α ∈ D, and let ψσ be a singular inner function on D

corresponding to the positive singular measure σ. Then the following hold.

(i) ψσ is a divisor of ψσ ◦ ϕα if and only if σα ≥ σ.(ii) ψσ ◦ ϕα is a divisor of ψσ if and only if σα ≤ σ.(iii) ψσ ◦ ϕα = λψσ, where λ is a unimodular constant, if and only if σα = σ.

Corollary 8.3. Let α ∈ D, ζ ∈ T and a > 0. Put

S(z) = exp

{−a ζ + z

ζ − z

}.

Then

(S ◦ ϕα)(z) = η exp

{−a

|ϕα−1(ζ)− α|2

1− |α|2 ·ϕα−1

(ζ) + z

ϕα−1(ζ)− z

},

where η is the unimodular constant

η = exp

{−i 2a

�(αϕα−1(ζ))

1− |α|2

}.

Proof. Put σ = aδζ . Then, by (8.2), we have

σα = a|w − α|21− |α|2 δw,

where w = ϕα−1(ζ). According to Theorem 8.1, σα is the measure that generates

S ◦ ϕα. The explicit formula for η follows from (8.7). �

Corollary 8.4. Let α ∈ D, and let ψ and ω be singular inner functions. Thenω is a divisor of ψ if and only if ω ◦ ϕα is a divisor of ψ ◦ ϕα.

Proof. Let σ and ν be the positive singular measures that generate respec-tively ψ and ω. Hence, ω is a divisor of ψ if and only if μ = σ − ν is a positivesingular measure on T. In this situation, by the main definition (8.1), we haveσα = να + μα. But, according to Theorem 8.1, σα and να are the measures thatgenerates ψ◦ϕα and ω◦ϕα. Therefore, if ω is a divisor of ψ, then ω◦ϕα is a divisorof ψ ◦ ϕα. The inverse implication follows by considering α−1 instead of α. �

8.1. Minimal discrete singular inner functions. In this section, our maingoal is to study the discrete singular inner solutions of the equation ψ ◦ ϕα = λψ,λ ∈ T. While this result is a special case of the result presented in Section 8, ithas two new characteristics. First, we provide an explicit formula for the innerfunction. Second, this construction leads to the minimal discrete singular solutionsof the functional equation. Hence, we are faced with a similar phenomenon as in[32] were the minimal Blaschke products solutions were completely characterized.

Before discussing the main case (case 3), let us consider two special cases cor-responding to the fixed points. Note that in all cases in this section, we assumethat ψ satisfies the functional equation ψ ◦ ϕα = λψ.


Case I, ζ = 1: By Corollaries 8.3 and 8.4,

S(z) = exp

{−a 1 + z

1− z

}, (z ∈ D),

is a divisor of ψ if and only if

(8.8) (S ◦ ϕα)(z) = η exp

{− a

Aα

1 + z

1− z

}, (z ∈ D),

is a divisor of ψ. Therefore, we must have Aα = 1, or equivalently α ∈ D1.

Case II, ζ = κα: By Corollaries 8.3 and 8.4,

S(z) = exp

{−a κα + z

κα − z

}, (z ∈ D),

is a divisor of ψ if and only if

(8.9) (S ◦ ϕα)(z) = η exp

{−aAα

κα + z

κα − z

}, (z ∈ D),

is a divisor of ψ◦ϕα. Therefore, we must again have Aα = 1, or equivalently α ∈ D1.

Case III, ζ ∈ T \ {1, κα}: For this situation, we need the following construction.If (σn)n is a sequence of positive numbers (finite or infinite, and repetition allowed)such that

∑n σn <∞, and (ζn)n is any sequence of points on T, then

(8.10) S(z) =∏n

exp

{−σn

ζn + z

ζn − z

}, (z ∈ D),

is a singular inner function with Dirac measures anchored at the points (ζn)n. Aparticular type of these singular inner functions (some necessary restrictions on σn

and ζn) appears below.

Lemma 8.5. Let α ∈ D, α = 0, and let ψ be a singular inner function suchthat ψ ◦ ϕα = λψ, λ ∈ T. Assume that

S0(z) = exp

{−σ0

ζ0 + z

ζ0 − z

}, (z ∈ D),

where ζ0 ∈ T \ {1, κα} and σ0 > 0, is a divisor of ψ. Then, for each n ∈ Z,

Sn(z) = exp

{−σn

ζn + z

ζn − z

},

where ζn = ϕαn(ζ0) and

(8.11)σn

σ0=

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

n−1∏k=0

1− |α|2|α− ζk|2

if n ≥ 1,

−1∏k=n

|α− ζk|21− |α|2 if n ≤ −1,

is also a divisor of ψ.


Proof. By Theorem 6.1, the equivalence class generated by ζ0 is precisely(ζn)n∈Z

=(ϕαn

(ζ0))n∈Z

. Since ζ0 ∈ T \ {1, κα}, the class(ζn)n∈Z

consists of

distinct points of T \ {1, κα}. Note that, by (5.8), we can also write

ζn−1 = ϕα−1(ζn), (n ∈ Z).

Therefore, by Corollary 8.3, we obtain

(Sn ◦ ϕα)(z) = λn exp

{−σn

|ζn−1 − α|21− |α|2 · ζn−1 + z

ζn−1 − z

},

where

(8.12) λn = exp

{−2iσn

�(αζn−1)

1− |α|2

}, (n ∈ Z).

Now, an elementary calculation shows that the explicit definition (8.11) is in factequivalent to the recursive relation

σn−1 = σn|ζn−1 − α|21− |α|2 , (n ∈ Z).

Thus, in short, we have

(8.13) Sn ◦ ϕα = λn Sn−1, (n ∈ Z).

Since ψ ◦ ϕα = λψ, by induction and Corollary 8.4, we deduce that Sn is a divisorof ψ. �

Lemma 8.5 paves the road for the following important construction. Fix α ∈ D,α = 0. Given ζ0 ∈ T \ {1, κα} and σ0 > 0, let C =

(ζn)n∈Z

be the equivalence class

generated by ζ0, and let σn be given by (8.12). Then define Sα,σ0,C(z) by (8.10),i.e.,

(8.14) Sα,σ0,C(z) =∞∏

n=−∞Sn(z) =

∞∏n=−∞

exp

{−σn

ζn + z

ζn − z

}, (z ∈ D).

The only obstacle in this definition that needs to be verified is that∑∞

n=−∞ σn <∞,but, as explained below, this always holds.

According to Theorem 6.1(iv), ζ±∞ = limn→±∞ ζn exists and, moreover, ζ±∞ ∈{1, κα}. Hence, by (8.11),

limn→+∞

|σn|1n =

1− |α|2|α− ζ+∞|2

and

limn→−∞

|σn|1

|n| =|α− ζ−∞|21− |α|2 .

Therefore, by (8.14) and the precise classification provided in part (iv) of Theorem6.1, we have

(8.15) limn→+∞

|σn|1n = lim

n→−∞|σn|

1|n| = min{Aα, 1/Aα} =: qα.

Thus, on the regions Aα > 1 and Aα < 1, there is no problem and∑∞

n=−∞ σn

is in fact comparable to a geometric series∑∞

n=0 qnα. However, on Aα = 1, or


equivalently α ∈ D1, a more subtle calculation is needed. In this case, by theformula in Theorem 6.1(iii) for ζk, we have

1− |α|2|α− ζk|2

=|1− α|2∣∣∣α− ζ0+kβ

1+kβ

∣∣∣2=

∣∣∣∣ (1− α)(1 + kβ)

α(1 + kβ)− (ζ0 + kβ)

∣∣∣∣2=

∣∣∣∣k + a

k + b

∣∣∣∣2 , (k ∈ Z),

where

a =1− α

α(ζ0 − 1)and b =

ζ0 − α

α(ζ0 − 1).

Hence,

σn

σ0=

n−1∏k=0

∣∣∣∣k + a

k + b

∣∣∣∣2 , (n ≥ 1).

Using the Gamma function, this formula can be rewritten as

(8.16) σn = σ0

∣∣∣∣Γ(a+ n)Γ(b)

Γ(a)Γ(b+ n)

∣∣∣∣2 , (n ≥ 1).

This formula clearly holds for n = 0. For n ≤ −1, we similarly have

σn

σ0=

−1∏k=n

|α− ζk|21− |α|2 =

−1∏k=n

∣∣∣∣ k + b

k + a

∣∣∣∣2 =

∣∣∣∣Γ(a+ n)Γ(b)

Γ(a)Γ(b+ n)

∣∣∣∣2 .Therefore, the formula (8.16) in fact holds for all values of n ∈ Z.

We know that

zb−a Γ(z + a)

Γ(z + b)∼ 1 +

(a− b)(a+ b− 1)

z+O(1/z2)

as z −→ ∞ along any curve joining 0 and ∞, and avoiding the points −a,−a −1,−a− 2, . . . and −b,−b− 1,−b− 2, . . . [1, Page 257]. Hence,

Γ(a+ n)

Γ(b+ n)∼ 1

n1/α

as n −→ ±∞. But, on D1, we can write

α =1

2+

1

2eiθ = cos(θ/2) eiθ/2, (θ = kπ).

Thus, 1/α = 1− i tan(θ/2), which implies

(8.17)

∣∣∣∣ Γ(a+ n)

Γ(b+ n)

∣∣∣∣2 ∼ 1

n2

as n −→ ±∞. Therefore,∑

n∈Zσn <∞, for any choice of ζ0 ∈ T \ {1}.

Theorem 8.6 ([11]). Let α ∈ D \ {0}, and let Sα,σ0,C be defined by (8.14).Then Sα,σ0,C satisfies the functional equation

Sα,σ0,C ◦ ϕα = λSα,σ0,C ,


where λ is a unimodular constant (depending on α, σ0 and ζ0), and no properdivisor of Sα,σ0,C which contains at least one of the factors

z �−→ exp

{−σn

ζn + z

ζn − z

}does so.

Proof. By the discussion before the theorem, Sα,σ0,C is a well-defined discretesingular inner function. Using the above notations, we write

Sα,σ0,C =

∞∏n=−∞

Sn.

Hence, by (8.13),

Sα,σ0,C ◦ ϕα =

∞∏n=−∞

Sn ◦ ϕα =

∞∏n=−∞

λnSn−1 = λSα,σ0,C .

Note that, based on (8.12) and (8.16), we see that λ = eicσ0 . However, we do notneed to calculate the precise value of c.

Finally, according to Lemma 8.5, If a divisor of ψ contains one the factors Sn,it contains all of them. Hence, no proper divisor can satisfy the functional equationψ ◦ ϕα = λψ. �

Based on the above three cases, we can now provide the following set of solu-tions. This result shows that the singular functions Sα,σ0,C behave like atoms inthe family of all solutions.

Corollary 8.7. Let α ∈ D\{0}, and let C = (Cm)m≥1 be a (finite or infinite, andrepetition allowed) collection of equivalence classes of ∼α on T. Let σ = (σm,0)m≥1,σm > 0, be such that

(8.18)∑m≥1

σm,0 <∞.

Put

(8.19) Sα,σ,C =∏m≥1

Sα,σm,0,Cm,

where Sα,σm,0,Cmis given by (8.14). Then Sα,σ,C ◦ ϕα = λSα,σ,C for some λ ∈ T.

Proof. By (8.15), (8.16), (8.17) and (8.18),∑n∈Z, m≥1

σm,n ≤ C∑m≥1

σm <∞.

Hence, the function Sα,σ,C given by (8.19) is a well-defined singular inner function.Then, by Theorem 8.6,

Sα,σ,C ◦ ϕα =∏m≥1

Sα,σm,0,Cm◦ ϕα =

∏m≥1

λmSα,σm,0,Cm= λSα,σ,C ,

where λ ∈ T. �


In the above results, we studied the discrete singular inner solutions of thefunctional equation ψ ◦ ϕα = λψ, λ ∈ T, and this led to a two-sided product. Inthis section, we study one-sided products and this naturally leads to the solutionsof the more general functional equation ψ ◦ ϕα × ω = ψ, where ω is a nonconstantinner function.

Fix m ∈ Z and, using the same notations as in (8.14), define

(8.20) ψ =

m∏n=−∞

Sn.

Hence, by the crucial identity (8.13), we have

ψ ◦ ϕα =

m∏n=−∞

Sn ◦ ϕα =

m∏n=−∞

λn Sn−1 =

m∏n=−∞

λn

m−1∏n=−∞

Sn.

Thus, ψ satisfies the equation

ψ ◦ ϕα × Sm = λψ,

where λ is a unimodular constant.On the other hand, if we define

(8.21) ψ =

∞∏n=m

Sn,

then, by a similar calculation,

ψ ◦ ϕα =

∞∏n=m

Sn ◦ ϕα =

∞∏n=m

λn Sn−1,

and thus ψ fulfills

ψ ◦ ϕα = λSm × ψ.

8.2. Discussion on the general solution. Let ψ be an inner function satis-fying ψ◦ϕα = λψ; denote its zero set on D by Z(ψ). Then the equation ψ◦ϕα = λψimplies ψ ◦ ϕα−1

= λ ψ, and by induction we obtain

ψ ◦ ϕαn= λn ψ, (n ∈ Z).

This identity reveals that if z1 is a zero of ψ, then in fact the whole equivalenceclass [zn]n∈Z, generated by ∼α, is in Z(ψ). Hence, we can write

Z(ψ) =⋃m

Cm,

where (Cm)m is a (finite or infinite, and repetition allowed) collection of equivalenceclasses of ∼α in D. Note that since ψ is a non-constant inner function, we musthave

(8.22)∑m

∑zmn∈Cm

(1− |zmn|) <∞.

Thus,

(8.23) Bα,(Cm)m =∏m

Bα,Cm


is a well-defined Blaschke product and, by Theorem 7.1, Bα,(Cm)m satisfies thefunctional equation

Bα,(Cm)m ◦ ϕα = λ′Bα,(Cm)m ,

where λ′ is an appropriate unimodular constant. These types of Blaschke productsform the main building blocks for a description of solutions of the equation ψ◦ϕα =λψ, λ ∈ T.

Again, thanks to Theorem 7.1, it is rather trivial that if we have a sequencewhich can be decomposed as above, then the corresponding Blaschke product is infact a solution of the functional equation.

Put S = ψ/Bα,(Cm)m . The discussion above shows that S is a zero free innerfunction (i.e. a singular inner function), which satisfies an equation of the formS ◦ϕα = λ′′S, λ′′ ∈ T. The classification of such functions is still an open question.However, we can at least deduce the following result.

Theorem 8.8. Fix α ∈ D, α = 0. If a Blaschke product B satisfies the func-tional equation B ◦ ϕα = λB, then its zero set is a union of equivalence classesgenerated by ∼α. Reciprocally, if a sequence (zn)n ⊂ D is such that:

i) as in (8.23), it can be decomposed as a union of equivalence classes generatedby ∼α, and

ii) satisfies (8.22),

then the corresponding Blaschke product B is a solution of the functional equationB ◦ ϕα = λB, with some unimodular constant λ. In particular, if α ∈ D1, thenλ = 1.

If ψ0 satisfies the equation ψ ◦ ϕα = ψ, and ω is any arbitrary inner function,then we also have

(ω ◦ ψ0) ◦ ϕα = (ω ◦ ψ0).

Hence, ψ = ω ◦ ψ0 is also a solution of the equation ψ ◦ ϕα = ψ. For example, ifB is any of the Blaschke products (8.23) for which γ = 1, then ω ◦B is a solution.What is rather surprising is that all solutions are obtained in this manner.

Theorem 8.9. Let α ∈ D, α = 0. Then the inner function ψ is a solution ofthe equation ψ ◦ ϕα = ψ if and only if there is an inner function ω and a Blaschkeproduct B of type (8.23) such that

ψ = ω ◦B.

Proof. Without loss of generality, assume that ψ is nonconstant. Then, by

a celebrated result of Frostman [33], there is a β ∈ D such that ψ = bβ ◦ ψ is aBlaschke product with simple zeros. As a matter of fact, in a sense (logarithmiccapacity), there are many such β’s, but just one choice is enough for us.

Surely, ψ satisfies ψ ◦ ϕα = ψ. By induction, we get

ψ ◦ ϕαn= ψ, (n ∈ Z).

If z0 is a zero of ψ, then the above identity shows that ϕαn(z0) is also a zero of ψ.

Hence, we can classify the zeros of ψ as a union of equivalence classes of ∼α, e.g.,(Cm)m. This observation immediately reveals that, up to a unimodular constant,

ψ is precisely a Blaschke product of type (8.23). Since ψ = b−1β ◦ ψ, the proof is

complete. �


It it important to keep in mind that the representation ψ = ω ◦ B, given inTheorem 8.9, is far away from being unique. For example, in the proof of thetheorem, we picked one of the Frostman shifts and then constructed B. Differentshifts give different sets of zeros and thus different Blaschke products.

9. Model spaces

The most distinguished operator in the whole theory of operators is probablythe shift operator. The unilateral forward shift operator on H2 is defined by

S : H2 −→ H2

f �−→ zf.

A celebrated theorem of Beurling [10] says that the nonzero shift invariant closedsubspaces of H2 are precisely of the form uH2, where u is an inner function. Thatis why subspaces uH2 are also known as Beurling subspaces of H2. Therefore, theproper closed subspaces of H2 which are invariant under the backward shift S∗ are(uH2)⊥ = H2 � uH2. These subspaces are so important in their own right thatthey have their own name. Due to their application in the theory of contractions,they are called model subspaces of H2 and denoted by Ku. A model space Ku isactually a reproducing kernel Hilbert space with kernel

kuλ(z) =1− u(λ)u(z)

1− λz, (λ, z ∈ D).

If M and N are closed subspaces of H2 such that for each f ∈ M we haveCϕf = f ◦ ϕ ∈ N , then the restricted mapping Cϕ : M −→ N is automaticallycontinuous. Even more can be said about this phenomenon. If E is a dense subsetof M and the property Cϕf = f ◦ ϕ ∈ N holds for all f ∈ E , then we can stillconclude that Cϕ :M−→ N is well-defined and continuous. This fact is exploitedbelow.

9.1. The model space generated by Cϕ(Ku). Given an inner functionϕ, we know that f is non-cyclic for S∗ if and only if f ◦ ϕ is non-cyclic for S∗

[26, Theorem 2.4.4]. In other words, if we put

K =⋃

u is inner

Ku,

then, for any inner symbol ϕ, the restricted mappings Cϕ : K −→ K and Cϕ :H2 \ K −→ H2 \ K are both well-defined. Theorem 9.1 describes the first mappingin a more precise fashion. More detailed treatment of this topic is available at[57,58] and in the forthcoming book [34].

Theorem 9.1 ([58]). Let ϕ and u be inner, and let

v(z) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩(u ◦ ϕ)(z), if u(0) = 0 and ϕ(0) = 0,

z (u ◦ ϕ)(z), if u(0) = 0 and ϕ(0) = 0,

z u(ϕ(z))ϕ(z) , if u(0) = 0.

Then the mapping

Cϕ : Ku −→ Kv


is well-defined and bounded. Moreover, Kv is the smallest closed S∗-invariant sub-space of H2 which contains the image Cϕ(Ku).

Proof. We just treat the first case, as other cases are similar. Write S forthe forward shift operator. For each h ∈ H2 and n ≥ 1, it is well known thatS∗nθ ∈ Kθ, n ≥ 1, and moreover the sequence (S∗nθ)n≥1 is dense in Kθ. Theverification of these facts is straightforward. We have

〈S∗nθ, θh〉 = 〈θ, Snθh〉 = 〈θ, znθh〉 = 〈1, znh〉 = 0.

Hence, S∗nθ ∈ (θH2)⊥ = Kθ, n ≥ 1. Moreover, assume f ∈ Kθ is such thatf ⊥ S∗nθ, n ≥ 1. Then

0 = 〈f, S∗nθ〉 = 〈znf, θ〉 = 〈zn, θf〉, (n ≥ 1),

which implies θf ∈ H2, or equivalently f ∈ θH2. Thus, f = 0, i.e., the sequence(S∗nθ)n≥1 is dense in Kθ. Therefore, to show that Cϕ maps Kθ into Kη, it isenough to verify that

Cϕ(S∗nθ) ∈ Kη, (n ≥ 1).

For this, we show that Cϕ(S∗nθ) ⊥ ηH2.

Write θ(z) =∑∞

k=0 akzk. Then

Cϕ(S∗nθ) =

∞∑k=n

akϕk−n =

(θ ◦ ϕ−

n−1∑k=0

akϕk

)ϕ−n.

Since ϕ and η are inner, for each h ∈ H2 and n ≥ 1, we have

〈Cϕ(S∗nθ), θh〉 =

⟨(θ ◦ ϕ−

n−1∑k=0

akϕk

)ϕ−n, ηh

⟩

=

⟨(η −

n−1∑k=0

akϕk

), ϕnηh

⟩

= 〈η, ϕnηh〉 −n−1∑k=0

ak〈ϕk, ϕnηh〉

= 〈1, ϕnh〉 −n−1∑k=0

ak〈1, ϕn−kηh〉 = 0.

Remember ϕ(0) = 0, which was exploited in the last line. In short, we showed thatCϕKθ ⊂ Kη, which immediately also implies 〈CϕKθ〉 ⊂ Kη.

It remains to show that the smallest closed subspace of H2 which contains therange of Cϕ, i.e., CϕKθ, is precisely Kη. To this end, note that

kθλ(z) =1− θ(λ)θ(z)

1− λz

is the reproducing kernel of the evaluation functional at the point λ ∈ D. Inparticular, kθλ ∈ Kθ for any values of λ ∈ D. Since kθ0 = 1 − θ(0)θ ∈ Kθ, we get

Cϕkθ0 = 1−θ(0)η ∈ CϕKθ. Thus, remembering the assumption θ(0) = 0, we deduce

S∗Cϕkθ0 = S∗η ∈ S∗CϕKθ ⊂ 〈CϕKθ〉.

But, we saw above that S∗η is a generator of Kη. Hence, Kη ⊂ 〈CϕKθ〉. �


Another proof of the above result is available in [32]. Since ϕ is inner, by[24, Theorem 3.8], CϕKu is a closed subspaces of H2. However, CϕKu mayfail to be invariant under S∗. For example, take u(z) = ϕ(z) = z2. Thenv(z) = z u(ϕ(z))/ϕ(z) = z3, Ku = span{1, z}, Kv = span{1, z, z2}, but CϕKu =span{1, z2}.

In the hierarchy of inner functions, for a given ϕ and u, the largest innerfunction v among the three possible situations in Theorem 9.1 is zu ◦ ϕ, and thisgives the largest model subspace Kz u◦ϕ among the three possible cases. Hence, ifwe ignore the smallest possible model subspace which contains the image of Cϕ, weobtain the following result.

Corollary 9.2. For every inner functions ϕ and u, the composition operatorCϕ : Ku −→ Kz u◦ϕ is well-defined and bounded.

9.2. When do we have Cϕ ∈ L(Ku)? Since in the mapping Cϕ : Ku −→ Kv,given in Theorem 9.1, the choice of v is optimal we naturally wonder when theinclusion Kv ⊂ Ku holds in order to obtain a composition operator which maps Ku

into itself.

Theorem 9.3. Let ϕ and u be inner functions on D. Then the mapping Cϕ :Ku −→ Ku is well-defined and bounded if and only if one of the the followingsituations holds:

(i) ϕ(z) = z and any inner u;(ii) u(z) = γ z, γ ∈ T, and any inner ϕ;(iii) u(z) = ϑ(zn), for some integer n ≥ 2 and an arbitrary inner function ϑ with

ϑ(0) = 0, and

ϕ = ρei2kπ/n , (1 ≤ k ≤ n);

(iv) u(z) = γ z(τp(z)

)m, where γ ∈ T, p ∈ D, m ≥ 1, and any hyperbolic rotation

ϕ = τp ◦ ρλ ◦ τp, (λ ∈ T);

(v) u(z) = z(τp(z)

)mψ((τp(z)

)n), where p ∈ D, m ≥ 0, n > 1, ψ is a non-

constant inner function, and

ϕ = τp ◦ ρei2kπ/n ◦ τp, (1 ≤ k ≤ n);

(vi) p, the Denjoy–Wolff point of ϕ, is on T, and u is of the form u(z) = zψ(z),where ψ fulfills

ψ(ϕ(z)

)= λψ(z), (z ∈ D),

for some unimodular constant λ;(vii) p, the Denjoy–Wolff point of ϕ, is on T, and

u(z) = γ z ψ(z)∞∏

n=0

ω(ϕ[n](z)

),

where ω is a non-constant inner function such that the product is convergent,and ψ fulfills

ψ(ϕ(z)

)= ψ(z), (z ∈ D).

Proof. If u(z) = γz, thenKu = C, for which each Cϕ is a well-defined operatoron Ku. It is also trivial that ϕ(z) = z gives the composition operator Cϕ = id oneach Ku.


According to Theorem 9.1, Cϕ ∈ L(Ku) if and only if Kv ⊂ Ku, and the latterhappens if and only if v divides u in the family of inner functions, i.e.

v(z) u1(z) = u(z), (z ∈ D),

where u1 is an inner function. To treat this equation, we should naturally con-sider three cases corresponding to the different definitions of v which were given inTheorem 9.1:

(i) If u(0) = 0 and ϕ(0) = 0, then v = u ◦ ϕ and we must have

u(ϕ(z)

)u1(z) = u(z), (z ∈ D).

This is Case I, Category III, with m = 0 and p = 0. Hence, there is an integern ≥ 1 and an inner function ϑ, with ϑ(0) = 0, such that u(z) = ϑ(zn) andϕ = ρei2kπ/n for 1 ≤ k ≤ n.

(ii) If u(0) = 0 and ϕ(0) = 0, then v(z) = z u(ϕ(z)

)and we must have

z u(ϕ(z)

)u1(z) = u(z), (z ∈ D).

Put z = 0 to see that this is impossible.

(iii) If u(0) = 0, then v(z) = z u(ϕ(z))/ϕ(z) and we must have

u(ϕ(z)

)ϕ(z)

u1(z) =u(z)

z, (z ∈ D),

Put u2(z) = u(z)/z. Hence, the above becomes

u2

(ϕ(z)

)u1(z) = u2(z), (z ∈ D).

According to the Grand Iteration Theorem, ϕ has a fixed point p in D = D∪T.Hence, we have the following three possibilities.(a) If p ∈ D, Category I gives

u(z) = γ z(τp(z)

)mwhere γ ∈ T, m ≥ 1, and

ϕ = τp ◦ ρλ ◦ τp,where λ ∈ T.

(b) If p ∈ D, Category II gives

u(z) = γ z(τp(z)

)mψ((τp(z))

n)

where γ ∈ T, m ≥ 1, n > 1, ψ is a nonconstant inner function, and

ϕ = τp ◦ ρei2kπ/n ◦ τp, (1 ≤ k ≤ n).

(c) If p ∈ T, then we are in Case II and thus, u is either of the form u(z) =zu2(z), where u2 fulfills

u2

(ϕ(z)

)= λu2(z), (z ∈ D),

for some unimodular constant λ, or

(9.1) u(z) = γ z u2(z)

∞∏n=0

u1

(ϕ[n](z)

),


where the product is convergent and u2 fulfills

u2

(ϕ(z)

)= u2(z), (z ∈ D).

�

Corollary 9.4. Let θ be a Blaschke product with θ(0) = 0. Assume that thereis α ∈ D, α = 0, such that ϕα maps Kθ into itself. Then the following hold:

(a) θ is of the form (9.1).(b) The zeros of Blaschke product B(z) = θ(z)/z can be decomposed as a union of

equivalence classes generated by ∼α.(c) If θ1 = 1, the composition mapping Cϕα

is actually an isomorphism from Kθ

into itself.

Proof. Most of the proof is done in the above discussions. In particular, wesaw that θ must have the form (9.1). We know that the Denjoy–Wolff point of ϕα

is either 1 or κα. This is because ϕα has just two fixed points on D and one ofthem has to be the Denjoy–Wolff fixed point. Therefore, by Theorem 8.8, the zerosof Blaschke product θ2 are decomposed as a union of equivalence classes generatedby ∼α and, by Case (iii), the operator Cϕα

maps Kθ into itself. To show that Cϕ

is surjective whenever θ1 = 1, note that

KzB = C⊕ Span{kzj : B(zj) = 0},where kzj is the Cauchy reproducing kernel

kzj (z) =1

1− zjz.

We have Cϕα1 = 1 and, by Theorem 5.1,

Cϕαkzj (z) =

1

1− zjϕα(z)=

A+Bz

1− ϕα−1(zj) z

,

where A and B are constants. Hence, kϕα−1(zj) belongs to the image of Cϕα

. We

assumed that the zeros of B can be decomposed as a union of equivalence classesgenerated by ∼α. Therefore, by Theorem 6.1(i), the image contains all Cauchykernels kzj , where zj runs through the zeros of B. In short, this means that themapping is surjective. �

We can also interpret Theorem 7.1 in order to say something about the pointspectrum of Cϕα

: if we write the functional equation as CϕαBα,z0 = γα,z0Bα,z0 , the

theorem says that Bα,z0 is an eigenvector of Cϕαcorresponding to the eigenvalue

γα,z0 .

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Syracuse University, Department of Mathematics, 215 Carnegie Building, Syracuse,

New York 13244


Universite Laval, Departement de mathematiques et de statistique, Quebec, QC,

G1V 0A6, Canada



A survey of some recent results on truncated Toeplitzoperators

Isabelle Chalendar, Emmanuel Fricain, and Dan Timotin

Abstract. Truncated Toeplitz operators are compressions of Toeplitz opera-tors on model spaces; they have received much attention in the last years. Thissurvey article presents several recent results, which relate boundedness, com-pactness, and spectra of these operators to properties of their symbols. Wealso connect these facts with properties of the natural embedding measuresassociated to these operators.

1. Introduction

Truncated Toeplitz operators on model spaces have been formally introducedby Sarason in [34], although some special cases have long ago appeared in literature,most notably as model operators for completely nonunitary contractions with defectnumbers one and for their commutant. This new area of study has been recentlyvery active and many open questions posed by Sarason in [34] have now beensolved. See [5,6,8,9,12,13,19–21,35,36]. Nevertheless, there are still basic andinteresting questions which remain mysterious.

The truncated Toeplitz operators live on the model spaces KΘ, which are theclosed invariant subspaces for the backward shift operator S∗ acting on the Hardyspace H2 (see Section 2 for precise definitions). Given a model space KΘ and afunction φ ∈ L2 = L2(T), the truncated Toeplitz operator AΘ

φ (or simply Aφ if

there is no ambiguity regarding the model space) is defined on a dense subspaceof KΘ as the compression to KΘ of multiplication by φ. The function φ is thencalled a symbol of the operator. An alternate way of defining a truncated Toeplitzoperator is by means of a measure; in case φ is bounded, then a possible choice ofthe defining measure for AΘ

φ is φ dm (with m Lebesgue measure).Note that the symbol or the associated measure are never uniquely defined by

the operator. From this and other points of view the truncated Toeplitz operatorshave much more in common with Hankel Operators than with Toeplitz operators.This point of view will be occasionally pursued throughout the paper.

We intend to survey several recent results that are mostly scattered in theliterature. They focus on the relation between the operator and the symbol or themeasure. Obviously the nonuniqueness is a main issue, and in some situations itmay be avoided by considering the so-called standard symbol of the operator. The

2010 Mathematics Subject Classification. Primary 30J05, 30H10, 46E22.Key words and phrases. Truncated Toeplitz operators, model spaces, compactness.


59

60 I. CHALENDAR, E. FRICAIN, AND D. TIMOTIN

properties under consideration are boundedness, compactness, and spectra. Mostof the results presented are known, and our intention is only to put them in contextand emphasize their connections, indicating the relevant references. Part of theembedding properties of measures have not appeared explicitely in the literature,so some proofs are provided only where references seemed to be lacking.

The structure of the paper is the following. After a preliminary section withgeneralities about Hardy spaces and model spaces, we discuss in section 3 Carlesonmeasures, first for the whole H2 and then for model spaces. Truncated Toeplitzoperators are introduced in Section 4, where one also discusses some boundednessproperties. Section 5 is dedicated to compactness of truncated Toeplitz operators,and Section 6 to its relation to embedding measures. The last two sections discussSchatten–von Neumann and spectral properties, respectively.

2. Preliminaries

For the content of this section, [17] is a classical reference for general factsabout Hardy spaces, while [26] can be used for Toeplitz and Hankel operators aswell as for model spaces.

2.1. Function spaces. Recall that the Hardy space Hp of the unit disk D ={z ∈ C : |z| < 1} is the space of analytic functions f on D satisfying ‖f‖p < +∞,where

‖f‖p = sup0≤r<1

(∫ 2π

0

|f(reit)|p dt2π

)1/p

, (1 ≤ p < +∞).

The algebra of bounded analytic functions on D is denoted by H∞. We denote alsoHp

0 = zHp and Hp− = zHp. Alternatively, Hp can be identified (via radial limits)

to the subspace of functions f ∈ Lp = Lp(T) for which f(n) = 0 for all n < 0. HereT denotes the unit circle with normalized Lebesgue measure m.

In the case p = 2, H2 becomes a Hilbert space with respect to the scalarproduct inherited from L2 and given by

〈f, g〉2 =

∫T

f(ζ)g(ζ) dm(ζ), f, g ∈ L2.

The orthogonal projection from L2 to H2 will be denoted by P+. The space H2−

is precisely the orthogonal of H2, and the corresponding orthogonal projection isP− = I − P+.

The Poisson transform of a function f ∈ L1 is

(2.1) f(z) =

∫T

f(ξ)1− |z|2|1− ξz|2 dξ, z ∈ D.

Suppose now Θ is an inner function, that is a function in H∞ whose radiallimits are of modulus one almost everywhere on T. Its spectrum is defined by

(2.2) s(Θ) := {ζ ∈ D : lim infλ∈D,λ→ζ

|Θ(λ)| = 0}.

Equivalently, if Θ = BS is the decomposition of Θ into a Blaschke product anda singular inner function, then ρ(Θ) is the union between the closure of the limitpoints of the zeros of B and the support of the singular measure associated to S.We will also define

ρ(Θ) = s(Θ) ∩ T.

RECENT RESULTS ON TRUNCATED TOEPLITZ OPERATORS 61

We define the corresponding shift-coinvariant subspace generated by Θ (also

called model space) by the formula KpΘ = Hp ∩ΘHp

0 , where 1 ≤ p < +∞. We willbe especially interested in the Hilbert case, that is when p = 2. In this case, we alsodenote by KΘ = K2

Θ and it is easy to see that KΘ is also given by the following

KΘ = H2 �ΘH2 ={f ∈ H2 : 〈f, g〉 = 0, ∀g ∈ H2

}.

The orthogonal projection of L2 onto KΘ is denoted by PΘ. It is well known (see[26, page 34]) that PΘ = P+−ΘP+Θ. Since P+ acts boundedly on Lp, 1 < p <∞,this formula shows that PΘ can also be regarded as a bounded operator from Lp

into KpΘ, 1 < p <∞.

The spaces H2 and KΘ are reproducing kernel spaces over the unit disc D. Therespective reproducing kernels are, for λ ∈ D,

kλ(z) =1

1− λz,

kΘλ (z) =1−Θ(λ)Θ(z)

1− λz.

Evaluations at certain points ζ ∈ T may also be bounded sometimes; thishappens precisely when Θ has an angular derivative in the sense of Caratheodory

at ζ [1]. In this case the function kΘζ (z) = 1−Θ(ζ)Θ(z)

1−ζzis in KΘ, and it is the

reproducing kernel for the point ζ.It is easy to check that, if f, g ∈ KΘ, then fg ∈ H1 ∩ zΘ2H1

− ⊂ K1Θ2 . In

particular, if f, g are also bounded, then fg ∈ KΘ2 . So (kΘλ )2 ∈ KΘ2 for all λ ∈ D.

The map CΘ defined on L2 by

(2.3) CΘf = Θzf ;

is a conjugation (i.e. CΘ is anti-linear, isometric and involutive), which has theconvenient supplementary property of mapping KΘ precisely onto KΘ.

2.2. One-component inner functions. In view of their main role in thestudy of operators on model spaces, we devote this subsection to a particular classof inner functions. Fix a number 0 < ε < 1, and define

(2.4) Ω(Θ, ε) = {z ∈ D : |Θ(z)| < ε}.

The function Θ is called one-component if there exists a value of ε for which Ω(Θ, ε)is connected. (If this happens, then Ω(Θ, δ) is connected for every ε < δ < 1.) One-component functions have been introduced by Cohn [15]. An extensive study ofthese functions appears in [3,4]; all results quoted below appear in [3].

The above definition is not very transparent. In fact, one-component functionsare rather special: a first immediate reason is that they must satisfy m(ρ(Θ)) = 0.This condition, of course, is not sufficient, but it suggests examining some simplecases.

The set ρ(Θ) is empty for finite Blaschke products, which are one-component.The next simplest case is when ρ(Θ) consists of just one point. One can prove easily

that the elementary singular inner functions Θ(z) = ez+ζz−ζ (for ζ ∈ T) are indeed

one-component.


Suppose then that Θ is a Blaschke product whose zeros an tend nontangentiallyto a single point ζ ∈ T. If

(2.5) infn≥1

|ζ − an+1||ζ − an|

> 0,

then Θ is one-component. So, in particular, if 0 < r < 1 and Θ is the Blaschkeproduct with zeros 1 − rn, n ≥ 1, then Θ is one-component. If condition (2.5) isnot satisfied, then usually Θ is not one-component. A detailed discussion of suchBlaschke products is given in [3], including the determination of the classes Cp(Θ)(see Subsection 3.2).

One-component inner functions can be characterized by an estimate on the H∞

norm of the reproducing kernels kΘλ . While for a general inner function Θ we have‖kΘλ ‖∞ = O(1−|λ|−1), this estimate can be improved for one-component functions:Θ is one-component if and only if there exists a constant C > 0 such that for everyλ ∈ D, we have

‖kΘλ ‖∞ ≤ C1− |Θ(λ)|1− |λ| .

2.3. Multiplication operators and their cognates. For φ ∈ L∞, we de-note by Mφf = φf the multiplication operator on L2; we have ‖Mφ‖ = ‖φ‖∞. TheToeplitz operator Tφ : H2 −→ H2 and the Hankel operator Hφ : H2 −→ H2

− =L2 �H2 are given by the formulae

Tφ = P+Mφ, Hφ = P−Mφ.

In the case where φ is analytic, Tφ is just the restriction of Mφ to H2. We haveT ∗φ = Tφ and H∗

φ = P+MφP−.It should be noted that, while the symbols of Mφ and Tφ are uniquely defined

by the operators, this is not the case with Hφ. Indeed, it is easy to check thatHφ = Hψ if and only if φ− ψ ∈ H∞. So statements about Hankel operators oftenimply only the existence of a symbol with corresponding properties.

The Hankel operators have the range and domain spaces different. It is some-times preferable to work with an operator acting on a single space. For this, weintroduce in L2 the unitary symmetry J defined by

J (f)(z) = zf(z).

We have then J (H2) = H2− and J (H2

−) = H2. Define Γφ : H2 → H2 by

(2.6) Γφ = JHφ.

Obviously properties of boundedness or compactness are the same for Hφ and Γφ.The definition of Mφ, Tφ and Hφ can be extended to the case when the symbol

φ is only in L2 instead of L∞, obtaining (possibly unbounded) densily definedoperators. Then Mφ and Tφ are bounded if and only if φ ∈ L∞ (and ‖Mφ‖ =‖Tφ‖ = ‖φ‖∞). The situation is more complicated for Hφ. Namely, Hφ is boundedif and only if there exists ψ ∈ L∞ with Hφ = Hψ, and

‖Hφ‖ = inf{‖ψ‖∞ : Hφ = Hψ}

This is known as Nehari’s Theorem; see, for instance, [24, p. 182]. Moreover (butwe will not pursue this in the sequel) an equivalent condition is P−φ ∈ BMO (and‖Hφ‖ is then a norm equivalent to ‖P−φ‖BMO).


Related results are known for compactness. The operators Mφ and Tφ arenever compact except in the trivial case φ ≡ 0. Hartman’s Theorem states that Hφ

is compact if and only if there exists ψ ∈ C(T) with Hφ = Hψ; or, equivalently,P−φ ∈ VMO. If we know that φ is bounded, then Hφ is compact if and only ifφ ∈ C(T) +H∞.

3. Carleson measures

3.1. Embedding of Hardy spaces. Let us discuss first some objects relatedto the Hardy space; we will afterwards see what analogous facts are true for thecase of model spaces.

A positive measure μ on D is called a Carleson measure if H2 ⊂ L2(μ) (suchan inclusion is automatically continuous). It is known that this is equivalent toHp ⊂ Lp(μ) for all 1 ≤ p < ∞. Carleson measures can also be characterized by ageometrical condition, as follows. For an arc I ⊂ T such that |I| < 1 we define

S(I) = {z ∈ D : 1− |I| < |z| < 1 and z/|z| ∈ I}.Then μ is a Carleson measure if and only if

(3.1) supI

μ(S(I))

|I| <∞.

Condition (3.1) is called the Carleson condition.The result can actually be extended (see [10]) to measures defined on D. Again

the characterization does not depend on p, and it amounts to the fact that μ|T isabsolutely continuous with respect to Lebesgue measure with essentially boundeddensity, while μ|D satisfies (3.1).

There is a link between Hankel operators and Carleson measures that has firstappeared in [29,39]; a comprehensive presentation can be find in [28, 1.7]. Let μbe a finite complex measure on D. Define the operator Γ[μ] on analytic polynomialsby the formula

〈Γ[μ]f, g〉 =∫D

zf(z)g(z) dμ(z).

Note that if μ is supported on T, then the matrix of Γ[μ] in the standard basis ofH2 is (μ(i+ j))i,j≥0, where μ(i) are the Fourier coefficients of μ.

Then the operator Γ[μ] is bounded whenever μ is a Carleson measure. Con-versely, if Γ[μ] is bounded, then there exists a Carleson measure ν on D such thatΓ[μ] = Γ[ν].

It is easy to see that if dμ = φdm for some φ ∈ L∞, then Γ[μ] = Γφ, where Γφ

has been defined by (2.6) and is the version of a Hankel operator acting on a singlespace.

Analogous results may be proved concerning compactness. In this case therelevant notion is that of vanishing Carleson measure, which is defined by theproperty

(3.2) lim|I|→0

μ(S(I))

|I| = 0.

Note that vanishing Carleson measures cannot have mass on the unit circle (inter-vals containing a Lebesgue point of the corresponding density would contradict thevanishing condition). Then the embedding Hp ⊂ Lp(μ) is compact if and only if μis a vanishing Carleson measure.


A similar connection exists to compactness of Hankel operators. If μ is avanishing Carleson measure on D, then Γ[μ] is compact. Conversely, if Γ[μ] iscompact, then there exists a vanishing Carleson measure ν on D such that Γ[μ] =Γ[ν].

3.2. Embedding of model spaces. Similar questions for model spaces havebeen developed starting with the papers [15,16] and [38]; however, the results inthis case are less complete. Let us introduce first some notations. For 1 ≤ p <∞,define

Cp(Θ) = {μ finite measure on T : KpΘ ↪→ Lp(|μ|) is bounded},

C+p (Θ) = {μ positive measure on T : KpΘ ↪→ Lp(μ) is bounded},

Vp(Θ) = {μ finite measure on T : KpΘ ↪→ Lp(|μ|) is compact},

V+p (Θ) = {μ positive measure on T : Kp

Θ ↪→ Lp(μ) is compact}.It is clear that Cp(Θ) and Vp(Θ) are complex vectorial subspaces of the complex

measures on the unit circle. Using the relations KΘ2 = KΘ⊕ΘKΘ and KΘ ·KΘ ⊂K1

Θ2 , it is easy to see that C2(Θ2) = C2(Θ), C1(Θ2) ⊂ C2(Θ), and V1(Θ2) ⊂ V2(Θ).It is natural to look for geometric conditions to characterize these classes.

Things are, however, more complicated, and the results are only partial. We startby fixing a number 0 < ε < 1; then the (Θ, ε)-Carleson condition asserts that

(3.3) supI

μ(S(I))

|I| <∞,

where the supremum is taken only over the intervals |I| such that S(I)∩Ω(Θ, ε) = ∅.(Remember that Ω(Θ, ε) is given by (2.4).)

It is then proved in [38] that if μ satisfies the (Θ, ε)-Carleson condition, then theembedding Kp

Θ ⊂ Lp(μ) is continuous. The converse is true if Θ is one-component ;in which case the embedding condition does not depend on p, while fulfilling of the(Θ, ε)-Carleson condition does not depend on 0 < ε < 1 (see Theorem 3.1 below).

As concerns the general case, it is shown by Aleksandrov [3] that if the converseis true for some 1 ≤ p <∞, then Θ is one-component. Also, Θ is one-component ifand only if the embedding condition does not depend on p. More precisely, the nexttheorem is proved in [3] (note that a version of this result for p ∈ (1,∞) alreadyappears in [38]).

Theorem 3.1. The following are equivalent for an inner function Θ:

(1) Θ is one-component.(2) For some 0 < p < ∞ and 0 < ε < 1, Cp(Θ) concides with the class of

measures that satisfy the (Θ, ε)-Carleson condition.(3) For all 0 < p < ∞ and 0 < ε < 1, Cp(Θ) concides with the class of

measures that satisfy the (Θ, ε)-Carleson condition.(4) The class Cp(Θ) does not depend on p ∈ (0,∞).

In particular, if Θ is one component, then so is Θ2, whence C1(Θ2) = C2(Θ2) =C2(Θ).

Note that a general characterization of C2(Θ) has recently been obtained in [22];however, the geometric content of this result is not easy to see.

The question of compactness of the embedding KpΘ ⊂ Lp(μ) in this case should

be related to a vanishing Carleson condition. In fact, there are two vanishingconditions, introduced in [14]. What is called therein the second vanishing condition


is easier to state. We say that μ satisfies the second (Θ, ε)-vanishing condition [7,14]if for each η > 0 there exists δ > 0 such that μ(S(I))/|I| < η whenever |I| < δ andS(I) ∩ Ω(Θ, ε) = ∅. The following result is then proved in [7].

Theorem 3.2. If the positive measure μ satisfies the second (Θ, ε)-vanishingcondition, then the embedding Kp

Θ ⊂ Lp(μ) is compact for 1 < p <∞.The converse is true in case Θ is one-component.

In other words, the theorem thus states that positive measures that satisfy thesecond vanishing condition are in V+

p (Θ) for all 1 < p < ∞, and the converse istrue for Θ one-component.

To discuss the case p = 1, we have to introduce what is called in [14] the firstvanishing condition. Let us call the supremum in (3.3) the (Θ, ε)-Carleson constantof μ. Define

(3.4) Hδ = {z ∈ D : dist(z, ρ(Θ)) < δ},and μδ(A) = μ(A ∩ Hδ). Then μδ are also Θ-Carleson measures, with (Θ, ε)-Carleson constants decreasing when δ decreases. We say that μ satisfies the first(Θ, ε)-vanishing condition if these Carleson constants tend to 0 when δ → 0.

It is shown in [7] that the first vanishing condition implies the second, and thatthe converse is not true: there exist measures which satisfy the second vanishingcondition but not the first.

The next theorem is proved in [14].

Theorem 3.3. If a positive measure μ satisfies the first (Θ, ε)-vanishing con-dition, then μ ∈ V+

p (Θ) for 1 ≤ p <∞.

In case μ ∈ Cp(Θ), we will denote by ιμ,p : KpΘ → Lp(|μ|) the embedding

(which is then known to be a bounded operator). Then μ ∈ Vp(Θ) means that ιμ,pis compact. We will also write ιμ instead of ιμ,2.

4. Truncated Toeplitz operators

Let Θ be an inner function and φ ∈ L2. The truncated Toeplitz operator Aφ =AΘ

φ , introduced by Sarason in [34], will be a densely defined, possibly unboundedoperator on KΘ. Its domain is KΘ ∩H∞, on which it acts by the formula

Aφf = PΘ(φf), f ∈ KΘ ∩H∞.

If Aφ thus defined extends to a bounded operator, that operator is called a TTO.The class of all TTOs on KΘ is denoted by T (Θ), and the class of all nonnegativeTTO’s on KΘ is denoted by T (Θ)+.

Although these operators are called truncated Toeplitz, they have more in com-mon with Hankel operators Hφ, or rather with their cognates Γφ, which act ona single space. As a first example of this behavior, we note that the symbol of atruncated Toeplitz operators is not unique. It is proved in [34] that

(4.1) Aφ1= Aφ2

⇐⇒ φ1 − φ2 ∈ ΘH2 +ΘH2.

Let us denote SΘ = L2� (ΘH2+ΘH2); it is called the space of standard symbols.It follows from (4.1) that every TTO has a unique standard symbol. One provesin [34, Section 3] that S is contained in KΘ +KΘ as a subspace of codimension atmost one; this last space is sometimes easier to work with.


It is often the case that the assumption Θ(0) = 0 simplifies certain calculations.For instance, in that case we have precisely S = KΘ + KΘ; we will see anotherexample in Section 7. Fortunately, there is a procedure to pass from a general innerΘ to one that has this property: it is called the Crofoot transform. For a ∈ D letΘa be given by the formula

Θa(z) =Θ(z)− a

1− aΘ(z).

If we define the Crofoot transform by

J(f) :=

√1− |a|21− aΘ

f,

then J is a unitary operator from KΘ to KΘa, and

(4.2) JT (Θ)J∗ = T (Θa).

In particular, if a = Θ(0), then Θa(0) = 0, and (4.2) allows the transfer of propertiesfrom TTOs on KΘa

to TTOs on KΘ.Especially nice properties are exhibited by TTOs which have an analytic sym-

bol φ ∈ H2 (of course, this is never a standard symbol). It is a consequence ofinterpolation theory [33] that

{AΘφ ∈ T (Θ) : φ ∈ H2} = {AΘ

z }′

(AΘz is called a compressed shift, or a model operator).One should also mentioned that other two classes of TTOs have already been

studied in different contexts. First, the classical finite Toeplitz matrices are TTOswith Θ(z) = zn written in the basis of monomials. Secondly, TTOs with Θ(z) =

ez+1z−1 correspond, after some standard transformations, to a class of operators alter-

nately called Toeplitz operators on Paley–Wiener spaces [31], or truncated Wiener–Hopf operators [11].

There is an alternate manner to introduce TTOs, related to the Carleson mea-sures in the previous section. For every μ ∈ C2(Θ) the sesquilinear form

(f, g) �→∫

fg dμ

is bounded, and therefore there exists a bounded operator AΘμ on KΘ such that

(4.3) 〈AΘμ f, g〉 =

∫fg dμ.

It is shown in [34, Theorem 9.1] that AΘμ thus defined is actually a TTO. In

fact, the converse is also true, as stated in Theorem 4.2 below. An interesting openquestion is the characterization of the measures μ for which Aμ = 0.

The definition of TTOs does not make precise the class of symbols φ ∈ L2 thatproduce bounded TTOs. A first remark is that the standard symbol of a boundedtruncated Toeplitz operator is not necessarily bounded. To give an example, con-sider an inner function Θ with Θ(0) = 0, for which there exists a singular pointζ ∈ T where Θ has an angular derivative in the sense of Caratheodory. It is shownthen in [34, Section 5] that kΘζ ⊗ kΘζ is a bounded rank one TTO with standard

symbol kΘζ + kΘζ − 1, and that this last function is unbounded.A natural question is therefore whether every bounded TTO has a bounded

symbol (such as is the case with Hankel operators). In the case of Tφ with φ


analytic, the answer is readily seen to be positive, being proved again in [33];moreover,

inf{‖ψ‖∞ : ψ ∈ H∞, AΘψ = AΘ

φ } = ‖AΘφ ‖.

The first negative answer for the general situation has been provided in [6], andthe counterexample is again given by the rank one TTO kΘζ ⊗ kΘζ . The following

result is proved in [6].

Theorem 4.1. Suppose Θ has an angular derivative in the sense of Caratheodoryin ζ ∈ T (equivalently, kΘζ ∈ L2), but kΘζ ∈ Lp for some p ∈ (2,∞). Then kΘζ ⊗ kΘζhas no bounded symbol.

A more general result has been obtain in [5], where one also makes clear therelation between measures and TTO. In particular, one characterizes the innerfunctions Θ which have the property that every bounded TTO onKΘ has a boundedsymbol.

Theorem 4.2. Suppose Θ is an inner function.

(1) For every bounded TTO A ≥ 0 there exists a positive measure μ ∈ C+2 (Θ)such that A = AΘ

μ .(2) For every bounded A ∈ T (Θ) there exists a complex measure μ ∈ C2(Θ)

such that A = AΘμ .

(3) A bounded TTO A ∈ T (Θ) admits a bounded symbol if and only if A = AΘμ

for some μ ∈ C1(Θ2).(4) Every bounded TTO on KΘ admits a bounded symbol if and only if C1(Θ2)

= C2(Θ2).

In particular, as shown by Theorem 3.1, the second condition is satisfied ifΘ is one-component (since then all classes Cp(Θ) coincide). It is still an openquestion whether Θ one-component is actually equivalent to C1(Θ2) = C2(Θ2). (Asmentioned previously, Θ is one-component if and only if Θ2 is one-component.)Such a result would be a significant strengthening of Theorem 3.1.

As a general observation, one may say that, if Θ is one-component, then TTOson KΘ have many properties analogous to those of Hankel operators. This is theclass of inner functions for which the current theory is more developed.

5. Compact operators

Surprisingly enough, the first result about compactness of TTOs dates from1970. In [1, Section 5] one introduces what are, in our terminology, TTOs withcontinuous symbol, and one proves the following theorem.

Theorem 5.1. If Θ is inner and φ is continuous on T, then AΘφ is compact if

and only if φ|ρ(Θ) = 0.

This result has been rediscovered more recently in [21]; see also [20].Thinking of Hartmann’s theorem, it seems plausible to believe that continuous

symbols play for compact TTOs the role played by bounded symbols for generalTTOs. However, as shown by Theorem 4.1, there exist inner functions Θ for whicheven rank-one operators might not have bounded symbols (not to speak about con-tinuous). So we have to consider only certain classes of inner functions, suggested bythe boundedness results in the previous section. In this sense one has the followingresult proved by Bessonov [8].


Theorem 5.2. Suppose that C1(Θ2) = C2(Θ) and A is a truncated truncatedToeplitz operator. Then the following are equivalent:

(1) A is compact.(2) A = AφΘ for some φ ∈ C(T).

In particular, this is true if Θ is one component.One can see that instead of C(T) the main role is played by ΘC(T). We give

below some ideas about the connection between these two classes.

Theorem 5.3. Suppose C2(Θ) = C1(Θ2) and m(ρ(Θ)) = 0. Then the followingare equivalent for a truncated Toeplitz operator A.

(i) A is compact.(ii) A = AΘ

φ for some φ ∈ C(T) with φ|ρ(Θ) = 0.

Proof. (ii) =⇒ (i) is proved in Theorem 5.1.Suppose now (i) is true. By Theorem 5.2 A = AΘψ for some ψ ∈ C(T). By the

Rudin–Carleson interpolation theorem (see, for instance, [18, Theorem II.12.6]),there exists a function ψ1 ∈ C(T) ∩ H∞ (that is, in the disk algebra) such thatψ|ρ(Θ) = ψ1|ρ(Θ). Then one checks easily that φ = Θ(ψ − ψ1) is continuous on T,and Aφ = AΘψ (since AΘψ1

= 0). �In particular, Theorem 5.3 applies to the case Θ one-component, since for such

functions we have C2(Θ) = C1(Θ2) and m(ρ(Θ)) = 0 [2, Theorem 6.4].We also have the following result which is contained in [8, Proposition 2.1];

here is a simpler proof.

Proposition 5.4. (i) If φ ∈ ΘC(T) + ΘH∞, then Aφ is compact.(ii) If φ ∈ C(T) +H∞, then the converse is also true.

Proof. First note that

Aφ = (ΘHΘφ −Hφ)|KΘ.(5.1)

By Hartmann’s Theorem we know that a Hankel operator with bounded symbol iscompact if and only if its symbol is in C(T)+H∞. Since C(T)+H∞ is an algebra,φ ∈ ΘC(T) + ΘH∞, that is, Θφ ∈ C(T) + H∞, implies φ ∈ C(T) + H∞. Thenapplying (5.1) proves (i).

On the other hand, if φ ∈ C(T) +H∞, again (5.1) proves (ii). �It is interesting to compare Theorem 5.2 to Proposition 5.4. Suppose that a

TTO Aφ is compact. Proposition 5.4 says that, if we know that φ ∈ C(T) +H∞,then it has actually to be in ΘC(T) +ΘH∞. So there exists ψ ∈ C(T) +H∞ suchthat φ = Θψ. This is true with no special assumption on Θ, but the symbol φ isassumed to be in a particular class.

On the other hand, suppose that Θ satisfies the assumption C2(Θ) = C1(Θ2),and again Aφ is compact. Without any a priori assumption on the symbol, applyingTheorem 5.2 yields the existence of ψ ∈ C(T)+H∞ such that Aφ = AΘψ. However,in this case we will not necessarily have φ = Θψ, but, according to (4.1), φ−Θψ ∈ΘH2 +ΘH2.

It would be interesting to give an example of a compact operator, with a symbolψ ∈ ΘC(T) + ΘH∞, that has no continuous symbol.

Since Aφ is compact if and only if A∗φ = Aφ is, any condition on the symbol

produces another one by conjugation. So one expects a definitive result to be


invariant by conjugation. This is not the case, for instance, with Proposition 5.4:by conjugation we obtain that if φ ∈ ΘC(T) + ΘH∞, then AΘ

φ is compact. Also,in Theorem 5.1 one could add a third equivalent condition, namely that A = AφΘ

for some φ ∈ C(T). From this point of view, Theorem 5.3 is more satisfactory.Naturally, if Θ is one-component would actually be equivalent to C1(Θ2) = C2(Θ)(the open question stated above), then Theorems 5.1 and 5.3 would turn out to beequivalent to a simple and symmetric statement for this class of functions.

6. Compact TTOs and embedding measures

In the present section we discuss some relations between compactness of TTOsand embedding measures. Let us first remember that a (finite) complex measureon the unit circle can be decomposed by means of nonnegative finite measures, asstated more precisely in the following lemma [32, chap. 6].

Lemma 6.1. If μ is a complex measure, one can write μ = μ1 − μ2 + iμ3 − iμ4

with 0 ≤ μj ≤ |μ| for 1 ≤ j ≤ 4.

We will also use the following simple result.

Lemma 6.2. If 0 ≤ ν1 ≤ ν2, then ν2 ∈ C+p (Θ) implies ν1 ∈ C+p (Θ), and

ν2 ∈ V+p (Θ) implies ν1 ∈ V+

p (Θ).

Proof. If 0 ≤ ν1 ≤ ν2, then we have a contractive embedding J : Lp(ν2) →Lp(ν1), and the lemma follows from the equality ιν1,p = Jιν2,p. �

The ultimate goal would be to obtain for compact TTOs statements similarto those for boundedness appearing in Theorem 4.2. But one can only obtainpartial results: measures in V2(Θ) produce compact TTOs, but the converse canbe obtained only for positive operators.

Theorem 6.3. Suppose A ∈ T (Θ).

(1) If there exists μ ∈ V2(Θ) such that A = Aμ, then A is compact.(2) If A is compact and positive, then there exists μ ∈ ν+2 (Θ) such that A =

Aμ.

Proof. (1) Take Aμ with μ ∈ V2(Θ). Writing μ = μ1 − μ2 + iμ3 − iμ4 asin Lemma 6.1, one has Aμ = Aμ1

− Aμ2+ iAμ3

− iAμ4. Since 0 ≤ μj ≤ |μ|, it

follows that μj ∈ V+2 (Θ) by Lemma 6.2. So we may suppose from the beginning

that μ ∈ V+2 (Θ).

To show that Aμ is compact, take a sequence (fn) tending weakly to 0 in KΘ,and g ∈ KΘ with ‖g‖2 = 1. Formula (4.3) can be written

〈Aμfn, g〉2 =

∫ιμ(fn)ιμ(g)dμ,

and thus

|〈Aμfn, g〉| ≤ ‖ιμ(fn)‖L2(μ)‖ιμ(g)‖L2(μ) ≤ ‖ιμ(fn)‖L2(μ)‖ιμ‖.Taking the supremum with respect to g, we obtain

‖Aμfn‖2 ≤ ‖ιμ(fn)‖L2(μ)‖ιμ‖.But fn → 0 weakly and ιμ compact imply that ‖ιμ(fn)‖L2(μ) → 0. So ‖Aμfn‖ → 0and therefore Aμ is compact.


(2) If A ≥ 0, by Theorem 4.2, there exists μ ∈ C+2 (Θ) such that A = Aμ. Wemust then show that μ ∈ V+

2 (Θ); that is, ιμ is compact.Take then a sequence fn tending weakly to 0 in KΘ; in particular, (fn) is

bounded, so we may assume ‖fn‖ ≤M for all n. Applying again formula (4.3), wehave

〈Aμfn, fn〉 =∫

ιμ(fn)ιμ(fn)dμ = ‖ιμ(fn)‖2L2(μ).

Therefore

‖ιμ(fn)‖2L2(μ) ≤ ‖Aμfn‖‖fn‖ ≤M‖Aμfn‖.Since Aμ is compact, ‖Aμfn‖ → 0. The same is then true about ‖ιμ(fn)‖2L2(μ);

thus ιμ is compact, that is, μ ∈ V+2 (Θ). �

This approach leads to an alternate proof of Theorem 5.1.

Proposition 6.4. If φ ∈ C(T) and φ|ρ(Θ) = 0, then the measure μ = φdm isin Vp(Θ) for every 1 ≤ p <∞. In particular, Aφ is compact.

Proof. Since φ ∈ L∞, the measure |μ| is an obvious Θ-Carleson measure.Now fix ε > 0. Since φ is uniformly continous on T, there exists η > 0 such that, ifζ ∈ T, dist(ζ, ρ(Θ)) < η, then |φ(ζ)| < ε. In other words, if ζ ∈ Hη, then |φ(ζ)| < ε(where Hη is defined by (3.4)).

Let δ < η and I be any arc of T. Then we have

|μ|δ(T (I)) = |μ|(T (I) ∩Hδ)

= sup

⎧⎨⎩∑i≥1

|μ(Ei)| :⋃i≥1

Ei = T (I) ∩Hδ, Ei ∩ Ej = ∅ for i = j

⎫⎬⎭Since Ei ⊂ Hδ ⊂ Hη, note that

|μ(Ei)| =∣∣∣∣∫

Ei

φ dm

∣∣∣∣≤∫Ei

|φ| dm ≤ εm(Ei).

Hence

|μ|δ(T (I)) ≤ ε|I|,which shows that the Θ-Carleson constant of |μ|δ is smaller than ε. We concludethe proof applying Theorem 3.3 and Theorem 6.3. �

The next theorem is a partial analogue of Theorem 4.2 (3).

Theorem 6.5. Suppose μ ∈ V1(Θ2). Then Aμ = AΘφ for some φ ∈ C(T).

Proof. By [8, Lemma 3.1] we know that K1zΘ2 ∩ zH1 is w*-closed when we

consider it embedded in H1 = C(T)/H10 . We define on K1

zΘ2 ∩ zH1 the linearfunctional by

(f) =

∫Θf dμ.

It is clear that is continuous, but we assert that it is also w*-continuous. Indeed,the w* topology is metrizable (since C(T) is separable), and therefore we can checkw*-continuity on sequences.


If fn → 0 w*, then, in particular, the sequence (fn) is bounded. Then, since ιμis compact, the sequence (ιμ(fn)) is compact in L1(μ), and a standard argumentsays that, in fact, ιμfn → 0 in L1(μ). Then

(fn) =

∫Θ(ιμfn) dμ→ 0.

It follows that there exists φ ∈ C(T), such that

(f) =

∫Θf dμ =

∫φf dm

for every f ∈ K1zΘ2 ∩ zH1, or, equivalently,∫

f dμ =

∫Θφf dm

for every f ∈ Θ(K1zΘ2 ∩ zH1). If g, h ∈ K2

Θ, then gh ∈ Θ(K1zΘ2 ∩ zH1) , so

〈Aμg, h〉 =∫

gh dμ =

∫Θφgh dm = 〈AΘφf, g〉,

which proves the theorem. �

In particular, it follows from Proposition 5.4 that if μ ∈ V1(Θ2) then AΘμ is

compact.

7. TTOs in other ideals

The problem of deciding when certain TTOs are in Schatten–von Neumannclasses Sp has no clear solution yet, even in the usually simple case of the Hilbert–Schmidt ideal. In [23] one gives criteria for particular cases; to convey their flavour,below is an example (Theorem 3 of [23]). Remember that Θ is called an interpo-lating Blaschke product if its zeros (zi) form an interpolation sequence, or, equiva-lently, if they satisfy the Carleson condition

infi∈N

∏j �=i

∣∣∣∣ zi − zj1− zizj

∣∣∣∣ > 0.

Theorem 7.1. Suppose Θ is an interpolating Blaschke product and φ is ananalytic function. Then:

(1) Aφ is compact if and only if φ(zi)→ 0.(2) For 1 ≤ p <∞, Aφ ∈ Sp if and only if (φ(zi)) ∈ p.

More satisfactory results are obtained in [23] in the case of Hilbert–Schmidtoperators, but even in this case an explicit equivalent condition on the symbol ishard to formulate. Let us start by assuming that Θ(0) = 0 (see the discussion ofthe Crofoot transform in Section 4); in this case the space of standard symbols S

is precisely KΘ +KΘ. We define then Φ = Θ2/z; Φ is also an inner function withΦ(0) = 0, and CΘ(KΘ +KΘ) = KΦ (remember that CΘ is given by formula (2.3)).

Let then K0Φ be the linear span (nonclosed) of the reproducing kernels kΦλ ,

λ ∈ D. It can be checked that for every λ ∈ D we have (kΘλ )2 ∈ KΦ, and therefore

the formulaD0k

Φλ = (kΘλ )

2

defines an (unbounded) linear operator D0 : K0Φ → KΦ.

The result that is proved in [23] is the following.


Theorem 7.2. With the above notations, the following assertions are true:

(1) D0 is a positive symmetric operator. Its Friedrichs selfadjoint extension(see [30, Theorem X.23]) will be denoted by D; it has a positive root D1/2.

(2) A TTO AΘφ , with φ ∈ KΘ + KΘ, is a Hilbert–Schmidt operator if and

only if CΘφ is in the domain of D1/2, and the Hilbert–Schmidt norm is‖D1/2(CΘφ)‖.

Since the square of a reproducing kernel is also a reproducing kernel, let us de-note by H2

Θ the reproducing kernel Hilbert space that has as kernels (kΘλ )2 (λ ∈ D).

It is a space of analytic functions defined on D, and it provides another characteri-zation of Hilbert–Schmidt TTOs obtained in [23].

Theorem 7.3. Define, for φ ∈ KΘ +KΘ,

(Δφ)(λ) = 〈CΘφ, (kΘλ )

2〉.

Then:

(1) Δφ is a function analytic in D, which coincides on K0Φ with D(CΘφ).

(2) An alternate formula for Δφ is

(Δφ)(λ) = (zα)′(λ)− 2Θ(λ)(zα2)′(λ),

where CΘφ = α = α1 +Θα2, with α1, α2 ∈ KΘ.(3) AΘ

φ is a Hilbert–Schmidt operator if and only if Δφ ∈ H2Θ, and the Hilbert–

Schmidt norm is ‖Δφ‖H2Θ.

The proof of these two theorems uses the theory of Hankel forms as developedin [27]. Admittedly, none of the characterizations is very explicit.

For the case of one-component functions, a conjecture is proposed in [8, 4.3]for the characterization of Schatten–von Neumann TTOs. It states essentially thata truncated Toeplitz operator is in Sp if and only if it has at least one symbol

φ in the Besov space B1/ppp (note that this would not necessarily be the standard

symbol). This last space admits several equivalent characterizations; for instance,if we define, for τ ∈ T, Δτf(z) = f(τz)− f(z), then

B1/ppp =

{f ∈ Lp :

∫T

‖Δτf‖pp|1− τ |2 dm(τ ) <∞

}.

The conjecture is suggested by the similar result in the case of Hankel operators [28,

Chapter 6]. It is true if Θ(z) = ez+1z−1 , as shown in [31].

Bessonov also proposes some alternate characterizations in terms of Clark mea-sures; we will not pursue this approach here.

8. Invertibility and Fredholmness

Invertibility and, more generally, spectrum of a TTO has been known sinceseveral decades in the case of analytic symbols. The main result here is stated inthe next theorem (see, for instance, [25, 2.5.7]). It essentially says that σ(AΘ

φ ) =

φ(s(Θ)), but we have to give a precise meaning to the quantity on the right, sinces(Θ) (as defined by (2.2)) intersects the set T, where φ ∈ H∞ is defined only almosteverywhere.


Theorem 8.1. If φ ∈ H∞, then

σ(AΘφ ) = {ζ ∈ C : inf

z∈D

(|Θ(z)|+ |φ(z)− ζ|) = 0},

{λ ∈ C : λ = φ(z), where z ∈ D, Θ(z) = 0} ⊂ σp(AΘφ ),

σe(AΘφ ) = {ζ ∈ C : lim inf

z∈D,|z|→1(|Θ(z)|+ |φ(z)− ζ| = 0}.

As noted above, the class of TTOs is invariant by conjugation, and thereforewe may obtain corresponding characterizations for coanalytic symbols. But whathappens for more general TTOs? Again a result in [1] seems to be historically thefirst one. It deals with the essential spectrum of a TTO with continous symbol.More precisely, it states that

σe(AΘφ ) = φ(ρ(Θ)).

There is a more extensive development of these ideas in [21], which, in particular,studies the C∗-algebra generated by TTOs with continuous symbols.

The above characterization of the essential spectrum is extended in [8] to sym-bols in C(T) + H∞. Since functions φ ∈ C(T) + H∞ are defined only almosteverywhere on T, one should explain the meaning of the right hand side. The fol-lowing is the precise statement of Bessonov’s result; its form is similar to that ofTheorem 8.1.

Theorem 8.2. Suppose φ ∈ C(T) +H∞. Then

σe(AΘφ ) = {ζ ∈ C : lim inf

z∈D,|z|→1(|Θ(z)|+ |φ(z)− ζ| = 0}

(for the definition of φ, see (2.1)).

It is harder to find criteria for invertibility of TTOs with nonanalytic symbols.The next part of the section uses embedding measures to obtain some partial results.We start with a statement which is essentially about bounded below TTOs.

Theorem 8.3. Let A be a (bounded) TTO, and let μ a complex measure suchthat A = Aμ.

(1) If A is bounded below, then ιμ is also bounded below, i.e. there existsC > 0 (depending only on μ and Θ) such that∫

T

|f |2dm ≤ C

∫T

|ιμ(f)|2d|μ|

for all f ∈ KΘ.(2) Suppose A ∈ T (Θ)+ and let μ ∈ C2(Θ)+ such that A = Aμ. The following

assertions are equivalent:(a) the operator A is invertible;(b) there exists C > 0 (depending only on μ and Θ) such that∫

T

|f |2dm ≤ C

∫T

|ιμ(f)|2dμ

for all f ∈ KΘ.

Proof. (1) By definition of Aμ, for all f, g ∈ KΘ, we have

|〈Aμf, g〉| =∣∣∣∣∫

T

ιμ(f)ιμ(g)dμ

∣∣∣∣ ≤ ∫T

|ιμ(f)||ιμ(g)|d|μ|.


The Cauchy–Schwarz inequality implies that

|〈Aμf, g〉| ≤ ‖ιμ(f)‖L2(|μ|)‖ιμ(g)‖L2(|μ|) ≤ ‖ιμ(f)‖L2(|μ|)‖ιμ‖‖g‖2.Then, taking the supremum over all g ∈ KΘ of unit norm, we get:

‖Aμf‖2 ≤ ‖ιμ‖‖ιμ(f)‖L2(|μ|).

Now, if Aμ is bounded below, there exists C > 0 such that

‖ιμ(f)‖L2(|μ|) ≥C

‖ιμ‖‖f‖2,

and thus ιμ is bounded below.(2) First, recall that since Aμ = A∗

μ, Aμ is invertible if and only if Aμ is boundedbelow. Thus (a) =⇒ (b) follows from part (1).

Conversely, assume that ιμ is bounded below and let C be the constant definedin (b). It remains to check that Aμ is bounded below.

For a nonzero f ∈ KΘ, we have:

‖Aμf‖2 ≥ |〈Aμf, f/‖f‖2〉| =1

‖f‖2‖ιμ(f)‖2L2(μ) ≥

1

‖f‖21

C‖f‖22 =

‖f‖2C

,

as expected. �

Volberg [37] proved that given ϕ ∈ L∞(T), and an inner function Θ, thefollowing are equivalent:

• there exist C1, C2 > 0 such that

C1‖f‖2 ≤ ‖f‖L2(|ϕ|dm) ≤ C2‖f‖2,for all f ∈ KΘ;

• there exists δ > 0 such that

|ϕ|(λ) + |Θ(λ)| ≥ δ,

for all λ ∈ D.

Volberg’s result allows the translation of the embedding conditions in Theo-rem 8.3 into concrete functional inequalities, leading to the following statement.

Theorem 8.4. Let ϕ ∈ L∞(T) and let Θ be an inner function.

(1) If AΘϕ is bounded below, then there exists δ > 0 such that

|ϕ|(λ) + |Θ(λ)| ≥ δ,

for all λ ∈ D.(2) If ϕ ≥ 0, the following assertions are equivalent:

(a) The operator AΘϕ is invertible;

(b) there exists δ > 0 such that

|ϕ|(λ) + |Θ(λ)| ≥ δ,

for all λ ∈ D.

Denote by σap(T ) the approximate point spectrum of a bounded operator T .

Corollary 8.5. Let ϕ ∈ L∞(T) and let Θ be an inner function. Then

{μ ∈ C : infλ∈D

( |ϕ− μ|(λ) + |Θ(λ)|) = 0} ⊂ σap(AΘϕ ).


Acknowledgements

The authors thank Richard Rochberg for some useful discussions. The authorswere partially supported by French-Romanian project LEA-Mathmode.

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Universite Paris Est Marne-la-Vallee 5 bd Descartes, Champs-sur-Marne, Marne-

la-Vallee Cedex 2, 77454, Frnace


Laboratoire Paul Painleve, Universite Lille 1, 59 655 Villeneuve d’Ascq Cedex,

France


Institute of Mathematics of the Romanian Academy, PO Box 1-764, Bucharest

014700, Romania.



Approximating z in the Bergman Space

Matthew Fleeman and Dmitry Khavinson

Abstract. We consider the problem of finding the best approximation toz in the Bergman Space A2(Ω). We show that this best approximation is

the derivative of the solution to the Dirichlet problem on ∂Ω with data |z|2and give examples of domains where the best approximation is a polynomial,or a rational function. Finally, we obtain the “isoperimetric sandwich” fordist(z,Ω) that yields the celebrated St. Venant inequality for torsional rigidity.

1. Introduction

Let Ω be a bounded domain in C with boundary Γ. Recall that the Bergmanspace A2(Ω) is defined by:

A2(Ω) := {f ∈ Hol(Ω) : ‖f‖2A2(Ω) =

ˆΩ

|f(z)|2dA(z) <∞}.

In [12] the authors studied the question of “how far” z is from A2(Ω) in the L2(Ω)-norm. They showed that the best approximation to z in this setting is 0 if and onlyif Ω is a disk, and that the best approximation is c

z if and only if Ω is an annuluscentered at the origin. In this paper, we examine the question of what the bestapproximation looks like in other domains. In section 2, we characterize the bestapproximation to z as the derivative of the solution to the Dirichlet problem on Γ

with data |z|2 . This shows an interesting connection between the Dirichlet prob-lem and the Bergman projection. Recently in [16], A. Legg noted independentlyanother such connection via the Khavinson-Shapiro conjecture. (Recall that thelatter conjecture states that ellipsoids are the only domains where the solution tothe Dirichlet problem with polynomial data is always a polynomial, cf. [17] and[20]. In [16, Proposition 2.1], the author showed that in the plane this happens ifand only if the Bergman projection maps polynomials to polynomials). In section3 we look at specific examples. In particular we look at domains for which the bestapproximation is a monomial Czk, some examples where the best approximation isa rational function with simple poles, as well as examples where the best approx-imation is a rational function with non-simple poles. In section 4, we prove twoisoperimetric inequalities, and obtain the St. Venant inequality.


79

80 MATTHEW FLEEMAN AND DMITRY KHAVINSON

2. Results

Unless specified otherwise, we consider domains bounded by finitely manysmooth Jordan curves. The following theorem is the high ground for the prob-lem.

Theorem 1. Let Ω be a bounded finitely connected domain. Then f(z) is the

projection of z onto A2(Ω) if and only if |z|2 = F (z) + F (z) on Γ = ∂Ω, whereF ′(z) = f(z).

(Although F can, in a multiply connected domain, be multivalued, Re(F ) can

be assumed to be single valued as a solution to the Dirichlet problem with data |z|2on Γ.)

Proof. First suppose that z − f(z) is orthogonal to A2(Ω) in L2(Ω). Then

for every z ∈ C\Ω we have thatˆΩ

(ζ − f(ζ))1

ζ − zdA(ζ) = 0 =

ˆΩ

(ζ − f(ζ))1

ζ − zdA(ζ).

Then, by Green’s Theorem, for any single valued branch of F , where F ′ = f , wehave that ˆ

Γ

(|ζ|2 − F (ζ))1

ζ − zdζ = 0.

Since F belongs to the Dirichlet space (F ′ = f ∈ A2), F also belongs to the Hardyspace H2, and therefore has well defined boundary values almost everywhere on Γ(cf. [9, p. 17] and [10, p. 88]). By the F. and M. Riesz Theorem (cf. [9, p. 41]and [10, pp. 62, 107]), vanishing of the Cauchy transform outside of Ω in the aboveformula occurs if and only if we have

|ζ|2 − F (ζ) = h(ζ)

almost everywhere on Γ, where h(ζ) is analytic in Ω and belongs to the Hardy spaceH2.

Now, since |ζ|2 is real and we have that |ζ|2 = F (ζ) + h(ζ) on Γ, then it mustbe that

F (ζ) + h(ζ) = F (ζ) + h(ζ),

which implies that h = F .Conversely, if |ζ|2 − F (ζ) = h(ζ) on Γ for some h(ζ) analytic in Ω, then we

have that for all z ∈ C\Ω,

0 =

ˆΓ

(|ζ|2 − F (ζ))1

ζ − zdζ

=

ˆΩ

(ζ − F ′(ζ))1

ζ − zdA(ζ),

and so we have that ζ − F ′(ζ) is orthogonal to A2(Ω). �

This argument is similar to that of Khavinson and Stylianopoulos in [15]. Thefollowing is an immediate corollary.

Corollary 2. The best approximation to z in A2(Ω) is a polynomial if and

only if the Dirichlet problem with data |z|2 has a real-valued polynomial solution.Similarly, the best approximation to z in A2(Ω) is a rational function if and only

APPROXIMATING z IN THE BERGMAN SPACE 81

if the Dirichlet problem with data |z|2 has a solution which is the sum of a ratio-nal function and a finite linear combination of logarithmic potentials of real pointcharges located in the complement of Ω.

The following theorem, loosely speaking, shows that increasing the connectivityof the domain essentially improves the approximation.

Theorem 3. Let Ω be a finitely connected domain and let f(z) be the bestapproximation to z in A2(Ω). Then f must have at least one singularity in everybounded component of the complement.

Proof. Suppose ∂Ω = Γ = ∪ni=1Γi where Γi is a Jordan curve for each i. By

Theorem 1, we must have that |z|2 − 2ReF = 0 on Γ where F ′ = f . Suppose thatthere is a bounded component K of the complement of Ω such that f is analyticin G := Ω ∪ K. Without loss of generality we will assume ∂G = ∪n−1

i=1 Γi. Then|z|2 − 2ReF is subharmonic in G and vanishes on ∂G. However since |z|2 − 2ReFcannot be constant in G, it must be that |z|2 − 2ReF < 0 in G. In particular itcannot vanish on Γn. �

The following noteworthy corollary is now immediate.

Corollary 4. If Ω is a finitely connected domain, and the best approximationto z is a polynomial, then Ω must be simply connected and ∂Ω is algebraic.

The converse to Corollary 4 is false. In Section 3, we will give an example of asimply connected domain where the best approximation to z is a rational function.Corollary 4 implies that if the best approximation to z is a polynomial then theboundary of Ω, Γ = ∂Ω, possesses the Schwarz function (cf. [21, p. 3]). Recall thatthe Schwarz function S(z) is the function, analytic in a tubular neighborhood ofΓ, which satisfies the condition that S(z) = z for all z ∈ Γ. There is a connectionbetween the best approximation to z in A2(Ω) and the Schwarz function of Γ. Werecord this connection in the following proposition.

Proposition 5. If Ω is a simply connected domain, and if the best approxima-tion to z is a polynomial of degree at least 1, then the Schwarz function of Γ = ∂Ωcannot be meromorphic in Ω. Further, when the best approximation is a polynomialthe Schwarz function of the corresponding domain must have algebraic singularitiesand no finite poles unless Ω is a disk.

Proof. Suppose that S(z) is the Schwarz function of Γ = ∂Ω and p(z), apolynomial of degree n − 1, is the best approximation to z in A2(Ω) with anti-

derivative P (z). By Theorem 1, zS(z) = P (z) + P (z) = P (z) + P#(S(z)) on Γ,

where P#(z) = P (z). If S has a pole of order k at some z0 = 0, then zS(z) has apole of order k at z0 while P#(S(z)) has a pole of order nk at z0. Thus n ≤ 1. Ifz0 = 0, and k ≥ 2, then the same argument applies. If z0 = 0 and k = 1, then p isconstant and Γ is a circle. Since S is meromorphic in Ω if and only if the conformalmap ϕ : D→ Ω is a rational function, this shows that if Ω is a quadrature domainwhich is not a disk, then the best approximation to z cannot be a polynomial (cf.[21, pp.17-19] for a quick background on quadrature domains). �

We now look at some examples illustrating the above results.


3. Examples

The following examples were generated using Maple by plotting the boundary

curve |z|2 − 1 = Const Re(F (z)) where, by Theorem 1, f(z) = F ′(z)2 is the best ap-

proximation to z in A2(Ω), and Re(F (z)) is the real part of F (z). Since F is uniqueup to a constant of integration, all such examples will be similar perturbations ofa disk.

Note in the next few examples with best approximation Czk, the associateddomains have the k + 1 fold symmetry inherited from the k fold symmetry of thebest approximation.

Figure 3.1. Here, the best approximation to z is3z2

10.


5.


(Note that by the domain we mean everywhere the bounded domain.)


14.

The following example shows that the best approximation may be a rationalfunction even when the domain is simply connected. Thus while Corollary 4 guar-antees that Ω is simply connected whenever the best approximation to z is an entirefunction, the converse is not true.

Figure 3.4. Here, the best approximation to z is f(z) =1

3z+

1

5(z − 12 )

.


The constant(s) involved also play a strong role in the shape, and even connec-tivity of the domain, as the following pictures shows.

Figure 3.5. Here, the best approximation to z is f(z) =1

7z+

1

10(z − 12 )

.

Note that in Figure 3.5, the best approximation has the same poles as the bestapproximation for the domain in Figure 3.4. Yet the domain in Figure 3.4 is simplyconnected, while the domain in Figure 3.5 is not.

Figure 3.6. Here, the best approximation to z is

f(z) = −3z2 − 2( 14 −

13 i)z −

18 + 1

12 i

40(z − 12 )

2(z − i3 )

2(z + 14 )

2.


In Figure 3.6, the domain is multiply connected with three holes.


f(z) = −3z2 − 2( 14 −

13 i)z −

18 + 1

12 i

10(z − 12 )

2(z − i3 )

2(z + 14 )

2.

In Figure 3.7, the best approximation has the same poles as the best approxi-mation in Figure 3.6, but the resulting domain has only two holes.


f(z) = −3z2 − 2( 14 −

13 i)z −

18 + 1

12 i

8(z − 12 )

2(z − i3 )

2(z + 14 )

2.


Note that we actually have here two simply connected domains where the bestapproximation to z in both domains is

f(z) = −3z2 − 2( 14 −

13 i)z −

18 + 1

12 i

8(z − 12 )

2(z − i3 )

2(z + 14 )

2.

(It should be noted that in all of the above examples, the poles lie outside of Ω.)As the order of the pole of the best approximation increases we see k − 1

symmetric loops separating the pole from the domain. (Here k is the order of thepole of the best approximation.)

Figure 3.9. Here, the best approximation to z is f(z) = −310z7 .

(It should be noted that the loops do not pass through 0. So 0 does not belongto Ω!)

4. Bergman Analytic Content

In [12] the authors expanded the notion of analytic content,

λ(Ω) := inff∈H∞(Ω)

‖z − f‖∞ ,

defined in [6] and [13], to Bergman and Smirnov spaces context. The following“isoperimetric sandwich” goes back to [13]:

2A(Ω)

Per(Ω)≤ λ(Ω) ≤

√A(Ω)

π,

where A(Ω) is the area of Ω, and Per(Ω) is the perimeter of its boundary. Herethe upper bound is due to Alexander (cf. [2]), and the lower bound is due to D.Khavinson (cf. [6], [11], and [13]).

Following [12], we define λA2(Ω) := inff∈A2(Ω) ‖z − f‖2.


Theorem 6. If Ω is a simply connected domain with a piecewise smooth bound-ary, then √

ρ(Ω) ≤ λA2(Ω) ≤ Area(Ω)√2π

,

where ρ(Ω) is the torsional rigidity of Ω (cf. [19, pg. 24]).

Proof. To see the lower bound, we note that by duality (cf. [9, p. 130])

(4.1) λA2(Ω) := inff∈A2(Ω)

‖z − f‖2 = supg∈(A2(Ω))⊥

∣∣∣∣ 1

‖g‖2

ˆΩ

zgdA(z)

∣∣∣∣ .By Khavin’s lemma cf., e.g., [5], [12] and [21, p. 26], we have that

(A2(Ω))⊥ :=

{∂u

∂z| u ∈W 1,2

0 (Ω)

},

where W 1,20 (Ω) is the standard Sobolev space of functions with square-integrable

gradients and vanishing boundary values. Thus, integrating by parts, (4.1) can bewritten as

λA2(Ω) = supu∈W 1,2

0 (Ω)

1∥∥∂u∂z

∥∥2

∣∣∣∣ˆΩ

udA(z)

∣∣∣∣ .Any particular choice of u(z) will thus yield a lower bound. Choose u(z) to be thestress function satisfying {

Δu = −2u|∂Ω = 0

(cf. [5] and [19, p. 24]). Then, since u(z) is real-valued, we have that∥∥∂u

∂z

∥∥2=

12 ‖∇u‖2 and

1∥∥∂u∂z

∥∥2

∣∣∣∣ˆΩ

udA(z)

∣∣∣∣ = 2∣∣´

ΩudA(z)

∣∣‖∇u‖L2(Ω)

=√ρ(Ω),

(cf. [5] and [18]). Thus,

(4.2) λA2(Ω) ≥√ρ(Ω).

To prove the upper bound, observe that

λ2A2(Ω) = ‖z‖2 − ‖P (z)‖2 ,

where P is the Bergman projection. Let Tz be the Toeplitz operator acting on A2(Ω)with symbol ϕ(z) = z, and let [T ∗

z , Tz] = T ∗z Tz − TzT

∗z be the self-commutator of

Tz. In [18], it was proved that

‖[T ∗z , Tz]‖ = sup

g∈A21(Ω)

(‖zg‖2 − ‖P (zg)‖2) ≤ Area(Ω)

2π,

where A21(Ω) = {g ∈ A2(Ω) : ‖g‖2 = 1}. Taking g = 1√

Area(Ω)yields

1

Area(Ω)(‖z‖2 − ‖P (z)‖2) ≤ Area(Ω)

2π,

and the upper bound follows. �

The celebrated St. Venant inequality (cf. [19, p. 121]) follows immediately.


Corollary 7. Let Ω be a simply connected domain. Then

ρ(Ω) ≤ Area2(Ω)

2π.

5. Concluding Remarks

Recall that for all u ∈W 1,20 (Ω), we may write

(5.1)

∣∣∣∣ˆΩ

u(z)dA(z)

∣∣∣∣ = ∣∣∣∣ˆΩ

−1π

ˆΩ

∂u

∂ζ

1

ζ − zdA(ζ)dA(z)

∣∣∣∣ .Applying Fubini’s Theorem and the Cauchy-Schwartz inequality, we find that

(5.2)

∣∣∣∣ˆΩ

u(z)dA(z)

∣∣∣∣ ≤ ∥∥∥∥∂u∂z∥∥∥∥2

∥∥∥∥ 1πˆΩ

dA(z)

z − ζ

∥∥∥∥2

.

In [7] and [8], (also cf. [3]) it was proved that the Cauchy integral operator C :L2(Ω)→ L2(Ω), defined by

Cf(z) =−1π

ˆΩ

f(ζ)

ζ − zdA(ζ),

has norm 2√Λ1

whenever Ω is a simply connected domain with a piecewise smooth

boundary, and Λ1 is the smallest positive eigenvalue of the Dirichlet Laplacian,{−Δu = Λu

u|∂Ω = 0.

Further, by the Faber-Krahn inequality, cf. [19, pp. 18, 98] and [4, p. 104], we havethat

2√Λ1

≤ 2

j0

√Area(Ω)

π,

where j0 is the smallest positive zero of the Bessel function J0(x) =∑∞

k=0(−1)k

(k!)2 (xk )2k.

Combining the above inequality with (5.2) we obtain

(5.3)1∥∥∂u

∂z

∥∥2

∣∣∣∣ˆΩ

udA(z)

∣∣∣∣ ≤ 2

j0

Area(Ω)√π

.

This together with (4.2) and (5.2), yields an isoperimetric inequality:

ρ(Ω) ≤ 4Area2(Ω)

j20π.

However, this is a coarser upper bound than that found above since 2j0≥ 1√

2. Since

this upper bound depends entirely on∥∥∥ 1π

´Ω

dA(z)z−ζ

∥∥∥2, and since in the case when Ω

is a disk D we find that∥∥∥ 1π

´D

dA(z)z−ζ

∥∥∥2= Area(D)√

2π, we conjecture, in the spirit of the

Ahlfors-Beurling inequality (cf. [1] and [11]), that∥∥∥∥ 1πˆΩ

dA(z)

z − ζ

∥∥∥∥2

≤ Area(Ω)√2π

.

If true, this would provide an alternate proof to the upper bound for Bergmananalytic content, as well as a more direct proof of the St. Venant inequality.

One is tempted to ask if any connection can be made between “nice” bestapproximations and the order of algebraic singularities of the Schwarz function. For


example when Ω is an ellipse, the Schwarz function has square root singularities atthe foci, and the best approximation to z is a linear function.

We would also like to find bounds on constants C which guarantee that thesolution to the equation |z|2 − 1 = C(zn + zn) is a curve which bounds a Jordandomain. This seems to depend on n.

It would also be interesting to examine similar questions for the Bergman spaces

Ap(Ω) when p = 2, as well as similar questions for the best approximation of |z|2 inL2h(Ω), the closed subspace of functions harmonic in Ω and square integrable with

respect to area. However, it’s not clear what the analog of Theorem 1 would be inthis case. Mimicking the proof of Theorem 1 runs aground quickly.

Acknowledgment

The final draft of this paper was produced at the 2015 conference “Complete-ness problems, Carleson measures and spaces of analytic functions” at the Mittag-Leffler institute. The authors gratefully acknowledge the support and the congenialatmosphere at the Mittag-Leffler institute. We would also like to thank Jan-FredrickOlsen for kindly pointing out to us his results that led to the proof of Theorem 6.

References

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[2] H. Alexander, Projections of polynomial hulls, J. Functional Analysis 13 (1973), 13–19.MR0338444

[3] J. M. Anderson, D. Khavinson, and V. Lomonosov, Spectral properties of some integral oper-ators arising in potential theory, Quart. J. Math. Oxford Ser. (2) 43 (1992), no. 172, 387–407,DOI 10.1093/qmathj/43.4.387. MR1188382

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[7] Milutin R. Dostanic, The properties of the Cauchy transform on a bounded domain, J. Op-erator Theory 36 (1996), no. 2, 233–247. MR1432117

[8] Milutin R. Dostanic, Norm estimate of the Cauchy transform on Lp(Ω), Integral EquationsOperator Theory 52 (2005), no. 4, 465–475, DOI 10.1007/s00020-002-1290-9. MR2184599


[10] Stephen D. Fisher, Function theory on planar domains, Pure and Applied Mathematics (NewYork), John Wiley & Sons, Inc., New York, 1983. A second course in complex analysis; AWiley-Interscience Publication. MR694693

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[12] Zdenka Guadarrama and Dmitry Khavinson, Approximating z in Hardy and Bergman norms,Banach spaces of analytic functions, Contemp. Math., vol. 454, Amer. Math. Soc., Providence,RI, 2008, pp. 43–61, DOI 10.1090/conm/454/08826. MR2408234

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[14] Dmitry Khavinson and Harold S. Shapiro, Dirichlet’s problem when the data is an entirefunction, Bull. London Math. Soc. 24 (1992), no. 5, 456–468, DOI 10.1112/blms/24.5.456.MR1173942


[15] Dmitry Khavinson and Nikos Stylianopoulos, Recurrence relations for orthogonal polynomi-als and algebraicity of solutions of the Dirichlet problem, Around the research of VladimirMaz’ya. II, Int. Math. Ser. (N. Y.), vol. 12, Springer, New York, 2010, pp. 219–228, DOI10.1007/978-1-4419-1343-2 9. MR2676175

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Department of Mathematics, Baylor University, One Bear Place #97328, Waco,

Texas, 76798

E-mail address: Matthew [email protected]

Department of Mathematics & Statistics, 4202 East Fowler Ave, CMC342, Tampa,

Florida, 33620



Real complex functions

Stephan Ramon Garcia, Javad Mashreghi, and William T. Ross

Abstract. We survey a few classes of analytic functions on the disk that havereal boundary values almost everywhere on the unit circle. We explore some oftheir properties, various decompositions, and some connections these functionsmake to operator theory.

1. Introduction

In this survey paper we explore certain classes of analytic functions on the openunit disk D that have real non-tangential limiting values almost everywhere on theunit circle T. These classes enjoy some remarkable analytic, algebraic, and struc-tural properties that connect to various problems in operator theory. In particular,these classes can be used to describe the kernel of a Toeplitz operator on the Hardyspace H2; to give an alternate description of the pseudocontinuable functions on H2

(alternatively the non-cyclic vectors for the backward shift operator); to define aclass of unbounded symmetric Toeplitz operators on H2; and to define an analogueof the classical Riesz projection operator for the Hardy spaces Hp when 0 < p < 1.

Much of this material originates in the papers [14–16], which, in turn, stemfrom seminal work of Aleksandrov [3, 4] and Helson [21, 22]. We do, however,provide many new examples and a few novel results not discussed in the worksabove. We also endeavor to make this survey accessible and thus include as manyproofs as reasonable. We hope the reader will be able to follow along and eventuallymake their own contributions.

2. Function spaces

In this section we review a few definitions and basic results needed for thissurvey. The details and proofs can be found in the well-known texts [11,17].

2.1. Lebesgue spaces. Let D denote the open unit disk in the complex planeC and let T denote the unit circle. We let m denote normalized Lebesgue measure

2010 Mathematics Subject Classification. Primary 30D55; Secondary 47A15.First author partially supported by National Science Foundation Grant DMS-1265973.Second author partially supported by NSREC.


91

92 STEPHAN RAMON GARCIA, JAVAD MASHREGHI, AND WILLIAM T. ROSS

on T and for 0 < p < ∞, we let Lp := Lp(T, dm) denote the space of Lebesguemeasurable functions on T for which

‖f‖p:=(∫

T

|f |p dm)1/p

<∞.

When 0 < p < 1, the preceding does not define a norm (the Triangle Inequalityis violated) although d(f, g) = ‖f − g‖pp defines a translation invariant metric withrespect to which Lp is complete. When 1 � p < ∞, the norm ‖·‖p induces a well-known Banach space structure on Lp. When p = 2, L2 is a Hilbert space equippedwith the standard inner product

〈f, g〉 =∫T

fg dm

and orthonormal basis {ζn : n ∈ Z}. When p = ∞, L∞ will denote the Banachalgebra of essentially bounded functions on T endowed with the essential supremumnorm ‖f‖∞.

If f ∈ L1, then the function Pf defined on D by

(2.1) (Pf)(z) =

∫T

f(ζ)Pz(ζ) dm(ζ),

denotes the Poisson extension of f to D, where

Pz(ζ) = Re

(ζ + z

ζ − z

), ζ ∈ T, z ∈ D.

The function Pf is harmonic on D and a theorem of Fatou says that

limr→1−

(Pf)(rξ) = f(ξ) a.e. ξ ∈ T.

We also let

(2.2) (Qf)(z) =

∫T

f(ζ)Qz(ζ) dm(ζ),

denote the conjugate Poisson extension of f , where

Qz(ζ) = Im

(ζ + z

ζ − z

), ζ ∈ T, z ∈ D.

The function Qf is also harmonic on D and

(2.3) f(ξ) := limr→1−

(Qf)(rξ)

exists for almost every ξ ∈ T, though the proof is more involved than for Pf .The function Qf is the harmonic conjugate of Pf . One often thinks in terms of

boundary functions and says that f is the harmonic conjugate of f . If f has Fourierseries

f ∼∑n∈Z

f(n)ζn, f(n) =

∫T

f(ζ)ζndm(ζ),

then the conjugate function f has Fourier series

(2.4) f ∼ −i∑n�=0

sgn(n)f(n)ζn.

A well-known theorem of M. Riesz ensures that if 1 < p < ∞ and f ∈ Lp then

f ∈ Lp. This is known to fail when p = 1 and p =∞.

REAL COMPLEX FUNCTIONS 93

2.2. Hardy spaces. For 0 < p <∞, the Hardy space Hp is the set of analyticfunctions f on D for which

‖ f ‖p:=(

sup0<r<1

∫T

|f(rζ)|p dm(ζ)

)1/p

<∞.

When 1 � p <∞, Hp is a separable Banach space while when 0 < p < 1, d(f, g) =‖f−g‖pp defines a translation invariant metric with respect to which Hp is completeand separable. We let H∞ denote the Banach algebra of all bounded analyticfunctions on D endowed with the norm

‖f‖∞= supz∈D

|f(z)|.

The Hardy spaces are nested in the sense that

Hq ⊆ Hp ⇐⇒ p � q.

For each f ∈ Hp with 0 < p <∞, the limit

f(ξ) := limr→1−

f(rξ)

exists for almost every ξ ∈ T and

(2.5) sup0<r<1

∫T

|f(rζ)|p dm(ζ) = limr→1−

∫T

|f(rζ)|p dm(ζ) =

∫T

|f |p dm.

Via these radial boundary values, Hp can be identified with a closed subspace ofLp. In fact, for p � 1,

Hp = {f ∈ Lp : f(n) = 0, ∀n � −1}.

By a theorem of M. Riesz, the integral operator

(2.6) f �→∫T

f(ζ)

1− ζzdm(ζ), f ∈ Lp,

maps Lp continuously onto Hp when 1 < p < ∞. In terms of Fourier series, thisRiesz projection is equivalently defined by∑

n∈Z

f(n)ζn �→∑n�0

f(n)ζn.

The Riesz projection does not define a bounded operator from L1 to H1 nor abounded operator from L∞ to H∞. We will revisit a version of this “Riesz projec-tion” in (2.6) later on for subspaces of Lp when 0 < p < 1.

2.3. The canonical factorization. Each f ∈ Hp can be factored as f = IF ,where I ∈ H∞ is an inner function and F ∈ Hp is an outer function. A generalouter function is of the form

F (z) = eiγ exp

(∫T

ζ + z

ζ − zlogw(ζ)dm(ζ)

),

where γ ∈ R, w � 0, and logw ∈ L1. When F ∈ Hp and outer, we have log|F |∈ L1,|F |∈ Lp, and w = log|F |. From (2.1) and (2.3), the radial boundary function Fbecomes

(2.7) F = exp(w + iw + iγ) a.e. on T,

which will be important later on. Specific examples of outer functions include: anyzero free function that is analytic in a neighborhood of the closed unit disk; f ∈ H1


with Re f > 0; and, in particular, functions of the form 1+ g where g is an analyticfunction with g(D) ⊆ D.

The inner factor I is a bounded analytic function on D with unimodular bound-ary values almost everywhere on T (the definition of an inner function) and can befactored further as

(2.8) I = BSμ.

In the above,

B(z) = ξzN∏n�1

an|an|

an − z

1− anz,

is a Blaschke product with zeros at the origin (of multiplicity N) and at {an}n�1 ⊆D \ {0} (repeated according to multiplicity) with∑

n�1

(1− |an|) <∞

(the Blaschke condition); ξ is a unimodular constant; and

Sμ(z) = η exp

(−∫T

ζ + z

ζ − zdμ(ζ)

),

called the singular inner factor, where μ is a finite positive measure on T withμ ⊥ m and η is a unimodular constant. Up to unimodular constants, the factors inthe canonical factorization are unique.

2.4. Smirnov class. If O(D) denotes the set of all analytic functions on D,we define the following sub-classes of analytic functions:

N =

{f

g: f ∈ H∞, g ∈ H∞ \ {0}

},

N = N ∩O(D),

N+ =

{f

g: f, g ∈ H∞, g outer

}.

The class N consists of the meromorphic functions of bounded type, N is called theNevanlinna class, and N+ is called the Smirnov class. Note that

N+ ⊆ N ⊆ N

and a standard theorem shows that⋃p>0

Hp ⊆ N+.

Functions in N have radial boundary values a.e. on T. Furthermore, the radialboundary function of an element of N \ {0} is log-integrable. As a consequence ofthe canonical factorization theorem, each f ∈ N \ {0} can be written as

(2.9) f = ξu1

u2F,

where ξ is a unimodular constant, u1, u2 are inner, and F is outer. If f ∈ N , thenu2 is a singular inner function. If f ∈ N+, we simply have

f = IF,


where I is inner and F is outer. In particular, all inner functions and all outerfunctions belong to N+. Focusing on N+, the quantity

(2.10) ρ(f, g) =

∫T

log(1 + |f − g|)dm

defines a metric on N+ which makes N+ a complete topological algebra.

Before leaving this subsection, we want to state a theorem of Smirnov whichsays that

(2.11) f ∈ N+ and f ∈ Lp =⇒ f ∈ Hp.

2.5. Classes of real complex functions. We now arrive at the main focusof this survey: analytic functions on D that have real boundary values a.e. on T. Weintroduce several classes of such functions and then focus on each particular classin a separate section. To get started, we define the following classes of functions:

R = {f ∈ N : f is real valued a.e. on T},R+ = R ∩N+,

RO = {f ∈ R+ : f is outer},Rp = R+ ∩Hp.

The class R is the real Nevanlinna class, R+ the real Smirnov class, RO the realouter functions, and Rp the real Hp functions. We will give plenty of examples ofthese “real” functions below. As one of the simplest examples, consider

f(z) = i1 + z

1− z∈ N+.

By direct computation, one shows that whenever θ = π/2 or 3π/2,

f(eiθ) = cot(θ/2) ∈ R

and thus f ∈ R+. In fact, f ∈ RO ∩ Rp for all 0 < p < 1.

3. Elementary observations

There are a number elementary observations that can be made about realSmirnov functions. Most of these involve standard results about Poisson inte-grals, linear fractional transformations, and inner-outer factorization in N+. Theseresults, however simple, set the stage for the deeper results that are to follow.

3.1. Helson’s representation. The following theorem of Helson [22] (seealso [21]) provides a concrete description of several classes of real Smirnov func-tions. Unfortunately, it is difficult to use in practice since it involves sums anddifferences of inner functions. For instance, it is difficult to determine the inner-outer factorization of a sum or difference of inner functions.

Theorem 3.1 (Helson). Let f ∈ N.

(a) If f ∈ R, then there are relatively prime inner functions ψ1 and ψ2 so that

(3.2) f = iψ1 + ψ2

ψ1 − ψ2.

Up to a common unimodular constant factor, the inner functions ψ1 andψ2 in (3.2) are uniquely determined by f .


(b) If f ∈ R+, then there are relatively prime inner functions ψ1 and ψ2 sothat ψ1 − ψ2 is outer and f is of the form (3.2).

(c) If f ∈ RO, then there are relatively prime inner functions ψ1 and ψ2 sothat ψ2

1 − ψ22 is outer and f is of the form (3.2).

Proof. (a) Suppose that f ∈ R. Observe that the linear fractional transfor-mation

z �→ z − i

z + i

maps R to T \ {1}. It follows thatf − i

f + i∈ N

and this function is unimodular a.e. on T. Then (2.9) ensures that there are rela-tively prime inner functions ψ1, ψ2, determined up to a common unimodular con-stant factor, so that

f − i

f + i=

ψ2

ψ1, ψ1 = ψ2.

After a little algebra, we obtain (3.2).

(b) Since R+ ⊆ R, (a) says that each f ∈ R+ enjoys a representation of the form(3.2), in which ψ1 and ψ2 are relatively prime inner functions. Suppose that u isan inner factor of the denominator ψ1−ψ2. Then, since f ∈ R+, u must also be aninner factor of the numerator ψ1+ψ2. This means that u is a common inner factorof both ψ1 and ψ2 (i.e., u must be a unimodular constant). We conclude that thefunction ψ1 − ψ2 has no non-constant inner factor, and is thus outer.

(c) Proceeding as in (b), we see that ψ1 + ψ2 is also outer. Thus

ψ21 − ψ2

2 = (ψ1 + ψ2)(ψ1 − ψ2)

is outer as well. �

Observe that the converses of statements (a), (b), and (c) trivially hold.

Example 3.3. Let

ψ1 =z + 1

2

1 + 12z

and ψ1 =z − 1

2

1− 12z

.

Then a short computation confirms that

(3.4) f = iψ1 + ψ2

ψ1 − ψ2=

3iz

2− 2z2.

If z = eiθ, then

f(eiθ) = −3

4csc θ,

so f maps D onto the complement of the rays (−∞,− 34 ] and [ 34 ,+∞). This is

illustrated in Figure 1.


r = 0.65 r = 0.75

r = 0.95 r = 0.999

Figure 1. Images of the disk |z|� r under the function (3.4) for severalvalues of r.

3.2. Koebe inner functions. We ultimately seek to replace the Helson rep-resentation (Theorem 3.1) with a more practical description of real Smirnov func-tions. The first step is to reduce the consideration of functions in R+ to the studyof real outer functions (i.e., RO). To this end, we require the following definition.

Definition 3.5. A Koebe inner function is a function of the form K(ϕ), whereϕ is an inner function,

K(z) = −4k(z),and

(3.6) k(z) =z

(1− z)2

is the Koebe function.

Recall that the Koebe function is a univalent map from D onto the complementof the half line (−∞,− 1

4 ] in C [12]. Thus K maps D onto the complement of thehalf line [1,∞) in C. In particular, K � 1 a.e. on T (Figures 2 and 3). The


i

−i

1−1

K(z)

∞1 2

Figure 2. Illustration of the function K(z) = −4k(z). It is a univalentmap from D onto the complement of the half line [1,∞) on the real axis.

following theorem provides an analogue of the canonical inner-outer factorizationthat is more suitable for real Smirnov functions.

Theorem 3.7. If f ∈ R+ (resp., Rp), then f = KfRf , where Kf is Koebeinner and Rf ∈ RO (resp., Rp). Moreover,

(a) |Rf |� |f | a.e. on T;

(b) f and Rf have the same sign a.e. on T.

Proof. Let f ∈ R+ (resp., Rp) with inner factor If and outer factor F . With-out loss of generality, we assume that If = 1. Otherwise, we may take If = i andreplace F by −iF (see the comment after the proof). Then

f =−4If

(1− If )2· (1− If )

2

−4 F = KfRf ,

where Kf = −4k(If ) is Koebe inner and

Rf = −1

4(1− If )

2F

is outer. Since Kf � 1 a.e. on T, the outer function Rf = f/Kf is real a.e. on T,so it belongs to RO. Moreover, |Rf |= |f/Kf |� |f | a.e. on T, so Rf ∈ Rp wheneverf ∈ Hp. Since f/Rf = Kf � 1 a.e. on T, the functions f and Rf have the samesign a.e. on T. �

In what sense is the factorization f = KfRf in Theorem 3.7 unique? Observethat the inner factor of f is hidden in Kf (they have the same inner factor up tounimodular constants). Modulo this constant, Kf , and hence Rf , is unique.

3.3. Growth restrictions. The following theorem tells us that Rp is of in-terest only when 0 < p < 1.

Theorem 3.8. If p � 1, then Rp = R.


r = 0.5 r = 0.7

r = 0.75 r = 0.800

Figure 3. Images of the disk |z|� r under the Koebe inner functionfunction K(ϕ), where ϕ(z) = exp[(z + 1)/(z − 1)], for several values ofr ∈ (0, 1). The image of D under K(ϕ) is the complement of {0}∪[1,∞).This sequence of images suggests how K(ϕ) “wraps” D around 0 withinfinite multiplicity. The function is univalent on |z|� r for small r; asr → 1−, the function K(ϕ) pulls D through the gap (0, 1) and beginswrapping it around 0.

Proof. When 1 � p <∞, we have Rp ⊆ R1 and so it suffices to prove R1 = R.If f ∈ R1 then f ∈ H1 and so g = if can be recovered from the analytic completionof its Poisson integral [11, Thm. 3.1 & p. 4], i.e.,

g(z) = iγ +

∫T

ζ + z

ζ − zRe g(ζ) dm(ζ)

for some real constant γ. The integral in the above expression is identically zerosince Re g = 0. Thus f = γ is a constant function. �

Example 3.9. There are many examples of functions in O(D) that have realnon-tangential boundary values a.e. on T, but which do not belong to N (that is,they are not of bounded type). Indeed, if f ∈ R+ (and non-constant), then ef

also has real (non-tangential) boundary values a.e. on T. However, ef ∈ R+ since


otherwise, Riesz’s theorem (log integrability of Nevanlinna functions on T) wouldimply that

log|ef |= Re f = f ∈ L1

and hence, by Smirnov’s theorem (2.11), f ∈ H1. Theorem 3.8 now ensures that fis constant, a contradiction. As an amusing consequence, we note that if f ∈ R+ isnon-constant, then

ef − i

ef + i

is meromorphic on D, unimodular a.e. on T, but not expressible as the quotient oftwo inner functions.

As mentioned earlier, if ϕ is inner, then 1 − ϕ is outer. As a result, the innerfactor of K(ϕ) is ϕ. Since the function z �→ (1− z)−1 belongs to Hp for 0 < p < 1[11, p. 13], the Littlewood Subordination Principle [11, p.10] ensures that

(3.10) (1− ϕ)−1 ∈ Hp ∀p ∈ (0, 1).

Thus, a non-constant Koebe inner function belongs to Rp for all 0 < p < 12 . This

exponent is sharp, as we will see in a moment. The following result is due to Helsonand Sarason [23] and to Neuwirth and Newman [26].

Theorem 3.11. If f ∈ R12 and f � 0 a.e. on T, then f is constant.

Proof. Suppose that f ∈ R12 and f � 0 a.e. on T. Theorem 3.7 says that

f = KfRf , in which Kf is Koebe inner and Rf ∈ R12 is outer and non-negative

a.e. on T. Then R12

f is outer and belongs to R1, so it is constant (Theorem 3.8).

Consequently, we may assume that f = K(ϕ) ∈ R12 is a Koebe inner function and

thus is non-negative a.e. on T. Observe that(i1 + ϕ

1− ϕ

)2

=−4ϕ− (1− ϕ)2

(1− ϕ)2= K(ϕ)− 1 ∈ R

12

is non-negative a.e. on T. Consequently, i(1 + ϕ)/(1 − ϕ) belongs to R1 so it isconstant (Theorem 3.8). Thus ϕ is constant as well. �

The proof above yields this interesting corollary.

Corollary 3.12. Any Koebe inner function belonging to R12 must be constant.

If f ∈ R+ is non-negative a.e. on T, it is not necessarily the case that f is thesquare of a function in R+. For instance, −4 times the Koebe function (3.6) is acounterexample: It has a zero of order 1 at z = 0 and hence cannot be the squareof any analytic function. On the other hand, the following theorem asserts that areal Smirnov function that is non-negative a.e. on T is the sum of two squares ofreal outer functions.

Theorem 3.13. If f ∈ R+ (resp., Rp) and f � 0 a.e. on T, then f = g21 + g22,

where g1, g2 ∈ RO (resp., RO ∩Hp2 ).


Proof. Suppose that f = IfF ∈ R+ and f � 0 a.e. on T, where If is innerand F is outer. Then f can be written in two different ways as

f =4If

(1 + If )2︸︷︷︸K(−If )

· (1 + If )2F

4=

−4If(1− If )2︸︷︷︸

K(If )

· (1− If )2F

−4 .

Note that K(−If ) and K(If ) are non-negative a.e. on T. It follows that

1

4(1 + If )

2F and − 1

4(1− If )

2F

are both outer and non-negative a.e. on T. Moreover, by direct verification,

[ 14 (1 + If )2F ] + [− 1

4 (1− If )2F ] = f.

This expresses f as the sum of two real outer functions that are non-negative a.e. onT. If f ∈ Hp, then F is as well. Thus the two summands above also belong to Hp.Consequently,

g1 =1

2(1 + If )

√F and g2 =

i

2(1− If )

√F

are real outer functions that satisfy f = g21 + g22 . They belong to Hp2 whenever

f ∈ Hp. �

3.4. Cayley Inner Functions. Theorem 3.7 reduces the study of real Smirnovfunctions to the study of real outer functions. The simplest real outer functions are,in essence, just Cayley transforms of inner functions. For reasons that will becomeclear much later, we actually require a certain variant of the Cayley transform thatturns out to be compatible with infinite products in a crucial way.

Consider the linear fractional transformation

(3.14) T (z) = i1− iz

1 + iz,

whose inverse is

(3.15) T−1(z) = iz − i

z + i.

One can verify that T satisfies the identities

(3.16) T−1(z) = T (1/z) =1

T (z)= −T (−z) = T (z)

and

(3.17) (T ◦ T )(z) = 1

zand (T ◦ T ◦ T ◦ T )(z) = z.

The mapping properties of T and T−1 are illustrated in Figure 4 and Table 1 below.

0 1 −1 i −i ∞ T R L (−∞, 0)T (·) i 1 −1 ∞ 0 −i R T (−∞, 0) L

T−1(·) −i 1 −1 0 ∞ i R T (−∞, 0) L

Table 1. Values of the linear fractional transformations (3.14) and(3.15). Here L denotes the open arc of the unit circle T running coun-terclockwise from i to −i.


∞

i

−i

1−1

T (z)

∞

i

−i

1−1

Figure 4. Illustration of the linear fractional transformation (3.14).The points 1 and −1 are fixed by T . The unit disk gets mapped to theupper half plane.

If ϕ is inner, then 1− iϕ and 1+ iϕ are both outer. Consequently, T (ϕ) belongsto RO. In fact, T (ϕ) belongs to Hp for all 0 < p < 1 (see (3.10)). This is sharp,for T (ϕ) cannot belong to H1 unless ϕ is constant (Theorem 3.8). Functions ofthe form T (ϕ), where ϕ is inner, form the basic building blocks from which all realouter functions can be built. This prompts the following definition.

Definition 3.18. A Cayley inner function is a function of the form T (ϕ),where ϕ is an inner function.

The properties of Cayley inner functions are almost self evident. For one, aCayley inner function is not inner at all, but outer! Since a non-constant innerfunction ϕ maps D into D and the transformation T maps D onto the upper halfplane, it follows that 0 < arg T (ϕ) < π on D, where arg refers to the principal branchof the argument. Since T (ϕ) has real boundary values a.e. on T, the precedingimplies that arg T (ϕ)(ζ) ∈ {0, π} for a.e. ζ ∈ T. In fact, arg T (ϕ)(ζ) = π onE = ϕ−1(L), where L denotes the open arc of T that runs counterclockwise fromi to −i (see Figure 5). With a little work, the preceding computations can bereversed.

For a Lebesgue measurable set E ⊆ T define

vE = P(χE),

the Poisson integral (2.1) of the characteristic function χE for E. Recall that vE isharmonic on D and its (non-tangential) boundary values agree with χE a.e. on T.Since χE only assumes the values 0 and 1, it follows that vE assumes only valuesin [0, 1] on D. Furthermore, vE(0) = m(E).

If we normalize the harmonic conjugate vE of vE so that vE(0) = 0, then

(3.19) fE = exp[π(−vE + ivE)]

is analytic on D, maps D into the upper half plane, and has real boundary valuesa.e. on T. In particular, fE ∈ RO and

ϕE := T−1(fE)


E

E

E

ϕ T

T (ϕ)

Figure 5. Illustration of the mapping properties of a Cayley inner function.

is a bounded analytic function on D with unimodular values a.e. on T, i.e., an innerfunction. Thus fE = T (ϕE) is a Cayley inner function and hence it belongs to RO.

The Cayley inner function fE obtained from (3.19) can alternatively be de-scribed as the exponential of a Herglotz integral, i.e.,

(3.20) fE(z) = exp

[iπ

∫E

ζ + z

ζ − zdm(ζ)

].

Indeed, vE is the Poisson integral of its boundary function χE and thus the inte-gral in the exponential in (3.20) can be obtained by analytic completion once onerecognizes that vE is its real part and vE(0) = 0.

Lemma 3.21. Let E ⊆ T be Lebesgue measurable. Then

(a) E = f−1E (−∞, 0) = ϕ−1

E (L), where L = {ζ ∈ T : Re ζ < 0};(b) arg fE = πχE a.e. on T;

(c) fE(0) = eiπm(E) and ϕE(0) = tan[π2 (12 −m(E))];

(d) f∅ ≡ 1, fT ≡ −1, ϕ∅ ≡ 1, ϕT ≡ −1;(e) If f ∈ RO satisfies (b), then f = |f(0)|fE;(f) If ϕ is inner and ϕ(0) ∈ R, then ϕ = ϕE, where E = ϕ−1(L).

Proof. (a), (b) The function fE from (3.19) is Cayley inner and satisfiesarg fE = πχE a.e. on T and f−1

E (−∞, 0) = E by construction. The mapping

properties of T ensure that ϕE = T−1(fE) satisfies ϕ−1E (L) = E (Figure 4).

(c) Since vE(0) = m(E) and vE(0) = 0, we have

fE(0) = eiπvE(0) = eiπm(E).

A somewhat tedious, but ultimately elementary, calculation reveals that

ϕE(0) = tan

(π

2

[1

2−m(E)

]).


(d) These identities come from the observations

v∅ ≡ v∅ ≡ 0, vT ≡ 1, vT ≡ 0.

(e) To prove this we first need a little detail. If g ∈ N+ and outer then

g(z) = eiγ exp

(∫T

ζ + z

ζ − zlog|g(ζ)|dm(ζ)

)and thus

log g(z) = iγ +

∫T

ζ + z

ζ − zlog|g(ζ)|dm(ζ).

Any analytic function on D whose range is contained in {Re z > 0} belongs to Hp

for all 0 < p < 1 [17]. One can show that the Cauchy transform∫1

1− ζzdμ(ζ)

of a positive measure μ on T satisfies this property and thus belongs to Hp, 0 < p <1. Writing any complex measure as a linear combination of four positive measuresshows that the Cauchy transform of any measure also belongs to Hp. From here itfollows that ∫

T

ζ + z

ζ − zlog|g(ζ)|dm(ζ)

belongs to Hp and, as a result, log g ∈ N+.

Apply this to the function g = f/fE , where f ∈ RO and satisfies (b). Thuslog g ∈ N+. Furthermore,∣∣∣∣log f

fE

∣∣∣∣2 =

(log

|f ||fE |

)2

+

(arg

f

fE

)2

=

(log

|f ||fE |

)2

a.e. on T

and so ∣∣∣∣log f

fE

∣∣∣∣ � |log|f |+ log|fE || ∈ L1.

Hence log g ∈ N+ with integrable boundary values and thus, by Smirnov’s Theorem(see (2.11)), log g ∈ H1. Since f and fE share the same sign a.e. on T we see thatlog f/fE is real a.e. on T. By Theorem 3.8 we conclude that log f/fE is a constantfunction and thus f and fE are positive scalar multiples of each other. Since|fE(0)|= 1 it follows that f = |f(0)|fE .(f) If ϕ is inner and ϕ(0) ∈ R then f = T (ϕ) satisfies |f(0)|= 1. Let E =f−1(−∞, 0) and observe that (e) ensures that f = fE . Consequently, ϕ = T−1(fE)= ϕE , where E = ϕ−1(L). �

A nice gem follows from the proof of this result: If f and g are outer functionswith arg f = arg g almost everywhere on T, then f = λg where λ > 0.

Example 3.22. The functions fE enjoy some convenient multiplicative proper-ties. For example, since χE+χF = χE∩F +χE∪F for any Lebesgue measurable setsE,F ⊆ T, we can use (3.20) to see that the corresponding Cayley inner functionssatisfy

fEfF = fE∩F fE∪F .

In particular, fEfF = fE∪F whenever E ∩ F = ∅. We also have

(3.23) fEfT\E ≡ −1.


These identities can be extended to collections of three or more sets in an analogousway.

Example 3.24. Suppose that β < α < β + 2π and

E = {eiθ : β < θ < α}is an arc in T, running counterclockwise from eiβ to eiα. Then we can obtain fEfrom (3.20):

fE(z) = exp

[i

2

∫ α

β

eiθ + z

eiθ − zdθ

].

Some routine calculations show that

(3.25) fE(z) = e−i(α−β2 )

(eiα − z

eiβ − z

)and confirm that

arg fE(eiθ) = arg

eiα/2 − e−iα/2eiθ

eiβ/2 − e−iβ/2eiθ

= argei(α−θ)/2 − e−i(α−θ)/2

ei(β−θ)/2 − e−i(β−θ)/2

= arg

[sin (

θ − α

2)/sin (

θ − β

2)

]= πχE(e

iθ),

as expected.

Since fE is a linear fractional transformation, it follows that the inner functionϕE = T−1(fE) is also a linear fractional transformation. This means it must be aunimodular constant multiple of a single Blaschke factor. In what follows, it willbe convenient to assume that 0 < α− β < π. This ensures that

(3.26) ϕE(0) = tan (14 [π − (α− β)]),

and so 0 < ϕE(0) < 1. Consequently,

ϕE(z) =|zE |zE

zE − z

1− zEz

for some zE ∈ D. From (3.26) we know that

|zE |= tan ( 14 [π − (α− β)]).

By symmetry, one expects zE to lie on the line segment joining the origin to themidpoint ei

12 (α+β) of the arc E. Since fE(zE) = T (0) = i, in which fE is given by

(3.25), we solve for zE in the equation

e−i(α−β2 )

(eiα − zEeiβ − zE

)= i

to obtain

zE =eiα/2 − ieiβ/2

e−iα/2 − ie−iβ/2

= ei(α+β

2 ) e−iβ/2 − ie−iα/2

e−iα/2 − ie−iβ/2


= ei(α+β

2 ) e−i β2 − ei

π2 −iα

2

e−iα2 + e−iπ

2 −i β2

= ei(α+β

2 ) ei(−π4 +α

4 + β4 )

−iei( π4 +α

4 + β4 )· e−i β

2 − eiπ2 −iα

2

e−iα2 + e−iπ

2 −iβ2

= ei(α+β

2 )−1−i

ei14 [π−(α−β)] − e−i 1

4 [π−(α−β)]

ei14 [π−(α−β)] + e−i 1

4 [π−(α−β)]

= ei(α+β

2 ) tan ( 14 [π − (α− β)])

as expected. In particular, ϕE is the single Blaschke factor with zero

(3.27) zE = ei12 (α+β) tan

[1

4(π − (α− β))

].

Example 3.28. Let E ⊆ T be a Lebesgue measurable and suppose thatm(E) =12 . Then Lemma 3.21 guarantees that ϕE(0) = 0 and hence we may write

ϕE(z) =∏n�1

|zn|zn

zn − z

1− znzexp

(−∫T

ζ + z

ζ − zdμ(ζ)

),

where {zn}n�1 is a Blaschke sequence and μ is a finite, non-negative, singularmeasure on T. Appealing again to Lemma 3.21, we find that

(3.29) tan

[π

2

(1

2−m(E)

)]= e−μ(T)

∏n�1

|zn|.

One can verify that the linear fractional transformation T defined by (3.14)satisfies the following algebraic identities:

T (z1z2) =T (z1)T (z2) + T (z1) + T (z2)− 1

1 + T (z1) + T (z2)− T (z1)T (z2),

T (z2/z1) =T (z1)T (z2)− T (z1) + T (z2) + 1

T (z1)T (z2) + T (z1)− T (z2) + 1,(3.30)

T (z1 + z2) =3T (z1)T (z2) + iT (z1) + iT (z2) + 1

3i+ T (z1) + T (z2) + iT (z1)T (z2).(3.31)

These often lead to some curious identities involving Cayley inner functions. Hereare two such examples.

Example 3.32. Suppose that f1 = T (ϕ1) and f2 = T (ϕ2) for some innerfunctions ϕ1 and ϕ2. Then f1 + f2 is real valued a.e. on R and maps D into theupper half plane. Consequently, there is an inner function ϕ so that f1+f2 = T (ϕ).Since (T ◦ T )(z) = 1/z by (3.16), it follows that T (f1) = 1/ϕ1, T (f2) = 1/ϕ2, andT (f1 + f2) = 1/ϕ. Then (3.31) reveals that

(3.33) ϕ =3iϕ1ϕ2 + ϕ1 + ϕ2 + i

3 + iϕ1 + iϕ2 + ϕ1ϕ2.

Although this does not look like an inner function, it is. The denominator

3 + iϕ1 + iϕ2 + ϕ1ϕ2 = (1 + iϕ1) + (1 + iϕ2) + (1 + ϕ1ϕ2)

is the sum of three outer functions, each of which assume values in the right half-plane, so it is outer. Thus ϕ is the inner factor of the numerator 3iϕ1ϕ2+ϕ1+ϕ2+i.


Example 3.34. A trivial consequence of Helson’s Theorem (Theorem 3.1) isthat each f ∈ R can be written as f = T (ψ2/ψ1), where ψ1 and ψ2 are relativelyprime inner functions. This fact, and little bit of algebra, show that every functionin R is a simple algebraic function of two Cayley inner functions. Indeed, if f1 =T (ψ1) and f2 = T (ψ2), then (3.30) reveals that

f =T (ψ1)T (ψ2)− T (ψ1) + T (ψ2) + 1

T (ψ1)T (ψ2) + T (ψ1)− T (ψ2) + 1

=f1f2 − f1 + f2 + 1

f1f2 + f1 − f2 + 1.

4. Unilateral Products of Cayley Inner Functions

We now consider the convergence of products of the form

(4.1)∏n�1

T (ϕn) =∏n�1

(i1− iϕn

1 + iϕn

),

where ϕn is a sequence of inner functions. We refer to such products as unilateralproducts to distinguish them from the bilateral products (i.e., analogous productswith indices ranging from −∞ to∞) that will be considered later. As we will see, acompletely satisfactory theory of unilateral products can be developed. In contrast,bilateral products pose a host of problems, not all of which have been resolved.

4.1. Bounded argument. Suppose that f ∈ RO has bounded argument. Itis instructive to consider this special case before considering the general setting.The approach below is essentially due to Poltoratski [27].

Since f has bounded argument, we may write

(4.2) f = |f(0)|exp[π(−v + iv)],

where v is non-negative, integer valued (to make f real valued almost everywhereon T), and bounded above by some integer N . For each positive integer n, let

En = {ζ ∈ T : v(ζ) � n}

and observe that

E1 ⊇ E2 ⊇ · · · ⊇ EN ⊇ EN+1 = ∅.

Then

v =∑

1�n�N

χEn

so that

exp[π(−v + iv)] =∏

1�n�N

exp[π(−χEn+ iχEn

)] =∏

1�n�N

fEn.

Returning to (4.2) and letting ϕn = ϕEn, the preceding yields

f = |f(0)|∏

1�n�N

T (ϕn) = |f(0)|∏

1�n�N

i

(1− iϕn

1 + iϕn

).

Combining this observation with Theorem 3.7 yields the following result.


Theorem 4.3. Suppose f ∈ R+ is factored as f = KfRf as in Theorem 3.7,where Kf is a Koebe inner function and Rf ∈ RO. If argRf is bounded, then thereare inner functions ϕ1, ϕ2, . . . , ϕN so that

f = Kf

∏1�n�N

T (ϕn).

Moreover, the product belongs to Hp whenever f belongs to Hp.

4.2. A convergence criterion. It turns out that practical necessary and suf-ficient conditions exist for determining when products of the form (4.1) converge.In fact, as we will see in a moment, any function f ∈ RO with semibounded ar-gument can be expanding in a product of the form (4.1) that converges absolutelyand locally uniformly on D. We first require a basic lemma. Before stating thislemma, we recall a definition.

Definition 4.4. For a sequence {an}n�1 of complex numbers we say that theproduct

∏n�1(1 + an) converges absolutely if

∏n�1(1 + |an|) converges.

This is equivalent to saying that∑

n�1|an| converges. See [1, p. 192] for details.

Lemma 4.5. Let {zn}n�1 be a sequence in C\{i}. Then∏

n�1 zn converges

absolutely if and only if∏

n�1 T (zn) converges absolutely.

Proof. If∏

n�1 zn converges absolutely, then zn → 1 and 1 + izn is boundedaway from zero. Since

(4.6) 1− T (zn) =1− i

1 + izn(1− zn),

the forward implication follows. If the product∏

n�1 T (zn) converges absolutely,

then T (zn) → 1. Since T (1) = 1, we conclude that zn → 1 and hence 1 + izn isbounded away from zero. Appealing to (4.6) yields the reverse implication. �

The following lemma reduces the consideration of products of Cayley innerfunctions to products of inner functions. This is a significant reduction, sincedetermining whether or not a product of inner functions converges is straightforward(see Lemma 4.10 below).

Lemma 4.7. Let ϕn be a sequence of inner functions satisfying ϕn(0) = 0 andlet fn = T (ϕn). The following are equivalent:

(a) The product∏

n�1 ϕn(0) converges absolutely;

(b) The product∏

n�1 ϕn converges absolutely and locally uniformly on D;

(c) The product∏

n�1 fn(0) converges absolutely;

(d) The product∏

n�1 fn converges absolutely and locally uniformly on D.

Proof. (a) ⇒ (b) Suppose that∏

n�1 ϕn(0) converges absolutely. For eachn ∈ N, the Schwarz’ Lemma yields

|ϕn(z)− ϕn(0)|� |z|∣∣∣1− ϕn(0)ϕn(z)

∣∣∣ , z ∈ D.

The above inequality can be rewritten as

|(1− ϕn(0))− (1− ϕn(z))|� |z|∣∣∣(1− ϕn(z)) + ϕn(z)(1− ϕn(0))

∣∣∣ ,


which implies

|1− ϕn(z)|−|1− ϕn(0)|� |z|(|1− ϕn(z)|+|1− ϕn(0)|).From here we deduce that

(4.8) |1− ϕn(z)|�(1 + |z|1− |z|

)|1− ϕn(0)|,

and hence the absolute convergence of∏

n�1 ϕn(0) implies the absolute and locally

uniform convergence of∏

n�1 ϕn on D.

(b) ⇒ (d) Suppose that product∏

n�1 ϕn converges absolutely and locally uni-

formly on D. Since fn = T (ϕn) and fn(0) = i, Lemma 4.5 implies that∏

n�1 fnconverges absolutely on D. The uniform continuity of T on compact subsets of Densures that this convergence is locally uniform.

(d) ⇒ (c) This implication is trivial.

(c) ⇒ (a) Suppose that∏

n�1 fn(0) converges absolutely. Since fn = T (ϕn), we

see from (3.16) that (T ◦T ◦T )(fn) = ϕn. Three successive applications of Lemma4.5 guarantee that

∏n�1 ϕn(0) converges absolutely. �

The preceding theorem reduces the study of unilateral products (4.1) of Cayleyinner functions to the study of infinite products of inner functions. This is well-understood territory. Indeed, let

(4.9) ϕn(z) = eiθn

⎛⎝zmn

∏k�1

|z(n)k |z(n)k

z(n)k − z

1− z(n)k z

⎞⎠ exp

(−∫T

ζ + z

ζ − zdμn(ζ)

)be a sequence of inner functions, where

(a) θn ∈ [−π, π);(b) mn ∈ N;

(c) z(n)1 , z

(n)2 , . . . is a Blaschke sequence for each n;

(d) μn is a finite, non-negative, singular measure on T.

In what follows, we allow the possibility that some of the zero sequences

z(n)1 , z

(n)2 , . . .

are finite. Since this does not affect our arguments in any significant way, exceptencumbering our notation, we proceed as if each such zero sequence is infinite. Thereader should have no difficulty in patching up the argument to handle the mostgeneral case.

Lemma 4.10. The product∏

n�1 ϕn of the inner functions (4.9) converges ab-solutely and locally uniformly on D if and only if

(a)∑

n�1 θn converges;

(b)∑

n�1 mn <∞;

(c) {z(n)k : n, k � 1} forms a Blaschke sequence:∑

n,k�1(1− |z(n)k |) <∞;

(d)∑

n�1 μn(T) <∞.


Proof. The necessity of (b) is self evident, so we may assume that (b) holdsand that ϕn(0) = 0 for n ∈ N. Lemma 4.7 says that

∏n�1 ϕn converges absolutely

and locally uniformly if and only if∏n�1

(e−μn(T)+iθn

∞∏k=0

|z(n)k |)

converges absolutely. This occurs if and only if

∑n�1

⎛⎝(μn(T)− iθn)− log∏k�1

|z(n)k |

⎞⎠converges absolutely. The series above converges absolutely if and only if each ofthe series

∑n�1 θn and

∑n�1 μn(T) converge absolutely, and if∑

n�1

(− log

∞∏k=0

|z(n)k |)

= −∑

n,k�1

log|z(n)k |

converge absolutely (that is, if (a) and (d) hold). However,∑n,k�1

log|z(n)k |

converges if and only if (c) holds. �

Suppose that ϕn is a sequence of non-constant inner functions and fn = T (ϕn).If En = f−1

n (−∞, 0), then Lemma 3.21 implies that fn = |fn(0)|fEn. Lemma 4.7

asserts that∏

n�1 fn converges absolutely and locally uniformly in D if and only if

∏n�1

fn(0) =∏n�1

(|fn(0)|e

12 im(En)

)= exp

⎛⎝1

2i∑n�1

m(En)

⎞⎠∏n�1

|fn(0)|

does. Consequently, when considering unilateral products of Cayley inner functions,it suffices to consider products of the form∏

n�1

fEn, En ⊆ T.

The following theorem addresses the convergence of such products.

Theorem 4.11. If En ⊆ T, then the following are equivalent:

(a)∑

n�1 m(En) <∞;

(b) v = π∑

n�1 χEn∈ L1;

(c)∏

n�1 fEnconverges absolutely and locally uniformly on D.

The product in (c) belongs to R+ if and only if v ∈ L logL.

Proof. The equivalence of (a) and (b) is immediate. Lemma 4.7 tells us that(c) is equivalent to the convergence of

∏n�1

fn(0) = exp

⎛⎝1

2i∑n�1

m(En)

⎞⎠ ,


which is equivalent to (a). If the product in (c) converges to

f = exp[π(−v + iv)],

then v � 0 so the conjugate v belongs to L1 if and only if v ∈ L logL [11, Thm. 4.4](Zygmund’s theorem on the integrability of the conjugate function). �

Example 4.12. Suppose that v ∈ L1 only assumes values in πZ and writev = v+−v−, where v+ and v− are non-negative L1 functions. Then two applicationsof Theorem 4.11 produce an analytic function f on D that is real-valued a.e. on T

(in the sense of non-tangential limiting values) and that satisfies arg f = v. If bothv+ and v− belong to L logL, then f belongs to R+.

Example 4.13. Suppose that E1, E2, . . . are Lebesgue measurable subsets of Tand that v = π

∑n�1 χEn

belongs to L1 but not L logL. Theorem 4.11 yields that

f =∏

n�1 fEnis well defined. However, Zygmund’s Theorem asserts that v ∈ L1

and so f = exp[π(−v + iv)] ∈ N .

Example 4.14. Recall from Example 3.24 that if E is the circular arc in T,running counterclockwise from eiβ to eiα and 0 < m(E) < 1

2 , then ϕE is the singleBlaschke factor with zero

zE = ei12 (α+β) tan

(π

2

[1

2−m(E)

]).

This provides a bijection E �→ zE between circular arcs E with 0 < m(E) < 12 and

D\{0}. Suppose that

(4.15)∏n�1

|zn|zn

zn − z

1− znz

is a Blaschke product. For each zero zn there is a unique circular arc En so thatzn = zEn

. The nth factor in the product (4.15) is precisely ϕEn. The above

discussion shows that the product (4.15) converges absolutely and locally uniformlyon D if and only if

∑n�1 m(En) < ∞. Consequently, the summability condition∑

n�1 m(En) <∞ must be equivalent to the Blaschke condition∑

n�1(1− |zn|) <∞. We can demonstrate this equivalence directly. Indeed, since

|zEn|= ϕEn

(0) = tan[π2 (12 −m(En))],

and

(4.16) limx→0

1− tan(π2 (12 − x))

x= π,

the limit comparison test shows that the series

(4.17)∑n�1

m(En) and∑n�1

(1− |zn|) =∑n�1

(1− tan( 14 (π −m(En))))

converge or diverge together.

Example 4.18. If E is an open subset of T, then E decomposes uniquelyas the countable union of disjoint circular arcs En. Let us assume that thereare infinitely many arcs involved in this decomposition. At most one can have


measure greater than 12 , so we may assume that 0 < m(En) <

12 for n ∈ N. Since

m(E) =∑

n�1 m(En), Theorem 4.11 ensures that

fE =∏n�1

fEn.

Since each En is an arc, Example 3.24 tells us that fEnis a linear fractional trans-

formation. Thus fE is an infinite product of linear fractional transformations.

Example 4.19. Suppose that E ⊆ T is a fat Cantor set, i.e., a Cantor setwith positive Lebesgue measure. Then T\E is an open set of positive measure, soExample 4.18 shows us that fT\E is a product of linear fractional transformations.Since (3.23) implies that fE = −1/fT\E , we conclude that fE is a product of linearfractional transformations.

Example 4.20. Consider the atomic inner function

ϕρ(z) = exp

(ρz + 1

z − 1

), ρ > 0.

Since ϕρ(0) = e−ρ > 0, Lemma 3.21 shows that

ϕρ = ϕE(ρ), E(ρ) = ϕ−1ρ (L).

A computation shows that z ∈ T belongs to E(ρ) if and only if ρ z+1z−1 lies in one

of the imaginary intervals (πi2 ,3πi2 ) (mod 2πin), where n ∈ Z. Since the linear

fractional transformation (z + 1)/(z − 1) is self-inverse, we see that z belongs toE(ρ) whenever z lies in one of the circular arcs In(ρ) connecting the points

(4.21)(4n+ 3)π − 2ρi

(4n+ 3)π + 2ρiand

(4n+ 1)π − 2ρi

(4n+ 1)π + 2ρi.

For n � 0, the arcs In(ρ) lie on the bottom half of T and shrink rapidly, approachingthe point 1 as n→∞. The arcs In(ρ) for n < 0 are the complex conjugates of thearcs In+1(ρ) and lie on the upper half of T. The total measure of the arcs In(ρ)can be computed with Lemma 3.21:∑

n∈Z

m(In(ρ)) = m(E(ρ)) =1

2− 2

πtan−1(e−ρ).

For any ρ > 0, Theorem 4.11 says that

fE(ρ) =∏n∈Z

fIn(ρ).

where the product converges either as a bilateral (i.e., symmetric) product or astwo separate unilateral products. Moreover, each term of the product is a linearfractional transformation (Example 3.24).

5. Herglotz A-integral representations

We wish continue our work from the previous section to obtain an infiniteproduct representation of an f ∈ RO. Recall that this starts by writing

f = |f(0)|exp[π(−v + iv)],


where v is integer valued. Unlike the cases considered in the preceding section, herev may be unbounded (above and below). To handle this situation, we require ageneralization of the classical Herglotz formula

(5.1) h(z) = i

∫T

ζ + z

ζ − zImh(ζ)dm(ζ), z ∈ D,

which holds when h ∈ H1. This formula permits us to recover an analytic functionh on D from the boundary values of its imaginary part.

This section is devoted to obtaining a suitable generalization of the Herglotzformula. We will resume our discussion of product representations of RO functionsin Section 6.

5.1. The A-integral. Unfortunately, as hinted in Example 4.12, we are notalways lucky enough to have h ∈ H1. The fact that f ∈ N+ only ensures thatv ∈ L1 while its harmonic conjugate v need not belong to L1. Thus we cannotimmediately recover h = −v + iv from its imaginary part. Consequently, we needto develop a suitable replacement for (5.1). This is where the A-integral comes in.

If h : T→ C is Lebesgue measurable, define

λh(t) = m({|h|> t}), t > 0,

to be the distribution function for h. We say that h belongs to L1,∞0 if

(5.2) λh(t) = o

(1

t

).

A short exercise will show that L1 ⊂ L1,∞0 . A classical theorem of Kolmogorov

[25, p. 131] says that if w ∈ L1 then its harmonic conjugate w belongs to L1,∞0 .

Definition 5.3. h : T→ C is A-integrable if it belongs to L1,∞0 and

limA→∞

∫{|h|�A}

h dm

exists. This limit is called the A-integral of h over T and is denoted by

(A)

∫T

h dm.

The theory of A-integrals was developed by Denjoy, Titchmarsh [31], Kol-mogorov, Ul′yanov [32], and Aleksandrov [3].

An analytic function on D is said to belong to the space H1,∞0 if it belongs to

N+ and its boundary function is in L1,∞0 . Aleksandrov showed that such a function

is the Cauchy A-integral of its boundary function [3]. That is, if h is in H1,∞0 , then

h(z) = (A)

∫T

h(ζ)

1− ζzdm(ζ), z ∈ D.

A detailed proof of Aleksandrov’s theorem can be found in [7]. In order to establishan infinite product expansion for real outer functions, we require a Herglotz integralanalogue of Aleksandrov’s theorem.


5.2. Herglotz A-integral representations. The following theorem drawsheavily on the work of Aleksandrov [3]. The proof given below can be found in[16].

Theorem 5.4. Let h = u+ iv ∈ H1,∞0 and u(0) = 0. For A > 0 let

vA =

⎧⎪⎨⎪⎩v if |v|� A,

A if v > A,

−A if v < −A.

Then

(5.5) h(z) = limA→∞

i

∫T

ζ + z

ζ − zvA(ζ)dm(ζ),

where the convergence is uniform on compact subsets of D.

Suppose that h = u + iv belongs to H1,∞0 and u(0) = 0. Without loss of

generality, we may assume that v(0) = 0, that is, h(0) = 0. Indeed, a shortargument shows that we may replace v with v − v(0) in the definition of vA. Thefact that the Herglotz integral of the constant function v(0) is indeed v(0) allowsthis reduction to go through.

For t > 0 let

ρh(t) = tλh(t).

Since h ∈ H1,∞0 , its boundary function belongs to L1,∞

0 , so (5.2) ensures thatρh → 0 as t→∞. For A > 0, let

(5.6) σh(A) = supt�A

ρh(t)

and observe that σh(A)→ 0 (as A→∞) as well.

The proof of Theorem 5.4 requires the following three technical lemmas.

Lemma 5.7. For A > 0 and h ∈ H1,∞0 ,

(5.8)

∣∣∣∣∣∫|h|�A

h dm

∣∣∣∣∣ � ρh(A) + 2√σh(0)σh(A).

Proof. Let A > 0 and let g be the outer function that satisfies

(a) g(0) > 0;

(b) |g|= 1 on {|h|� A} ⊆ T;

(c) |g|= A/|h| on {|h|> A} ⊆ T.

By construction, the analytic function gh vanishes at the origin and satisfies |gh|� Aa.e. on T. Consequently,

0 =

∫T

gh dm =

∫|h|�A

gh dm+

∫|h|>A

gh dm

=

∫|h|�A

h dm+

∫|h|�A

(g − 1)h dm+

∫|h|>A

gh dm,


and so

(5.9)

∫|h|�A

h dm = −∫|h|>A

gh dm︸︷︷︸I1(A)

+

∫|h|�A

(1− g)h dm︸︷︷︸I2(A)

.

The definition of g implies that

I1(A) �∫|h|>A

|gh| dm = A

∫|h|>A

dm = Aλh(A) = ρh(A),

which yields the first term on the right-hand side of (5.8).

To estimate I2(A) from (5.9), we first use the Cauchy–Schwarz inequality:

(5.10) |I2(A)|2�(∫

|h|�A

|h|2 dm)(∫

|h|�A

|1− g|2 dm).

By the distributional identity (see [7, p. 50 - 51]) and (5.6) we obtain

(5.11)

∫|h|�A

|h|2 dm = 2

∫ A

0

tλh(t) dt � 2σh(0)A.

This already provides one of the terms required for (5.8). The second integral in(5.10) is more troublesome. Let w = log|g|, so that

g = exp(w + iw).

The definition of g says that w ≡ 0 on the set {|h|� A} and so

|1− g|= |1− eiw|� |w|.Using the fact that the L2 norm of w is dominated by that of w (see (2.4)), we findthat ∫

|h|�A

|1− g|2 dm �∫T

|w|2 dm �∫T

|w|2 dm

=

∫T

(log|g|)2 dm =

∫|h|>A

(log|h|A

)2

dm

= −∫ ∞

A

(log

t

A

)2

dλh(t).

Integrating by parts leads to

2

∫ ∞

A

λh(t) logtA

tdt � 2σh(A)

∫ ∞

A

log tA

t2dt

=2σh(A)

A

∫ ∞

1

log s

s2ds

=2σh(A)

A.(5.12)

Returning to (5.10) with the bounds (5.11) and (5.12) we obtain

|I2(A)|� 2√σh(0)σh(A).

This completes the proof of the lemma. �


Lemma 5.13. For h ∈ H1,∞0 ,

h(z) = limA→∞

∫|h|�A

h(ζ)

1− ζzdm(ζ)

uniformly on compact subsets of D.

Proof. For z ∈ D and ζ ∈ T, let

(5.14) hz(ζ) =ζ(h(ζ)− h(z))

ζ − z

and observe that∫|h|�A

hz dm =

∫|h|�A

ζh(ζ)

ζ − zdm(ζ)−

∫|h|�A

ζh(z)

ζ − zdm(ζ)

=

∫|h|�A

h(ζ)

1− ζzdm(ζ)−

∫|h|�A

h(z)

1− ζzdm(ζ).

Lemma 5.7 tells us that

(5.15)

∣∣∣∣∣∫|hz |�A

hz dm

∣∣∣∣∣ � 2√σhz

(0)σhz(A) + ρhz

(A).

We claim that the right-hand side of the preceding tends to 0 uniformly on compactsubsets of D as A→∞.

For each r ∈ (0, 1) let

Mr = max|z|�r

|h(z)|.

For |z|� r and ζ ∈ T, we read from (5.14) that

(5.16) |hz(ζ)|�|h(ζ)|+|h(z)|

1− |z| � |h(ζ)|+Mr

1− r,

which implies that

(5.17) λhz(t) � λh((1− r)t−Mr).

If t > 2Mr/(1− r), then

t <2

1− r((1− r)t−Mr).

Multiplying (5.17) by the preceding we obtain

(5.18) tλhz(t) � 2

1− r((1− r)t−Mr)λh((1− r)t−Mr).

Therefore,

(5.19) σhz(t) � 2

1− rσh((1− r)t−Mr).

Now suppose that

(5.20)2Mr

1− r� A,

so that the inequality (5.19) is valid for t = A. Then

(1− r)A

2� (1− r)A−Mr,


so that

ρhz(A) � σhz

(A) � 2

1− rσh

((1− r)A

2

)and

σhz(0) � 2

1− rmax{Mr, σh(Mr)}.

The preceding two estimates show that the right-hand side of (5.15) tends to zerouniformly on |z|� r as A→∞. This proves our claim.

For A that satisfy (5.20), we let

Ar = (1− r)A−Mr.

We claim that the difference between

(5.21)

∫|hz|�A

hz dm and

∫|h|�Ar

hz dm

tends to 0 uniformly on |z|� r as A→∞. If |z|� r, then (5.16) shows that

{|h|� Ar} ⊆ {|hz|� A}.Consequently, the difference between the two integrals in (5.21) is bounded in ab-solute value by

(1− r)−1Aλh(Ar),

which, in turn, is bounded by (A

(1− r)Ar

)ρh(Ar).

Since A/Ar remains bounded as A→∞, the preceding tends to 0, as desired.

We showed that ∫|h|�A

hz dm→ 0

uniformly on |z|� r for each r in (0, 1); this was our claim above. Now observe thatthe difference between h(z) and∫

|h|�A

h(z)

1− ζzdm(ζ)

is bounded in absolute value by

(1− |z|)−1|h(z)|λh(A)

and hence tends to zero uniformly on |z|� r. This concludes the proof of thelemma. �

Lemma 5.22.

limA→∞

∫|h|�A

h(ζ)

1− ζzdm(ζ) = 0

uniformly on compact subsets of D.

Proof. This result can be obtained by applying Lemma 5.7 to the function

h(ζ)

1− zζ

and then arguing as in the proof of Lemma 5.13. The details are largely identical.�


We are now ready to conclude the proof of Theorem 5.4. From Lemmas 5.13and 5.22 we have

h(z) = limA→∞

∫|h|�A

h(ζ)− h(ζ)

1− ζzdm(ζ)

= limA→∞

i

∫|h|�A

2v(ζ)

1− ζzdm(ζ)(5.23)

uniformly on compact subsets of D. Since v(0) = 0, Lemma 5.7 ensures that

limA→∞

i

∫|h|�A

v(ζ)dm(ζ) = 0.

Subtract this from (5.23) to obtain

h(z) = limA→∞

i

∫|h|�A

(ζ + z

ζ − z

)v(ζ)dm(ζ)

uniformly on compact subsets of D. In light of the fact that

{|h|� A} ⊆ {|v|� A},we see that the difference between∫

|h|�A

(ζ + z

ζ − z

)v(ζ)dm(ζ) and

∫|v|�A

(ζ + z

ζ − z

)v(ζ)dm(ζ)

is bounded in absolute value by(1 + |z|1− |z|

)Aλh(A).

Finally, the difference between∫|v|�A

(ζ + z

ζ − z

)v(ζ)dm(ζ) and

∫T

(ζ + z

ζ − z

)vA(ζ)dm(ζ)

is bounded in absolute value by(1 + |z|1− |z|

)Aλv(A)

so

h(z) = limA→∞

i

∫T

ζ + z

ζ − zvA(ζ)dm(ζ),

which is the desired result (5.5). This concludes the proof of Theorem 5.4. �

6. Bilateral products

From Theorem 4.3 we can write every RO function with bounded argument asan infinite product. Our aim in this section is to establish a similar factorizationtheorem for RO functions with possibly unbounded argument. Suppose that f ∈ ROand write

f = |f(0)|exp[π(u+ iv)],

where u ∈ L1. Although this is not enough to say that v ∈ L1, now that we haveTheorem 5.4 at our disposal, this does not pose an insurmountable obstacle.


6.1. Factorization of RO functions.

Theorem 6.1. If f ∈ RO, then there exist inner functions ϕ+n and ϕ−

n so that

(6.2) f = |f(0)|∏n�1

(1− iϕ+

n

1 + iϕ+n

)(1 + iϕ−

n

1− iϕ−n

),

where the product converges locally uniformly on D.

Proof. Write f = |f(0)|exp[π(u+ iv)], where u ∈ L1. Without loss of gener-ality, we may assume that |f(0)|= 1; that is, u(0) = 0. For each positive integer n,let

E+n = {v � n}, E−

n = {v � −n},so that

E+1 ⊇ E+

2 ⊇ · · · and E−1 ⊇ E−

2 ⊇ · · · .Let

f+n = fE+

n, ϕ+

n = ϕE+n, f−

n = fE−n, and ϕ−

n = ϕE−n,

where ϕ±En

are the inner functions described in Subsection 3.4 and

fE±n= T (ϕE±

n)

are the corresponding Cayley inner functions (recall Definition 3.18).

Since v ∈ L1,∞0 , an application of Theorem 5.4 to the function h = u + iv

implies that for each r ∈ (0, 1) the harmonic extension of h to D satisfies

h = limA→∞

i

∫T

ζ + z

ζ − zvA(ζ)dm(ζ)

= limA→∞

∑1�n�|A|

(−χE+n+ iχE+

n+ χE−

n− iχE−

n)

uniformly on |z|� r. That is to say, the series representation

h =∑n�1

(−χE+n+ iχE+

n+ χE−

n− iχE−

n)

is valid on |z|� r and the convergence is uniform. Consequently,

f = expπh = exp

⎛⎝π∑n�1

(−χE+n+ iχE+

n+ χE−

n− iχE−

n)

⎞⎠(6.3)

=∏n�1

exp[π(−χE+n+ iχE+

n)]

exp[π(−χE−n+ iχE−

n)]

=∏n�1

f+n

fE−n

uniformly on |z|� r. �

Corollary 6.4. Suppose that f = IfF is a non-constant function in R+,where If is inner and F is outer. Then there exist inner functions ϕ+

n and ϕ−n so

that

(6.5) f = |f(0)|K(If )∏n�1

(1− iϕ+

n

1 + iϕ+n

)(1 + iϕ−

n

1− iϕ−n

),


where the product converges uniformly on compact subsets of D. The factor K(If )is to be ignored if If is constant. Moreover, if f belongs to Rp, then the infiniteproduct belongs to Rp.

Proof. This follows from Theorem 3.7 and 6.1. �

6.2. Absolute Convergence. Theorem 6.1 does not guarantee absolute con-vergence and it is not clear whether absolute convergence occurs in general. If

(6.6)∑n�1

(m(E+n ) +m(E−

n )) <∞,

then v ∈ L1 so Theorem 4.11 ensures that∏n�1

fE+n

and∏n�1

fE−n

converge separately – the convergence is absolute and locally uniform on D. So inthis case, the product (6.2) converges absolutely.

The Nth partial product of (6.3) is∏1�n�N

exp[π(−χE+n+ iχE+

n)]

exp[π(−χE−n+ iχE−

n)].

Since the harmonic conjugates χE±nvanish at the origin, the value of the preceding

product at 0 is

exp[πi(m(E+

n )−m(E−n ))]

.

Consequently, the product in (6.2) converges absolutely at 0 if and only if

(6.7)∑n�1

∣∣m(E+n )−m(E−

n )∣∣ <∞.

It is not clear if there is a function in RO for which (6.7) fails. This question wasposed in [16] and, frustratingly, remains open. We hope that some spirited readerwill someday be able to resolve this.

This question can be viewed in terms of distribution functions. Write v =v+− v−, where v+ and v− are non-negative. Then λv+ is equal to m(E+

n+1) on the

interval (n, n+1] and λv− is equal to m(E−n+1) on that same interval. Consequently,

(6.6) converges ⇐⇒∫ ∞

0

(λv+(t)− λv−(t)

)dt converges,

where the integral above is regarded as an improper Riemann integral. In otherwords, we have

(6.7) converges ⇐⇒ (λv+ − λv−) ∈ L1.

The next observation is from [16]. It follows by noting that (i) for any real-valuedmeasurable function w on T, the absolute integrability of λw+

− λw− on [0,∞) isequivalent to the same condition for any bounded perturbation of w; (ii) any w asin (b) (see below) is a bounded perturbation of such an integer-valued w

Theorem 6.8. The following are equivalent: (a) For every function f ∈ RO,the infinite product in the theorem converges absolutely at 0; (b) If w is the conjugateof a real-valued function in L1, then λw+

− λw− is absolutely integrable on [0,∞).


Example 6.9. In this example, we produce an f ∈ RO with nonintegrableargument, but such that the product in Theorem 6.1 converges absolutely on D.Suppose that f = exp[π(u+ iv)] ∈ RO is such that

(a) v(ζ) = −v(ζ) for a.e. ζ ∈ T;

(b) v is positive on the upper half of T;

(c) v is nonincreasing (with respect to θ) on the upper half of T.

Conditions (a), (b), and (c) ensure that E+n is an arc in the upper half of T with

one endpoint at 1 and E−n is the reflection of E+

n across the real line. Let the otherendpoint be denoted eiαn . By (3.25), we have

f+n (z)

f−n (z)

=(z − eiαn)(z − e−iαn)

(1− z)2.

Therefore,

f+n (z)

f−n (z)

− 1 =2z(1− cosαn)

(1− z)2

=4z sin2 αn

2

(1− z)2

= O

(α2n

(1− z)2

).

Since f ∈ RO, we know that u ∈ L1 and v ∈ L1,∞0 , so (5.2) tells us that

αn = m(En) = o

(1

n

).

Thus, ∑n�1

∣∣∣∣f+n (z)

f−n (z)

− 1

∣∣∣∣ <∞,

with uniform convergence on compact subsets of D. This establishes the absoluteconvergence of the product.

6.3. A sufficient condition. Theorem 6.1 shows that any function in ROenjoys a locally uniformly convergent bilateral product representation. As we haveseen above, this does not necessarily provide us with absolute convergence. Weinvestigate here a simple criterion which, given inner functions ϕ+

n and ϕ−n , imply

the absolute and locally uniform convergence of the bilateral product

(6.10)∏n�1

T (ϕ+n )

T (ϕ−n )

To this end, we require the following simple lemma.

Lemma 6.11. For z = i and w = −i,

(6.12) 1− T (z)

T (w)=

2i(z − w)

(1 + iz)(1− iw).

Proof. This is a straightforward computation:

1− T (z)

T (w)= 1−

(i1− iz

1 + iz

)(i1− iw

1 + iw

)−1


= 1−(1− iz

1 + iz

)(1 + iw

1− iw

)=

(1 + iz)(1− iw)− (1− iz)(1 + iw)

(1 + iz)(1− iw)

=(1 + iz − iw + zw)− (1− iz + iw + z + w)

(1 + iz)(1− iw)

=2i(z − w)

(1 + iz)(1− iw). �

When considering bilateral products (6.10), it is natural to assume that forr ∈ (0, 1), the inner functions ϕ+

n and ϕ−n are bounded away from i and −i as

n → ∞, respectively, on |z|� r. This guarantees that T (ϕ+n ) and T (ϕ−

n ) arebounded away from ∞ and 0 as n→∞, respectively, on |z|� r.

Theorem 6.13. Suppose that ϕ+n and ϕ−

n are two sequences of inner functionsso that for r ∈ (0, 1), the inner functions ϕ+

n and ϕ−n are bounded away from i and

−i on |z|� r as n→∞, respectively. Then (6.10) converges absolutely and locallyuniformly on D if and only if

∑n�1|ϕ+

n − ϕ−n | converges locally uniformly on D.

Proof. Fix r ∈ (0, 1) and suppose that

0 < δ < sup|z|�r

|1± iϕ±n |

for |z|� r. Since |1± iϕ±n |� 2 on D, (6.12) tells us that

1

2

∑n�1

|ϕ+n − ϕ−

n |�∑n�1

∣∣∣∣1− T (ϕ+n )

T (ϕ−n )

∣∣∣∣ � 2

δ2

∑n�1

|ϕ+n − ϕ−

n |. �

Example 6.14. Suppose that ϕ+n is a sequence of singular inner functions so

that for r ∈ (0, 1), ϕ+n is bounded away from ±i on |z|� r as n → ∞. Now recall

that the Blaschke products are uniformly dense in the set of all inner functions[17, Cor. 6.5]. Let ϕ−

n be a sequence of Blaschke products for which∑n�1

‖ϕ+n − ϕ−

n ‖∞<∞.

Then the product (6.10) converges.

7. Real complex functions in operator theory

We end this survey with a few connections the real complex functions makewith operator theory. Since these vignettes are applications and not the mainstructure results of these real functions, as was the rest of the survey, we will be abit skimpy on the details, referring the interested reader to the original sources inthe literature.


7.1. Riesz projections for 0 < p < 1. At first glance, the very title ofthis subsection refers to an absurdity. Any serious analyst knows that the Rieszprojection operator ∑

z∈Z

f(n)ζn �→∑n�0

f(n)ζn

cannot be properly defined for functions in Lp when 0 < p < 1. Indeed, one cannoteven speak of Fourier series for such functions. Bear with us.

For f ∈ L1 with Fourier series

f ∼∑n∈Z

f(n)ζn,

we may consider the “analytic part”∑n�0

f(n)ζn

of this series. When 1 < p <∞ and f ∈ Lp, the function

Pf =∑n�0

f(n)ζn

belongs to Hp and the linear transformation f �→ Pf is a bounded projection fromLp onto Hp and is called the Riesz projection. In fact, [24] tells us that

‖P ‖Lp→Hp= csc

(π

p

).

As mentioned earlier, this is no longer true when p = 1 or p =∞.

In this section we follow [14] and show that an analog of the Riesz projectioncan be defined on Lp when 0 < p < 1 by working modulo the complexificationof Rp. In other words, the functions in Rp are the culprit since their presence isthe source of the unboundedness of the Riesz projection. Indeed, if f ∈ Rp, thenf = f a.e. on T and so the intersection Hp∩Hp (in terms of boundary values on T)contains many non-constant functions. This, it turns out, is the only obstruction.

The set R+ of all real Smirnov functions is a real subalgebra of N+. It is naturalto consider the complexification of R+:

(7.1) C+ := {a+ ib : a, b ∈ R+}.This is a complex subalgebra of N+. With respect to the translation invariantmetric (2.10) it inherits from N+ [8], C+ is both a complete metric space and atopological algebra.

It is evident from the definition (7.1) that the set of boundary functions cor-responding to the elements of C+ is closed under complex conjugation. Indeed, ifa, b ∈ R+, then, in terms of boundary functions defined for a.e. ζ ∈ T,

a(ζ) + ib(ζ) = a(ζ)− ib(ζ) ∈ C+.

Consequently, C+ carries a canonical involution. Indeed, if f = a + ib, wherea, b ∈ R+, then we define

f = a− ib ∈ C+.

The · operation is a conjugation on C+: It is conjugate linear, involutive, andisometric. Moreover, · preserves outer factors since(7.2) |f |2= |a|2+|b|2= |f |2 a.e. on T.


We now consider the intersection of the algebra C+ with the Hardy spaces Hp.For each p let

Cp := C+ ∩Hp.

Theorem 3.8 implies that Cp contains non-constant functions only when p ∈ (0, 1).It turns out that Cp is the appropriate complex analogue of Rp needed to define a“Riesz projection” on Lp for 0 < p < 1.

The metric on Hp dominates the metric (2.10) on N+, so we conclude that Cp

is a closed subspace of Hp for each p ∈ (0, 1). Moreover, Cp is closed under theconjugation · since (7.2) ensures that it is isometric on Hp.

We leave it to the reader to verify the following theorem from [14].

Theorem 7.3. For 0 < p < 1, the following sets are identical.

(a) Cp.

(b) Rp + iRp.

(c) Hp ∩Hp (as boundary functions).

Let 0 < p < 1. Since Cp is a closed subspace of Lp, the quotient Lp/Cp isan F -space under the standard quotient metric. In other words, if [f ] denotes theequivalence class modulo Cp of a function f ∈ Lp, then

‖ [f ] ‖p:= infσ∈Cp

‖ f − σ ‖pp

induces a translation invariant metric

ρ([f ], [g]) := ‖ [f ]− [g] ‖ppwith respect to which Lp/Cp is complete. Similarly, we can regard Hp/Cp as aclosed subspace of Lp/Cp with respect to this metric.

A simple modification of a theorem of Aleksandrov [4] says that we can decom-pose each f ∈ Lp, 0 < p < 1, as

f = u+ v, u ∈ Hp, v ∈ Hp,

with some control over ‖u‖p and ‖v‖p [14]. It is easily seen that this decompositionis unique modulo Cp and hence each equivalence class [f ] ∈ Lp/Cp decomposesuniquely as

[f ] = [u] + [v].

We can therefore define the Riesz projection

(7.4) P : Lp/Cp → Hp/Cp, P [f ] := [u].

In a way, this map is an analogue of the Riesz projection operator from Lp to Hp

for p ∈ (1,∞). Indeed, if we regard equivalence classes as collections of boundaryfunctions, then P [f ] ⊆ Hp and the Riesz projection returns the “analytic part” of[f ]. The main theorem here is from [14].

Theorem 7.5. The Riesz projection from (7.4) is bounded for each p ∈ (0, 1).For every f ∈ Lp we have ‖P [f ] ‖p� Kp‖ [f ] ‖p.


7.2. Kernels of Toeplitz operators. For each ϕ ∈ L∞ one can define theToeplitz operator [6]

Tϕ : H2 → H2, Tϕf = P (ϕf),

where P : L2 → H2 is the Riesz projection. When ϕ ∈ H∞, Tϕf = ϕf is amultiplication operator (a Laurent operator). The kernel kerTϕ has been wellstudied [13,18–20,29] and relates to the broad topic of “nearly invariant” subspacesof H2. In particular there is the following theorem of Hayashi [18].

Theorem 7.6. If kerTϕ = {0}, then there is an outer function F ∈ H2 suchthat kerTϕ = kerTzF/F .

The connection to R+ is the following [15]:

Theorem 7.7. If F ∈ H2 is outer then

kerTzF/F = {(a+ ib)F : a, b ∈ R+} ∩H2.

7.3. A connection to pseudocontinuable functions. A widely studiedtheorem of Beurling [11,17] says that the invariant subspaces of the shift operator

S : H2 → H2, (Sf)(z) = zf(z),

take the form uH2 where u is an inner function. Taking annihilators shows thatthe invariant subspaces of the backward shift operator

S∗ : H2 → H2, (S∗f)(z) =f(z)− f(0)

z,

are of the form (uH2)⊥. Functions in (uH2)⊥ are often called the pseudo-continuable functions due to a theorem in [10] (see also [28]) which relates eachf ∈ (uH2)⊥ with a meromorphic function on the exterior disk via matching radialboundary values. Along the lines of our discussion of Toeplitz operators, we havethe following result [15].

Theorem 7.8. For an inner function u and ζ ∈ T for which

limr→1−

u(rζ) = u(ζ)

exists, define

kζ(z) =1− u(ζ)u(z)

1− ζz.

Then we have(uH2)⊥ = {(a+ ib)kζ : a, b ∈ R+} ∩H2.

We point out that similar results hold in Hp when 1 � p <∞.

7.4. A connection to unbounded Toeplitz operators. For ϕ ∈ R+ onecan define the unbounded Toeplitz operator by first defining its domain

D = {f ∈ H2 : ϕf ∈ H2}and then defining the operator Tϕ by

Tϕ : D ⊆ H2 → H2, Tϕf = ϕf, f ∈ D .

Helson [21] showed that the domain D of Tϕ is dense in H2: If Sf = zf is the shiftoperator on H2 then SD ⊆ D and so SD− ⊆ D−. By Beurling’s classification of


the S-invariant subspaces of H2 we have D− = IH2 for some inner function I. Ifg ∈ H2 and Ig ∈ D , then |ϕg|2= |ϕ(gI)|2 and this last quantity is integrable sinceϕ(gI) ∈ H2. This means that g ∈ D and so the common inner divisor I of D isequal to one, making D− = H2.

In [30] Sarason identified D as follows: One can always write ϕ as

ϕ =b

a,

where a, b ∈ H∞, a is outer, a(0) > 0, and |a|2+|b|2= 1 a.e. on T. Sarason showedthat

D = aH2

and so, since a is outer, one can see, via Beurling’s theorem [11, p. 114], that D isdense in H2. This verifies what was shown by Helson above.

Since ϕ ∈ R+, we have

〈Tϕf, g〉 = 〈f, Tϕg〉, f, g ∈ D ,

and thus Tϕ is an unbounded symmetric operator on H2. Furthermore, Tϕ is alsoa closed operator. General theory of symmetric operators [2] says that Tϕ−wI hasclosed range for every w ∈ R. Furthermore, if

η(w) = dim((ran(Tϕ − wI))⊥),

where ran denotes the range, then η is constant on each of the half planes {Im z > 0}and {Im z < 0}. The numbers η(i) and η(−i) are called the deficiency indices ofTϕ. The following comes from Helson [21].

Theorem 7.9.

(a) If ϕ ∈ R+ and η(i) and η(−i) are finite, then ϕ is a rational function.

(b) Given any pair (m,n), where m,n ∈ N∪{∞}, there is a ϕ ∈ R+ such thatη(i) = m and η(−i) = n.

Cowen [9] showed that two analytic Toeplitz operators Tϕ1, Tϕ2

, where ϕ1, ϕ2 ∈H∞, are unitarily equivalent if and only if ϕ1 = ϕ2 ◦ψ for some automorphism ψ ofD. A similar result was shown in [5] when ϕ1, ϕ1 ∈ R+ and Tϕ1

, Tϕ2have deficiency

indices (1, 1).

7.5. Value distributions. For ϕ ∈ R+ the connection to unbounded Toeplitzoperators (from the previous section) points out a useful connection to

card{z : ϕ(z) = β}, β ∈ R.

The unbounded symmetric Toeplitz operator Tϕ has a densely defined adjoint T ∗ϕ

and the Cauchy kernels

kλ(z) :=1

1− λzbelong to the domain of T ∗

ϕ. Furthermore, standard arguments show that

T ∗ϕkλ = ϕ(λ)kλ.

Since(ran(Tϕ − wI))⊥ = ker(T ∗

ϕ − wI),

we see thatker(T ∗

ϕ − wI) =∨{kλ : ϕ(λ) = w},


where∨

denotes the closed linear span, and

dim(ker(T ∗ϕ − wI)) = card{λ : ϕ(λ) = w}.

Furthermore, from our earlier discussion of the deficiency indices of (unbounded)symmetric operators, the function

w �→ card{λ : ϕ(λ) = w}is constant on each of the connected regions {Im z > 0} and {Im z < 0}.

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Department of Mathematics, Pomona College, Claremont, California 91711


Departement de Mathematiques et de Statistique, Universite Laval, Quebec, QC,

G1K 7P4, Canada


Department of Mathematics and Computer Science, University of Richmond, Rich-

mond, Virginia 23173



Thin Interpolating Sequences

Pamela Gorkin and Brett D. Wick

Abstract. We look at thin interpolating sequences and the role they play inuniform algebras, Hardy spaces, and model spaces.

“One of the striking successes of the function algebra viewpoint has been in thestudy of the algebra H∞ of all bounded holomorphic functions on the unit disk D.Not only are the results which have been obtained deep but the questions raisedhave also enriched the classical study of the boundary behavior of holomorphicfunctions.” R. G. Douglas, Mathematical Reviews, (MR0428044 and MR428045)

1. The beginning of interpolation in Hardy spaces

R. C. Buck proposed the idea of characterizing interpolating sequences for H∞,the algebra of bounded analytic functions on the open unit disk D, via an explicitcondition on the sequence, [30]. Buck conjectured that if a sequence of points in D

approached the boundary quickly enough, it would be interpolating for the algebraH∞; that is, for all {wn} ∈ ∞ there exists f ∈ H∞ such that f(zn) = wn for alln. We discuss briefly the background on interpolating sequences before turning tothin interpolating sequences.

In 1958, W. Hayman [19] proved the following theorem.

Theorem 1.1 (Hayman). A necessary condition for a sequence {zn} to be aninterpolating sequence is that there exist a constant C > 0 such that

infn

∏m:m �=n

∣∣∣∣ zm − zn1− znzm

∣∣∣∣ ≥ C.(1.1)

A sufficient condition is that there exist λ < 1 and C1 > 0 so that

infn

∏m:m �=n

[1−(1−∣∣∣∣ zm − zn1− znzm

∣∣∣∣λ)]≥ C1.

2010 Mathematics Subject Classification. Primary 46E22; Secondary 30D55, 47A, 46B15.Key words and phrases. Reproducing kernel, thin sequences, interpolation, asymptotic or-

thonormal sequence.The first author’s research was supported in part by Simons Foundation Grant 243653.The second author’s research was supported in part by National Science Foundation DMS

grants # 0955432 and #1500509.


129

130 P. GORKIN AND B. D. WICK

Hayman wrote, “It seems quite possible that (1.1) is in fact sufficient as well asnecessary, but I have been unable to prove this.” Hayman’s proof was constructiveand provided other very useful estimates on the sequence. Independently and alsoin 1958, Carleson [5] presented the condition Buck anticipated: If {zn} is a Blaschkesequence and B the corresponding Blaschke product, then {zn} is interpolating ifthere exists δ > 0 such that

infn

∏m �=n

∣∣∣∣ zm − zn1− zmzn

∣∣∣∣ = infn(1− |zn|2)|B′(zn)| ≥ δ > 0.(1.2)

In 1961, Shapiro and Shields [30] considered interpolation in the Hardy spaceHp and described it as a weighted interpolation problem. For f ∈ Hp, they definedan operator Tp by

Tpf ={f(zj)(1− |zj |p)1/p

}∞

j=1,

and asked when Tp(Hp) = p, where p is the space of p-summable sequences; when

p =∞, this is precisely the requirement that {zn} be interpolating for H∞. In thetheorem below, δjk = 1 if j = k and 0 otherwise.

Theorem 1.2 (Shapiro, Shields, 1961, [30]). For 1 ≤ p ≤ ∞ the necessary andsufficient condition for TpH

p = p is that there exist functions fk ∈ Hp such that

(1) fk(zj)(1− |zj |2)1/p = δjk;(2) ‖fk‖p ≤ 1/δ.

They note that in H1 an interpolating function can be given explicitly. If we

assume that

∞∑k=1

|wk|(1 − |zk|2) < ∞, then letting B denote the Blaschke product

corresponding to {zn},

f(z) =∑k

B(z)

B′(zk)

(1

z − zk+

zk1− zkz

)solves the problem.

In addition, it was also known that the interpolating function f could be chosento satisfy

|f(z)| ≤ 2

δ5

(1 + 2 log

1

δ

)supn|wn|.

Three natural questions then arise: First, when interpolating in Hp what is thebest bound on the p-norm of an interpolating function? We see that, as δ → 0, thecorresponding upper bounds approach infinity, which is expected, but as δ → 1,this point approaches twice what one would hope it would approach. Second, inview of Shapiro and Shield’s explicit demonstration of a function in H1 that doesthe interpolation, another natural question is whether we can exhibit the functionexplicitly. Finally, we might ask if these two questions can be combined; that is,can we explicitly exhibit the function of best norm that does the interpolation?

2. Best bounds and best functions

If we have an interpolating sequence for H∞, then the map T : H∞ → ∞

defined by T (f) = {f(zn)} is a bounded surjective map. If we let B denote the cor-responding Blaschke product, then T induces a bijective map T : H∞/BH∞ → ∞.

THIN INTERPOLATING SEQUENCES 131

Thus, as a consequence of the open mapping theorem, given w := {wn} ∈ ∞, thereis a constant C such that

‖f‖∞ ≤ C‖w‖∞.

The smallest such constant is called the constant of interpolation and depends onthe separation constant δ, so it is often denoted by M(δ). So we are after threethings: An estimate on M(δ), an explicit expression for a function that does theinterpolation, and the connection between the two. We will begin with Earl’sestimates.

In 1970, J. P. Earl [11] showed that if an interpolating sequence {zn} satisfies(1.2) then for any M such that

M >2− δ2 + 2(1− δ2)1/2

δ2supn|wn|(2.1)

there exists a Blaschke product B such that

MeiαB(zj) = wj for all j.

Though Earl’s result shows roughly where the zeros of the Blaschke product lie,it does not give an explicit expression for the function that does the interpolation.Now, if we restrict our sequences {wj} so that ‖w‖∞ ≤ 1, let δ = δ(B) = infn(1−|zn|2)|B′(zn)| and use a normal families argument, we obtain a function of norm atmost

2− δ2 + 2(1− δ2)1/2

δ2=

(1 + (1− δ2)1/2

δ

)2

that does the interpolation. In particular, Earl’s theorem gives us an estimate onM(δ).

If the original sequence {zj} has the property that the corresponding Blaschkeproduct C satisfies δj(C) = |Cj(zj)| → 1, then the function that does the inter-polation may be chosen to have this property as well; that is, it can be chosen tobe a so-called thin Blaschke product (this result was stated in [10] with a generalplan of attack; details appear in [25]). The proof involves adapting the proof ofJ. P. Earl to this situation. In general, however, the “best” function that does theinterpolation may not be a unimodular constant times a Blaschke product, but wesee that the closer δ is to 1, the closer the norm of f is to 1. Thus, interpolatingBlaschke products for which δj(B)→ 1 as j →∞ would seem to have particularlyinteresting properties and, indeed, they have been closely studied. Such sequencesare called thin sequences and, if we require that they also be interpolating, they arethin interpolating sequences.

Definition 2.1. A Blaschke product B with zero sequence {zn} satisfying

limn|Bn(zn)| = lim

n

∏m �=n

∣∣∣∣ zm − zn1− zmzn

∣∣∣∣ = limn(1− |zn|2)|B′(zn)| = 1

is called a thin Blaschke product and the sequence {zn} is said to be a thin sequence.

Note that thin sequences may have finitely many points that appear finitelymany times, but they cannot repeat infinitely many points. We will assume, unlessotherwise stated, that our thin sequences are interpolating sequences, so that pointsare distinct. We now present some examples of thin sequences. Recall that for

z, w ∈ D the pseudohyperbolic distance between z and w in D is ρ(z, w) =∣∣∣ z−w1−wz

∣∣∣.


Example 1. Every Blaschke sequence has a thin subsequence.

Proof. Let εj be an increasing sequence with 0 < εj < 1 so that δj =∏k �=j εk → 1. Beginning with zn1

= z1, choose zn2so that ρ(z1, zn2

) ≥ ε2. Assum-

ing zn1, . . . , znj

have been chosen, we choose znj+1with ρ(znk

, znj+1) > εj+1 for all

k ≤ j. If C is the Blaschke product corresponding to {znj}, then

|Cj(znj)| ≥

∏k �=j

ρ(znk, znj

) ≥∏k �=j

εk → 1,

as j →∞. �

The next example below is due to W. Hayman who, in working towards estab-lishing a condition that a sequence be interpolating, proved the following theorem:

Theorem 2.2 (W. Hayman). A sufficient condition for a sequence of distinctpoints, {zn}, to be interpolating is that

lim supn→∞

1− |zn+1|1− |zn|

< 1.

If 0 < zn < 1 and {zn} is increasing, then the condition is also necessary.

This theorem follows from the previous theorem due to Hayman, Theorem 1.1,above. A careful inspection of his proof establishes the following.

Example 2. Let {zn} be a sequence of distinct points satisfying

1− |zn+1|1− |zn|

→ 0.

Then {zn} is a thin sequence.

Here is a rough idea of why this is true (see [7, Proposition 4.3 (i)] for a differentproof): Let k ∈ N and suppose

1− |zn+1| ≤ ck(1− |zn|) for n ≥ k.

Note that we are assuming that ck → 0. Now for points z, w in D the pseudohy-perbolic distance between z and w satisfies

ρ(z, w) ≥ |z| − |w|1− |zw| .

Therefore, if we break the product∏

j �=k

∣∣∣ zk−zj1−zjzk

∣∣∣ into two pieces, for j > k we will

have1− |zj | ≤ cj−k

k (1− |zk|).Consequently

|zj | − |zk| ≥ (1− cj−kk )(1− |zk|).

But1− |zjzk| ≤ (1 + cj−k

k )(1− |zk|).Thus, ∏

j>k

∣∣∣∣ zk − zj1− zjzk

∣∣∣∣ ≥ 1− cj−kk

1 + cj−kk

.

For j < k, we have1− |zj+1| ≤ cj(1− |zj |).


Thus,

1− |zk| ≤k−1∏l=j

cl(1− |zj |) and |zk| − |zj | ≥

⎛⎝1−k−1∏l=j

cl

⎞⎠ (1− |zj |),

while

1− |zkzj | ≤

⎛⎝1 +

k−1∏l=j

cl

⎞⎠ (1− |zj |).

So ∏j<k

∣∣∣∣ zk − zj1− zjzk

∣∣∣∣ ≥ 1−∏k−1

l=j cl

1 +∏k−1

l=j cl.

From this, we conclude that the sequence {zn} is thin.

Our last example is reminiscent of a result of Naftalevitch that says that anyBlaschke sequence can be rotated to form an interpolating sequence. Naftalevitch’stheorem is also a consequence of the following theorem, which can be found in[7, Proposition 4.3 (ii)].

Example 3. Let {zn} be a Blaschke sequence. Then there is a thin interpo-lating sequence {wn} with |wn| = |zn|.

The proof actually constructs the sequence. Supposing that {rn} is increasing,choose a sequence of positive numbers bn with 1−rn

bn+1→ 0. Without loss of generality,

we may assume that∞∑

n=1

bn <π

2. Let θn =

n∑k=1

bk and λn = rneiθn . Then, as shown

in [7], this sequence is thin.This example is quite similar to an example developed by Joel H. Shapiro in

the context of composition operators. So recall that if T is an analytic self-mapof the unit disk, a composition operator on the Hardy space H2 is defined byCT (f) = f ◦ T . Once one has checked that the operator is bounded, which itis (see, for example, [28]), it is natural to study when it is compact. It is nowknown [28] that if CT is compact, then the angular derivative of T exists at nopoint of the unit circle. Shapiro’s example was developed to show that the angularderivative condition is not sufficient to imply that the operator is compact. Thatthe Blaschke product Shapiro constructed is actually thin was first noticed by D.Suarez and proved in [13], but the application of ideas in [7] simplifies the proofsignificantly.

Example 4. [29, p. 184] Wrap intervals Ik of length 1k about the unit circle,

placing a zero of the Blaschke product B at the point(1− 1

k2

)eiθk , where eiθk is

the center of the arc Ik. Then B is a thin Blaschke product for which the angularderivative does not exist at any point of the unit circle.

Though out of sequence chronologically, we complete this section with answersto the questions we discussed above. The first is an explicit description of thefunctions that solve the interpolation problem.

P. Beurling [6] showed that given an interpolating sequence {zj}, there existfunctions fj in H∞ with the property that fj(zj) = 1 and fj(zk) = 0 for j = k


such that

supz∈D

∑j

|fj(z)| <∞.(2.2)

In fact, the bound in (2.2) can be connected to the separation constant δ, but thefunctions are not given explicitly. Instead, the first explicit description is due toPeter Jones and appeared in 1983, [22]. The functions are given as follows:

fj(z) :=Bj(z)

Bj(zj)

(1− |zj |2

1− zjz

)2

e

(− 1

2C(δ)

∑|zm|≥|zj |

(1+zmz1−zmz−

1+zmzj1−zmzj

)(1−|zm|2)

).

It is easy to see that fj(zj) = 1 and fj(zk) = 0 for j = k. It is, of course, muchharder to see that

∞∑j=1

|fj(z)| < M for all z ∈ D,

but it is true and can be shown using a computation that culminates in a Riemannsum that yields the result.

Thus for any a ∈ ∞, if we let f(z) =∞∑j=1

ajfj(z) ∈ H∞(D) we see that

f(zj) = aj ; |f(z)| ≤ ‖a‖�∞

⎛⎝ ∞∑j=1

|fj(z)|

⎞⎠ ≤ C(δ)‖a‖�∞ .

In 2004, Nicolau, Ortega-Cerda, and Seip [26], modifying the explicit formulasgiven by Jones, were able to provide sharp upper and lower bounds on the interpo-lation constant. For thin sequences, the following version of P. Beurling’s theoremwas proven in [16] using the commutant lifting theorem.

Definition 2.3. In what follows, we let

‖a‖N,�∞ = supj≥N

|aj |

Theorem 2.4. Let {zn} be a thin sequence. Then for every ε > 0 there existsN such that for n ≥ N there exist fn ∈ H∞ such that for j, k ≥ N we have

fn(zn) = 1 and fn(zk) = 0, for j = k,

and

supz∈D

∑n≥N

|fn(z)| < (1 + ε).

In particular, for every sequence a ∈ ∞ with ‖a‖�∞ ≤ 1 the function ga defined byga(z) :=

∑n≥N anfn(z) ∈ H∞ satisfies ‖ga‖∞ ≤ (1 + ε)‖a‖N,�∞ and ga(zj) = aj

for j ≥ N .

This theorem can be proved using Earl’s estimate, but the proof requires a briefintroduction to the maximal ideal space of H∞. We turn to that introduction nowand return to Theorem 2.4 once we have established the basics.


3. The maximal ideal space

One of the tools that is most useful in the study of H∞ as a uniform algebrais its maximal ideal space, M(H∞), or the space of nonzero multiplicative linearfunctionals. It is called the maximal ideal space because the kernel of a nonzeromultiplicative linear functional is a maximal ideal and, conversely, every maximalideal is the kernel of a multiplicative linear functions.

When endowed with the weak-∗ topology, M(H∞) is a compact Hausdorffspace. By studying certain partitions of the maximal ideal space, mathematicianswere able to shed light on the behavior of functions in H∞. The disk, D, can beidentified with a subset of the maximal ideal space, by identifying the point z withthe functional that is point evaluation at z and Carleson’s corona theorem says thatthe disk is dense in M(H∞).

That, plus the following theorem, are the last two ingredients that we need inour proof of Theorem 2.4.

Theorem 3.1. Let A be a uniform algebra on a compact space X and let{x1, . . . , xn} be a finite set of points in X. If

M = sup‖a‖∞≤1

inf {‖g‖A : g ∈ A, g(xj) = aj , j = 1, 2, . . . , n} ,

then for every ε > 0 there are functions fj ∈ A for which

fj(xj) = 1 and fj(xk) = 0 for k = j

and

supx∈X

n∑j=1

|fj(x)| ≤M2 + ε.

In our case, the compact space X is M(H∞) and our points xj will be zj ∈ D.Note that in this case – that is, when A = H∞ – a normal families argumentimplies that we can find a sequence fj such that fj(zj) = 1, fj(zk) = 0 and∑

z∈D|fj(z)| ≤M2.

Proof of Theorem 2.4. Recall that δn = |Bn(zn)|. Let ε > 0 be given.Then, since δn → 1, there exists N such that

(3.1)

(2− δ2n + 2(1− δ2n)

1/2

δ2n

)2

< 1 + ε for n ≥ N.

Consider the sequence {zj}j≥N and let δ be the separation constant for this se-quence. By Earl’s estimate, (2.1), we know that given w ∈ ∞ with ‖w‖∞ ≤ 1,there exists a function f ∈ H∞ such that f(zj) = wj for all j ≥ N and ‖f‖∞ ≤2−δ2+2(1−δ2)1/2

δ2 . By (3.1) we see that ‖f‖∞ ≤ 1 + ε. Therefore, by Theorem 3.1and a normal families argument, we know that there are functions fj such that forj, k ≥ N we have

fj(zj) = 1, fj(zk) = 0 for j = k and supz∈D

∞∑j=N

|fj(z)| ≤ 1 + ε.

�

The maximal ideal space is particularly useful here, but in spite of our famil-iarity with the open dense set D, the set M(H∞) \D is difficult to understand. As


luck would have it, and as Sarason showed, the space H∞+C, consisting of sums of(boundary) functions inH∞ and continuous functions on the unit circle T is a closedsubalgebra [28] of L∞ and the maximal ideal space is M(H∞ +C) = M(H∞) \D– precisely the set we don’t understand well.

It turns out that the analytic structure that we have in D does, in a certainsense to be made precise, carry over to M(H∞) as we see from the Gleason parts.For ϕ1, ϕ2 ∈M the pseudohyperbolic distance is

ρ(ϕ1, ϕ2) = sup{|f(ϕ2)| : f ∈ H∞, ‖f‖∞ ≤ 1, f(ϕ1) = 0}.Points are in the same Gleason part if ρ(ϕ1, ϕ2) < 1 and this defines an equivalencerelation on M(H∞); the equivalence classes are the Gleason parts. One equivalenceclass is the unit disk and the others lie in M(H∞+C). In trying to understand theparts in M(H∞) \ D, Hoffman considered, for each point α ∈ D, the map Lα(z) =(z + α)(1 + αz)−1. Given a net of points (αβ) converging to a point ϕ in M(H∞),the corresponding maps Lαβ

converge to a one-to-one map Lϕ : D→M(H∞) andLϕ(D) = P (ϕ), the Gleason part of ϕ. The map Lϕ imparts an analytic structure

on P (ϕ): If f ∈ H∞, we can define the Gelfand transform of f , denoted f , on

M(H∞) by f(ϕ) = ϕ(f) and then f ◦ Lϕ is an analytic function on D. It is

customary to drop the “hat” and refer to f even when using the function f .One of Hoffman’s goals was to show that a point ϕ ∈ M(H∞ + C) is in the

closure of an interpolating sequence if and only if the map Lϕ is not constant.Though Hoffman’s work allowed mathematicians to use the analytic structure ofthe parts as a tool, that does not mean that the parts are tractable. For example,parts may or may not look like the disk, but the points ψ that lie in the closureof a thin part are always homeomorphic to the disk, as noted by Hoffman in hisseminal paper, [21].

Proposition 3.2. Let {αn} denote a thin sequence and B the correspondingBlaschke product. Then for any point ϕ ∈M(H∞ +C) in the closure of {αn}, theGleason part is homeomorphic to the unit disk.

Proof. By assumption, we know that limn(1− |αn|2)|B′(αn)| = 1. Then (B ◦

Lαn)(0) = 0 for all n and |(B ◦ Lαn

)′(0)| = |B′(αn)|(1 − |αn|2) → 1. So whatever

B ◦ Lϕ is, we know that (B ◦ Lϕ) : D→ D, B ◦ Lϕ(0) = 0, and |(B ◦ Lϕ)′(0)| = 1.

Since B ◦ Lϕ is also analytic, Schwarz’s lemma shows that B ◦ Lϕ(z) = λz forsome λ of modulus 1. Therefore if ϕ is in the closure of a thin sequence, Lϕ is a

homeomorphism and its inverse is a (unimodular) constant multiple of B. �

To the best of our knowledge, this is the first appearance of thin sequencesin the literature, and this has interesting implications. By Hoffman’s work, itturns out that a Blaschke product for which the zeros form a thin interpolatingsequence are indestructible; that is, if you take an automorphism Ta(z) = λ z−a

1−az

(with a ∈ D and λ in the unit circle, T) and consider Ta ◦ B, this will again bea thin Blaschke product (though finitely many zeros may be repeated). This factand Proposition 3.2 imply that if a thin Blaschke product is of modulus less thanone on a part, then it has exactly one zero on that part.

Proposition 3.3. If B = B1B2 is a factorization of a thin Blaschke product,then for each ϕ ∈M(H∞+C) either |B1◦Lϕ(z)| = 1 for all z ∈ D or |B2◦Lϕ(z)| =1 for all z ∈ D.


Proof. Let ϕ ∈ M(H∞ + C). If |B1(ϕ)| < 1, then C1 := TB1(ϕ) ◦ B1 is still

thin and C1 ◦ Lϕ(z) = λ1z. Similarly, if |B2(ψ)| < 1 for some ψ ∈ P (ϕ), then

for the corresponding Blaschke product C2, we have that C2 ◦ Lψ(z) = λ2z. Inparticular, B would have two zeros (counted according to multiplicity) in P (ϕ) andthat is impossible. �

Thus, thin sequences have zeros that are pseudo-hyperbolically far apart in thedisk as well as in M(H∞+C) and it is this separation that made them particularlyinteresting sequences for the study of interpolation.

4. Thin sequences, interpolation and uniform algebras

The strength of the separation of points in the closure of thin sequences isillustrated by a result of T. Wolff. To place it in its proper context, we need tounderstand what happened in the study of closed subalgebras of L∞ containingH∞; the so-called Douglas algebras, in honor of R. G. Douglas who conjecturedthat every such algebra is generated by H∞ and the complex conjugates of innerfunctions invertible in the algebra. That this is true for H∞ + C is Sarason’stheorem: the only invertible inner functions in H∞+C are finite Blaschke productsand Sarason showed that H∞ + C = H∞[z]. After Douglas made his conjecture,he and Rudin [9] showed that L∞ is of the right form. The final result was evenbetter than what Douglas conjectured: Chang and Marshall [8, 24] showed thatevery such algebra was generated by H∞ and the conjugates of the interpolatingBlaschke products invertible in that algebra.

The proof is divided into two pieces. Chang showed that if two Douglas alge-bras had the same maximal ideal space, then they were the same Douglas algebra,while Marshall showed that if A is a Douglas algebra and AI is the (closed) algebragenerated by H∞ and the complex conjugates of the interpolating Blaschke prod-ucts invertible in A, then the maximal ideal space of A, denoted M(A), is equalto the maximal ideal space of AI . Their work requires an understanding of howelements ϕ ∈ M(H∞) “work.” Each ϕ ∈ M(H∞) can be defined by integrationagainst a positive measure with closed support in the maximal ideal space of L∞;that is,

ϕ(f) =

∫suppϕ

fdμϕ,

and given a Douglas algebra, we may think of M(A) as a subset of M(H∞); M(A)can be identified with the multiplicative linear functionals in M(H∞) for whichthe representing measures are multiplicative on A. (See, for example, [14, ChapterIX].)

Sticking with the uniform algebra point of view for a moment, one might wonderwhat happens when one looks at the closed algebra A of H∞ and the conjugatesof all thin interpolating Blaschke products. Hedenmalm [20] showed that an innerfunction is invertible in A if and only if it is a finite product of thin interpolatingBlaschke products.

As this suggests, thin interpolating sequences are very well behaved. Wolffand Sundberg [33], [31] showed, among other things, that these sequences are theinterpolating sequence for the (very small) algebra QA = H∞ + C ∩ H∞ (herethe bar denotes the complex conjugate). This algebra acts, in many ways, likethe disk algebra (for this, [33] is a good resource). We start with the algebra of


quasi-continuous functions: let QC = (H∞ + C) ∩ H∞ + C. The algebra QA isthen QA := QC ∩H∞. The theorem we concentrate on here is the following:

Theorem 4.1 (Wolff, Wolff-Sundberg). The following are equivalent for aninterpolating sequence {zn}.

(1) For any {λn} ∈ ∞ there is f ∈ QA with f(zn) = λn;(2) For any {λn} ∈ ∞, ε > 0, then there is an f ∈ H∞ with ‖f‖∞ <

lim supn→∞

|λn|+ ε and f(zn) = λn all but finitely many n;

(3) limn→∞

∏m �=n

∣∣∣∣ zn − zm1− zmzn

∣∣∣∣ = 1.

Thus, thin sequences are interpolating sequences for a very small algebra and,therefore, they must have a strong separation property. One way to think of thisseparation property is in the maximal ideal space. We first describe the mostnatural partition of M(H∞ + C), namely the fibers.

Definition 4.2. For λ ∈ T, let Mλ = {ϕ ∈M(H∞ + C) : ϕ(z) = λ}. The setMλ is called the fiber over λ.

It is easy to see that the identity function f(z) = z is constant on each fiber.It follows that each continuous function is constant on each fiber as well. But thealgebra QC is strictly larger than C and not all QC functions are continuous oneach fiber. For QC, we need to refine this partition.

Definition 4.3. For each ϕ ∈M(H∞ + C), define Eϕ = {ψ ∈M(H∞ + C) :ϕ(q) = ψ(q) for all q ∈ QC}. The set Eϕ is called the QC-level set correspondingto ϕ.

Note that if f ∈ QC, then f is constant on a QC-level set.

Proposition 4.4. A thin sequence can have at most one cluster point in aQC-level set.

Proof. Suppose {αn} is a thin sequence with two cluster points in Eϕ. Thenthere are two distinct points, ψ1 and ψ2, in the closure of the sequence. ButM(H∞ +C) is a Hausdorff space and therefore we can separate the two points byopen sets U1 and U2 with disjoint closures and choose two disjoint subsets Λ1 andΛ2 of this sequence contained in U1 and U2, respectively. Now using Theorem 4.1,we obtain a function f such that f(αn) = 0 if αn ∈ Λ1 and f(αn) = 1 if αn ∈ Λ2.In particular ψ1(f) = 0 while ψ2(f) = 1. But f ∈ QC and therefore f must beconstant on the QC-level set. Since ψ1 and ψ2 belong to the same level set, this isimpossible. �

The fact that the zeros of a thin Blaschke product that lie in M(H∞+C) mustlie in different QC-level sets is a very strong separation property. This paved theway for further study of the interpolation properties of thin sequences: Can we, asShapiro and Shields did, transfer the study to the Hilbert space H2? What aboutother Hp spaces?

5. Extending the definition of thin to Hp spaces

We have already hinted that thin sequences are the ones for which interpolationcan be done with a very good bound on the norm. If we relax the interpolation


condition a bit, we can study when functions do approximate interpolation withthe best norm possible. To make this precise, we provide a definition that makessense in a wider context – for example, for general uniform algebras. (For moreinformation, see [15].)

Definition 5.1. A sequence {αn} is said to be an asymptotic interpolatingsequence for H∞ if for every sequence {wn} in the ball of ∞, there is an H∞

function f such that ‖f‖∞ ≤ 1 and |f(zn)− wn| → 0.

Following the work in [15], Dyakonov and Nicolau showed that an interpolatingsequence is thin if and only if there is a sequence {mj}, 0 < mj < 1 andmj → 1 suchthat every interpolation problem F (zj) = wj with |wj | ≤ mj has a solution f ∈ H∞

with ‖F‖∞ ≤ 1, [10]. In fact, this happens if and only if there exists a sequenceof positive numbers {εj} with εj < 1 and εj → 0 such that every interpolationproblem with 1 ≥ |aj | ≥ εj for all j has a nonvanishing solution g ∈ H∞. Thus,if the sequence {wn} grows slowly enough, we can do interpolation with the bestnorm possible. In fact, the solution can be chosen to be a thin Blaschke product,as noted in [10]. (For the details of the proof, see [25]).

What are some other possible ways of defining thin sequences in theHp context?We provide two possible alternative definitions below.

Definition 5.2. Let 1 ≤ p ≤ ∞. A sequence {zn} is an eventual 1-interpolatingsequence for Hp (EISp) if the following holds: For every ε > 0 there exists N suchthat for each {an} ∈ p there exists fN,a ∈ Hp with

fN,a(zn)(1− |zn|2)1/p = an for n ≥ N and ‖fN,a‖p ≤ (1 + ε)‖an‖N,�p .

Definition 5.3. Let 1 ≤ p ≤ ∞. A sequence {zj} is a strong AISp-sequenceif for all ε > 0 there exists N such that for all sequences {aj} ∈ p there exists afunction GN,a ∈ Hp such that ‖GN,a‖p ≤ ‖a‖N,�p and

‖GN,a(zj)(1− |zj |2)1/p − aj‖N,�p < ε‖aj‖N,�p .

It turns out that both of these “new” definitions are equivalent to a sequencebeing thin, see [16].

6. Maximal ideal space and operator theory

For h ∈ L∞ define the Toeplitz operator on H2 by Thf = Phf , where P is theorthogonal projection from L2 to H2. The Hankel operator is Hhf = (I−P )hf, f ∈H2. In 1963, Brown and Halmos [4] showed that if f, g ∈ L∞, then TfTg = Tfg if

and only if f ∈ H∞ or g ∈ H∞. A natural question is the following: For whichsymbols f, g is TfTg a compact perturbation of a Toeplitz operator? In [2], Axler,Chang and Sarason showed that if H∞[f ] ∩ H∞[g] ⊂ H∞ + C, then H�

fHg iscompact. Though they proved necessity for a large class of functions, the theoremwas completed in 1982 by A. Volberg [32]. These proofs relied on the maximalideal space structure. There is a reason for this and it goes back to something wecan see directly from the statement of the Chang-Marshall theorem.

Corollary 6.1 (Corollary to the Chang-Marshall Theorem). Let A and B beDouglas algebras. Then M(A) ⊆M(B) if and only if B ⊆ A.

Proof. Suppose M(A) ⊆ M(B). Let b be an interpolating Blaschke productinvertible in B. Then b cannot be in a maximal ideal of B. Therefore, since maximal


ideals are precisely the kernels of the nonzero multiplicative linear functionals onB, we see that b cannot vanish at any point of M(B) – and therefore the sameis true on M(A). Now b ∈ H∞ ⊆ A and since b does not vanish on M(A), b isinvertible in A. Thus b ∈ A. Now we use the Chang-Marshall theorem to concludethat since B is generated by H∞ and the conjugates of the interpolating Blaschkeproducts invertible in B – all of which are invertible in A as well, we have B ⊆ A.

For the other direction, suppose B ⊆ A. Let ϕ ∈ M(A). Then for everyBlaschke product b invertible in B, we see that b is also invertible in A. Therefore,1 = ϕ(bb) = ϕ(b)ϕ(b) = |ϕ(b)|2. Thus, |ϕ(b)| = 1 and since ϕ(b) is given byintegration against a positive measure μ supported on a subset of the maximalideal space, we see that b must be constant on the support of ϕ. Thus, if f, g ∈ B,we know that f and g are limits of functions of the form

∑j hjbj with bj Blaschke

products invertible in B. By our argument above, the conjugates of the Blaschkeproducts are all constant on the support of ϕ, and therefore – as far as ϕ is concerned– they act like H∞ functions; that is,

ϕ(fg) =

∫suppϕ

fgdμϕ = ϕ(f)ϕ(g).

Thus, ϕ is (or can be identified with) a nonzero multiplicative linear functionalon B. �

So let us return to what Axler, Chang, and Sarason and, later, Volberg wantedto do. They each wanted to show something about the algebra H∞[f ] ∩ H∞[g].Since H∞[f ] and H∞[g] are each Douglas algebras and the intersection is again aDouglas algebra, we expect the Chang-Marshall theorem to come into play here;that is, we expect a proof that relies on the techniques that were developing at thetime. And that is precisely what happened – their results depended on a distri-bution function inequality as well as maximal ideal space techniques and Volberg’sproof used some of these same techniques.

7. Asymptotically orthonormal sequences

Volberg’s paper not only answered the question of whether the converse of theAxler, Chang, Sarason result was valid, it also looked at so-called asymptoticallyorthonormal sequences and their connection to thin sequences and properties of theassociated Gram matrix. We first recall some definitions.

Let {xn} be a sequence in a complex Hilbert space H.

Definition 7.1. The sequence {xn} is said to be a Riesz sequence if there arepositive constants c and C for which

c∑n≥1

|an|2 ≤

∥∥∥∥∥∥∑n≥1

anxn

∥∥∥∥∥∥2

H

≤ C∑n≥1

|an|2

for all sequences {an} ∈ 2.

We are interested in the following setting: Let Kz(w) = 11−zw denote the

reproducing kernel for H2 for z ∈ D, kz the normalized reproducing kernel, andgiven a sequence of points {zj}, recall that G denotes the Gram matrix with entrieskij = 〈kzi , kzj 〉. Riesz sequences correspond to the ones for which the associated


Gram matrix is invertible. We are now ready to introduce our asymptoticallyorthonormal sequences.

Definition 7.2. A sequence {xn} is an asymptotically orthonormal sequence(AOS) in a Hilbert space H if there exists an integer N0 such that for all N ≥ N0

there are constants cN and CN such that

cN∑n≥N

|an|2 ≤

∥∥∥∥∥∥∑n≥N

anxn

∥∥∥∥∥∥2

H

≤ CN

∑n≥N

|an|2,

where limN→∞

cN = limN→∞

CN = 1.

If we can take N0 = 1, the sequence is an asymptotically orthonormal basicsequence, or AOB.

Volberg showed (see also [7]) that the following is true.

Theorem 7.3 (Volberg, Theorem 2 in [32]). The following are equivalent:

(1) {zn} is a thin interpolating sequence;(2) The sequence {kzn} is a complete AOB for its span;(3) There exist a separable Hilbert space K, an orthonormal basis {en} for K

and U,K : K → KB, U unitary, K compact, U +K invertible, such that

(U +K)(en) = kzn for all n ∈ N;

(4) The Gram matrix associated to the sequence defines a bounded invertibleoperator of the form I +K with K compact.

The proof used the main lemma from [2] as well as Hoffman’s theory. Volbergalso showed that G − I ∈ S2 where S2 denotes the Hilbert-Schmidt operators ifand only if

∏j δj converges. Thus, G− I is in the Schatten class S2 if and only if∑

j(1− δj) <∞. What about 2 < p <∞?Using Earl’s theorem and results that are essentially in Shapiro and Shields

(see also [1]) J. E. McCarthy, S. Pott, and the authors [17] showed the following:

Theorem 7.4. Let 2 ≤ p <∞. Then G−I ∈ Sp if and only if∑

n(1−δ2n)p/2 <

∞.

This theorem extends Volberg’s result to the cases between 2 and infinity andsimplifies the proof for the case p =∞.

8. Carleson measures and thin sequences

It is possible to characterize thin sequences in terms of a certain vanishing Car-leson measure condition. This Carleson measure condition has strong connectionsto the notions of eventual interpolating sequences and the property of strong AISp.

For z ∈ D, we let Iz denote the interval in T with center z|z| and length 1− |z|.

For an interval I in T, we let

SI =

{z ∈ D :

z

|z| ∈ I and |z| ≥ 1− |I|}.

For A > 0, the interval AI denotes an interval with the same center as I and lengthA|I|. Given a positive measure μ on D, let us denote the (possibly infinite) constant

C(μ) = supf∈H2,f �=0

‖f‖2L2(D,μ)

‖f‖22


as the Carleson embedding constant of μ on H2 and

R(μ) = supz∈D

‖Kz‖L2(D,μ)

‖Kz‖2as the embedding constant of μ on the reproducing kernel of H2. We use Kz forthe non-normalized kernel later, and kz for the normalized kernel.

The Carleson Embedding Theorem asserts that the constants are equivalent.In particular, there exists a constant c such that

R(μ) ≤ C(μ) ≤ cR(μ),

with best known constant c = 2e, [27].We recall the following result from [31]; for a generalized version, see [7]. This

result provides a direct connection between thin sequences and a certain measurebeing a vanishing Carleson measure.

Theorem 8.1 (See Sundberg, Wolff, Lemma 7.1 in [31] or Chalendar, Fricain,Timotin, Proposition 4.2 in [7]). Suppose Z = {zn} is a sequence of distinct points.Then the following are equivalent:

(1) Z is a thin interpolating sequence;(2) for any A ≥ 1,

limn→∞

1

|Izn |∑

k �=n,zk∈S(AIn)

(1− |zk|) = 0.

Using this result it is possible to prove the following.

Theorem 8.2 ([16]). Suppose Z = {zn} is a sequence. For N > 0, let

μN =∑k≥N

(1− |zk|2)δzk .

Then the following are equivalent:

(1) Z is a thin sequence;(2) C(μN )→ 1 as N →∞;(3) R(μN )→ 1 as N →∞.

The proof of (1)⇒ (2) uses Volberg’s characterization of thin sequences as thosethat are asymptotic orthonormal bases [32], while (3)⇒ (1) is a computation withthe Weierstrass inequality. And, of course (2)⇒ (3) is immediate.

With this characterization of thin sequences it is possible to provide the fol-lowing list of equivalent conditions for a sequence to be thin.

Theorem 8.3 ([16]). Let {zn} be a Blaschke sequence of distinct points in D.The following are equivalent:

(1) {zn} is an EISp sequence for some p with 1 ≤ p ≤ ∞;(2) {zn} is thin;(3) {kzn} is a complete AOB in KB;(4) {zn} is a strong-AISp sequence for some p with 1 ≤ p ≤ ∞;(5) The measure

μN =∑k≥N

(1− |zk|2)δzk

is a Carleson measure with Carleson embedding constant C(μN ) satisfyingC(μN )→ 1 as N →∞;


(6) The measure

νN =∑k≥N

(1− |zk|2)δk

δzk

is a Carleson measure with embedding constant RνNon reproducing ker-

nels satisfying RνN→ 1.

Moreover, if {zn} is an EISp (strong-AISp) sequence for some p with 1 ≤ p ≤ ∞,then it is an EISp (strong AISp) sequence for all p.

9. Future Directions: Model Spaces

We conclude with a discussion of thin sequences in other contexts. Given a(nonconstant) inner function Θ, one can also study thin sequences in model spaces,where the model space for Θ, an inner function, is defined by KΘ = H2 � ΘH2.The reproducing kernel in KΘ for λ0 ∈ D is

KΘλ0(z) =

1−Θ(λ0)Θ(z)

1− λ0z

and the normalized reproducing kernel is

kΘλ0(z) =

√1− |λ0|2

1− |Θ(λ0)|2KΘ

λ0(z).

Finally, note thatKλ0

= KΘλ0

+ΘΘ(λ0)Kλ0.

We let PΘ denote the orthogonal projection of H2 onto KΘ.Asymptotically orthonormal sequences were studied in [12] and [7]. We men-

tion here one theorem that encompasses many of these results. Proofs or referencesfor proofs can be found in [18]. We remark that we get Theorem 4.6 of [16] whenwe let Θ = B in the proof below (which is simply Theorem 8.3 above).

Theorem 9.1 (Theorem 3.5 in [18]). Let {λn} be an interpolating sequencein D and let Θ be an inner function. Suppose that κ := supn |Θ(λn)| < 1. Thefollowing are equivalent:

(1) {λn} is an EISH2 sequence;(2) {λn} is a thin interpolating sequence;(3) Either

(a) {kΘλn}n≥1 is an AOB, or

(b) there exists p ≥ 2 such that {kΘλn}n≥p is a complete AOB in KΘ;

(4) {λn} is an AISH2 sequence;(5) The measure

μN =∑k≥N

(1− |λk|2)δλk

is a Carleson measure for H2 with Carleson embedding constant C(μN )satisfying C(μN )→ 1 as N →∞;

(6) The measure

νN =∑k≥N

(1− |λk|2)δk

δλk

is a Carleson measure for H2 with embedding constant RνNon reproducing

kernels satisfying RνN→ 1.


Further, (7) and (8) are equivalent to each other and imply each of thestatements above. If, in addition, Θ(λn)→ 0, then (1) - (8) are equivalent.

(7) {λn} is an EISKΘsequence;

(8) {λn} is an AISKΘsequence.

There are many directions for future research. For example, connections totruncated Toeplitz operators have been studied by Lopatto and Rochberg [23] aswell as R. Bessonov [3]. In addition, we mention two questions below.

Question 1. One can define thin sequences in other spaces (for example,Bergman spaces) and see whether the results that we have discussed here extend tothose spaces: If a sequence is a thin sequence in a space X, is there a particularlygood bound on the interpolation constant?

Question 2. Finally, we note that thin sequences are those satisfying δj → 1and they are interpolating sequences for an important space of functions, QA. If∑

j(1− δj)p <∞, is the sequence interpolating for some natural function space?

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Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837


Department of Mathematics, Washington University - St. Louis, One Brookings

Drive, St. Louis, Missouri 63130-4899



Kernels of Toeplitz operators

Andreas Hartmann and Mishko Mitkovski

Abstract. Toeplitz operators are met in different fields of mathematics suchas stochastic processes, signal theory, completeness problems, operator theory,etc. In applications, spectral and mapping properties are of particular interest.In this survey we will focus on kernels of Toeplitz operators. This raises two

questions. First, how can one decide whether such a kernel is non trivial? Wewill discuss in some details the results starting with Makarov and Poltoratskiin 2005 and their succeeding authors concerning this topic. In connectionwith these results we will also mention some intimately related applications tocompleteness problems, spectral gap problems and Polya sequences. Second,if the kernel is non-trivial, what can be said about the structure of the kernel,and what kind of information on the Toeplitz operator can be deduced from itskernel? In this connection we will review a certain number of results startingwith work by Hayashi, Hitt and Sarason in the late 80’s on the extremalfunction.

1. Introduction

Toeplitz operators are natural generalizations of so-called Toeplitz matrices. In thestandard orthonormal basis of 2(N) = {a = (an)n≥0 : ‖a‖22 :=

∑n≥0 |an|2 < ∞},

a Toeplitz operator is represented by the infinite matrix

(1.1) T =

⎛⎜⎜⎜⎜⎜⎝u0 u−1 u−2 u−3 · · ·

u1 u0 u−1 u−2. . .

u2 u1 u0 u−1. . .

.... . .

. . .. . .

...

⎞⎟⎟⎟⎟⎟⎠ ,

where (un)n∈Z is a given sequence. If we identify 2(N) with the usual Hardy spaceH2 of functions f(eit) =

∑n≥0 ane

int with (an)n ∈ 2(N), and if we associate

(formally) u with the Fourier series ϕ(eit) =∑

n∈Zune

int then

(Ta)k = ϕf(k), k ≥ 0,

2010 Mathematics Subject Classification. Primary 30J05, 30H10, 46E22.Key words and phrases. Hardy spaces, model spaces, Toeplitz operators, Toeplitz kernels,

rigid functions, Muckenhoupt condition, injectivity, Beurling-Malliavin density, completeness, gapproblem, uncertainty principle, Polya sequences.


147

148 A. HARTMANN AND M. MITKOVSKI

whenever that makes sense. Hence, Ta defines the sequence of Fourier coefficientsof the function

P+(ϕf),

where P+ : L2(T) −→ L2(T), P (∑

n∈Zane

int) =∑

n≥0 aneint is the so-called Riesz

(or Szego) projection. We thus may define the Toeplitz operator with symbol ϕ by

Tϕ : H2 −→ H2, Tϕf = P+(ϕf),

which is a continuous operator on H2 when ϕ ∈ L∞(T), and the associated matrixin the orthonormal basis {zn}n≥0 of H2 is given by (1.1).

Two special, but very important, Toeplitz operators are the shift operator Sf =P+(zf) = zf and its adjoint S∗f(z) = P+(zf) — the so-called backward shiftoperator. Any Toeplitz operator satisfies the following “almost commuting” relation

(1.2) S∗TϕS = Tϕ,

and as a matter of fact this latter operator equality — which is a kind of displace-ment condition in the matrix reflecting the constance along diagonals — character-izes Toeplitz operators.

A closely related class of operators which are also called Toeplitz operators canbe defined in the so-called “continuous case”. For most parts these theories areparallel to each other and studying one or the other case mostly depends on theperson’s taste. However, it should be noted that some problems are much morenatural to be considered in the discrete setting while others are more suitable forthe continuous one. For this reasons in this survey we will switch from one case tothe other whenever appropriate.

In the continuous case one starts with a function u ∈ L∞(R) and associates to itthe convolution operator T : L2(R+)→ L2(R) defined, as usual, by Tf = u ∗ f . Ifwe want to consider T as an operator from L2(R+) to itself it is natural, as in the“discrete case” above, to project back from L2(R) onto L2(R+) using the Riesz pro-jection P+ (now considered on the real line). The operator Tu = P+T : L2(R+)→L2(R+) obtained in this way is again called a Toeplitz operator with symbol u.It should be noted here that in this form Tu is also known as a Wiener-Hopf op-erator. In order to recover the form of the Toeplitz operators discussed above itsuffices to apply the Fourier transform. The “Fourier transformed” Tu becomes anoperator on the Hardy space in the upper half plane H2(C+) = FL2(R+), givenby TΦf = P+Φf , where Φ = u and P+ : L2(R) → H2(C+) is the Riesz (or Szego)projection introduced above. In the continuous case the role of the shift oper-ator S is played by the translation semigroup S(t) : L2(R+) → L2(R+), t ≥ 0,defined by (S(t)f)(x) = f(x + t). In the Fourier domain this semigroup be-comes S(t) : H2(C+) → H2(C+), t ≥ 0, S(t)f = eitxf(x). Notice again thatfor each fixed t ≥ 0 both S(t) and S∗(t) are in fact Toeplitz operators with verysimple symbols eitz and e−itz. Several results about Toeplitz operators can beviewed as perturbation results for these two special (but important) classes ofToeplitz operators. Finally, notice that the analog of (1.2) in the continuous caseis S(t)∗TΦS(t) = TΦ, t ≥ 0, which can again be taken as defining property forToeplitz operators in the sense that any operator which satisfies this identity mustbe unitarily equivalent to a Toeplitz operator.

KERNELS OF TOEPLITZ OPERATORS 149

Toeplitz operators, as introduced above, have numerous applications in signal the-ory, stochastic processes, interpolation and sampling problems, etc., some of whichwill be overviewed in Subsection 1.3 below and discussed in more details in Section 8(see also [33,34,41]). We refer to the monograph by Bottcher and Silbermann [9]as a general reference on Toeplitz operators and their applications. In this surveywe will be interested in a very specific topic in connection with Toeplitz operators,namely the kernels of these Toeplitz operators — so-called — Toeplitz kernels. Wewill try to give an overview of results about Toeplitz kernels — some of which arevery recent — connected to the following two topics:

(1) Triviality of Toeplitz kernels: Given a Toeplitz operator Tϕ, how to tellfrom the symbol ϕ whether its kernel is trivial or non-trivial?

(2) Structure of Toeplitz kernels: In case of non-triviality we would like tounderstand the structure of a given Toeplitz kernel as a subset of theHardy space H2 or more generally Hp.

Besides being interesting objects to study intrinsically, as shown by Makarov andPoltoratski [33], Toeplitz kernels are also interesting to study because of theirintimate connections to numerous problems in complex and harmonic analysis, aswell as mathematical physics.

In the remaining part of this introduction we would like to give a brief overview ofthe topics that will be discussed in this paper.

1.1. Triviality of Toeplitz kernels. As just mentioned, the first naturalproperty that one would like to understand about Toeplitz kernels is whether theyare trivial or not. As far as we know, currently there is no explicit, easily check-able, criterion for triviality of a Toeplitz kernel for general symbols ϕ. However,if we restrict the class of symbols ϕ, then the important results of Makarov andPoltoratski [34] provide an “almost solution” of the triviality problem. We willdiscuss these results in some detail in the last section of the paper. The class ofsymbols which can be treated by Makarov-Poltoratski techniques consists of uni-modular symbols (unimodularity is no restriction of generality in the discussion ofkernels of Toeplitz operators as we will see later) of the form Bb for inner functionsB and b having their zeros accumulating only at one common point and which bothhave only one possible singular point mass at that point (this corresponds to mero-morphic inner functions in the upper half plane). We will discuss more thoroughlya list of problems related with this triviality question in Subsection 1.3.

1.2. Structure of Toeplitz kernels. Once the question of non-triviality clar-ified one is interested in descriptions of kernels of Toeplitz operators, and the in-formation that can be deduced from them. To motivate the results that follow,consider the special case of Toeplitz operators Tϕ with symbols of the form ϕ = θ,where θ is some inner function. Then clearly, KerTϕ = Kθ, where Kθ = H2 � θH2

is the model space generated by θ. It is a well-known result of Beurling that modelspaces can be characterized exactly as those subspaces of H2 which are invariantunder the backward shift. This raises the following natural question: Does thereexist some analogous characterization for Toeplitz kernels with general symbols?The first step towards the solution of this problem was made by Hitt [26] who


characterized the class of so-called nearly invariant subspaces. From (1.2) it is easyto check that every Toeplitz kernel is nearly invariant. However, not every nearlyinvariant subspace can be represented as a Toeplitz kernel. Recall that a nearly in-variant subspace (with respect to S∗) is a subspace M of H2 satisfying the followingproperty

f ∈M, f(0) = 0 =⇒ S∗f ∈M.

Hitt showed that nearly invariant subspaces (in H2) are of the form

M = gK2I ,

whereK2I is the model space generated by the inner function I and g is the extremal

function of M , meaning that it maximizes the real part at 0 among all the functionsinM with norm one. Extremal functions played a crucial role in the work of Hayashi[24] who used them to identify those nearly invariant subspaces which are exactlykernels of Toeplitz operators (see Section 2 for precise definitions and Section 4for results). Later, in [45] Sarason gave an approach to Hayashi’s result using deBranges-Rovnyak spaces.

As we will see below, one important consequence of this line of results is the factthat for Toeplitz kernels KerTϕ which are non-trivial we can always assume thatϕ is a unimodular function which can be represented as

ϕ =Ig

g.

This result represents the initial point in the Makarov-Poltoratski treatment ofinjectivity (see Lemma 8.1 below).

The extremal function g appearing in Hitt’s description of nearly invariant sub-spaces can also be used to decide whether the corresponding Toeplitz operator isonto [21]. One can view the surjectivity problem for a Toeplitz operator Tϕ asa “strong-injectivity” problem for the adjoint Toeplitz operator Tϕ. Namely, it iswell known that the surjectivity of any operator is equivalent to the left-invertibilityof its adjoint. Left-invertibility, on the other hand, being equivalent to injectivitywith closed range, can be naturally viewed as a type of “strong injectivity”. Sothe Toeplitz operator Tϕ is surjective if and only if the Toeplitz operator Tϕ is“strongly injective”, i.e., the corresponding Toeplitz kernel is “strongly trivial”.The properties of rigidity and exposed points (for which there is still no meaningfulcharacterization available) as well as the Muckenhoupt (A2) condition are centralnotions here when considering the Hilbert-space situation p = 2. The problem wasalso considered for the non-hilbertian situation where the extremal function doesnot have the same nice properties [22] (see also Hitt’s unpublished paper [25]).Related results for p = 2 were discussed by Camara and Partington and will bepresented in Section 7.

Bourgain factorization allowed Dyakonov to give another description of Toeplitzkernels on Hardy spaces Hp: for every unimodular symbol ϕ (unimodularity is norestriction of generality in the discussion of kernels of Toeplitz operators as we willsee later) the kernel Kerp Tϕ of Tϕ considered now on Hp, is of the form

(1.3) Kerp Tϕ =g

b(Kp

B ∩ bHp),


where the triple (B, b, g) (using Dyakonov’s terminology) consists of two Blaschkeproducts B and b, and a bounded analytic function g which is also boundedlyinvertible. Being in a more general situation here than the hilbertian case, we shalldefine Kp

I = Hp ∩ IzHp (on T) for an inner function I. Clearly, in that situationwe can replace the symbol ϕ by

bBg

g.

We will give some interesting consequences of this representation in Section 6.Dyakonov is actually able to construct symbols such that the kernel of the Toeplitzoperator takes precise given dimensions in Hp depending on the values of p.

The representation (1.3) makes a connection with the completeness problems dis-cussed above. Completeness of a system of reproducing kernels is via duality equiv-alent to uniqueness in the dual space. Then, if Kerp Tϕ = {0}, where ϕ = bBg/g,then this means that the zeros of b form a zero sequence of Kp

B. Though inDyakonov’s result, B is a Blaschke product, one can consider more general innerfunctions I.

Finally we would like to mention that a new connection between Toeplitz kernelsand multipliers between model space has been established in the recent preprint[19]. In this topic one is for instance interested in knowing whether the Toeplitzkernel, if non trivial, contains bounded functions. This relates to Dyakonov’s resultabove as well as to another result in the work of Makarov and Poltoratski (see[33, Section 4]). They discuss criteria which ensure that if a given Toeplitz kernelis non-trivial in some Hp (and more generally in the Smirnov class N+) then ina certain sense, increasing the size of Kp

B in the Dyakonov representation (1.3)ensures non-triviality for the smallest kernel in the Hp-chain, i.e. in H∞.

1.3. Applications of Toeplitz kernels. Since the list of problems which canbe translated into injectivity problems of Toeplitz operators is quite long, we willconcentrate here our discussion to three specific problems. In the solution of eachof these problems the Beurling-Malliavin densities play a very important role. Aswill become clear below the reason for this is the fact that the injectivity of manyToeplitz operators is closely dependent on these densities.

The first problem concerns the geometry or basis properties of reproducing kernelsin model spaces. It is a well-known idea that one can use Toeplitz operators tostudy the basis properties of normalized reproducing kernels in model spaces. Thisidea goes back at least to the seminal paper by Hruschev, Nikolski, and Pavlov [27]who used the Toeplitz operator approach to finally settle the Riesz basis problemfor non-harmonic complex exponentials. They also proved that most of the basisproperties of sequences of normalized reproducing kernels in model spaces can bedescribed in terms of invertibility properties of an appropriate Toeplitz operator.Let us briefly recall their well-known idea. Given a model space KI = H2 � IH2,where I is an inner function, and a sequence Λ ⊂ D (or in C+) the aim is todecide when a sequence of normalized reproducing kernels{kIλ}λ∈Λ is complete, aRiesz sequence, a Riesz basis, . . . in KI . It can easily be shown that under certainconditions on Λ (see [39, Chapter D4]) this happens if and only if the Toeplitzoperator TIBΛ

has dense range, is injective with closed range (i.e. left invertible),invertible, . . . , where BΛ denotes the Blaschke product with zero set Λ. Indeed, if


PI is the orthogonal projection from H2 to KI , then the basis properties translateinto mapping properties of PI : KBΛ

−→ KI (under certain conditions on Λ). Itis not difficult to check that those mapping properties are reflected in those of theToeplitz operator TIBΛ

(see [39, Lemma D4.4.4]). Note that TIBΛhaving dense

range is equivalent to TIBΛbeing injective. Now, the completeness problem for

non-harmonic complex exponentials [42] can be viewed as a completeness problemfor normalized reproducing kernels in a suitable model space. Therefore, it canbe restated as an injectivity problem for a suitable Toeplitz operator. This verywell-known notoriously difficult problem inspired a great deal of results in math-ematical analysis in the first half of the 20th century. Levinson gave in 1936 asufficient condition for completeness in Lp(I). After many unsuccessful attempts,the problem was finally settled by Beurling and Malliavin in 1967 [5] relying heav-ily on their previous results [4]. There are several different proofs that appearedsince [14,29,30,35] but none of them is much simpler than the original one. Theachievement of Makarov and Poltoratski here was that they were able to adapt thedeep ideas of Beurling and Malliavin to give a metric characterization of injectivityfor Toeplitz operators with very general symbols (much more general than symbolsneeded to solve the classical completeness problem for complex exponentials). As aconsequence they provided a solution to the completeness problem for normalizedreproducing kernels in a very general class of model spaces.

The second problem we would like to mention here is the spectral gap problem whichis related to the uncertainty principle in harmonic analysis [23]. One of the broadestformulations of the uncertainty principle says that a function (measure) and itsFourier transform cannot be simultaneously small. There are many mathematicallyprecise versions of this heuristic principle depending on what kind of smallness oneis interested in. In the classical gap problem one is interested in the gaps in thesupport of the measure and its Fourier transform. The heuristics says that thesegaps cannot be simultaneously too big. In other words, if the support of the measurehas gaps that are too big then the support of its Fourier transform cannot have toolarge gaps. As shown by the second author and Poltoratski, for certain sets X, thegap [0, a] where μ vanishes for a measure μ supported on X can be measured byBeurling-Malliavin densities. More precise results and extensions will be discussedin Section 8.

The third problem we want to present here is the Polya problem. Here we makea connection with the area of entire functions which received much interest in thepast due to its intimate connections to the spectral theory of differential opera-tors, signal processing, as well as analytic number theory. The Polya problem isa uniqueness problem and asks for a description of separated real sequences withthe property that there is no non-constant entire function of exponential type zero(entire functions that grow more slowly than exponentially in each direction) whichis bounded on this sequence. Such sequences are called Polya sequences. Thisproblem was resolved only recently by Poltoratski and the second author in [36]and its solution played a crucial role in the resolution of several important problemsin recent years [1,2]. As in the problems discussed above Toeplitz kernels play animportant role in the solution of this classical problem. Again the characterizationof Polya sequences is expressed in terms of Beurling-Malliavin densities.


2. Basic definitions

We denote by Hp, 0 < p < ∞, the classical Hardy space of analytic functions onthe unit disk D = {z ∈ C : |z| < 1}, for which

‖f‖pp := sup0<r<1

1

2π

∫ π

−π

|f(reit)|pdt <∞,

and H∞ are the bounded analytic functions on D equipped with the usual norm‖f‖∞ := supz∈D |f(z)|. As usual, we identify functions f in Hp with their non-tangential boundary limits on T also denoted by f . More precisely, associating withf ∈ Hp its non-tangential boundary function enables us to identify isometrically

Hp = {f ∈ Lp(T) : f(n) = 0, n < 0}.In view of this observation, we will mostly not distinguish between f and its bound-ary function, and when speaking about Hp in this paper we indifferently mean theHardy space of holomorphic functions on the unit disk or its boundary values inLp(T).

Observe that every function f ∈ Hp, as an analytic function on D, may be writtenas its Taylor expansion

f(z) =∑n≥0

anzn.

At least when p ≥ 1, we can associate with f on T its Fourier series, and as amatter of fact, the coefficients an turn out to be the Fourier coefficients of f on T.This identifies isometrically H2 with 2 by associating with f =

∑n≥0 anz

n ∈ H2

the sequence (an)n≥0 of its Fourier (or Taylor) coefficients.

2.1. Projection and Toeplitz operators. Given any function from Lp(T),1 < p < ∞, which is expressed as Fourier series f(eit) =

∑n∈Z

aneint, we can

associate its truncation to non-negative Fourier coefficients:

P+ : Lp(T) −→ Lp(T),

f(eit) =∑n∈Z

aneint �−→ f(eit) =

∑n≥0

aneint.

This projection is called the Riesz (or Szego) projection. As mentioned above, forp = 2, the norm in L2(T) is expressed as the 2-norm of its Fourier coefficients, andit is plain that in this situation P+ is continous (orthogonal projection). It turnsout that this projection is also continuous for 1 < p <∞.

Another way of defining the Riesz projection is via reproducing kernels. For λ ∈ D,set

kλ(z) =1

1− λz, z ∈ D.

Then, for every f ∈ Hp and for every λ ∈ D,

(2.1) f(λ) = 〈f, kλ〉 :=1

2π

∫ π

−π

f(eit)kλ(eit)dt,

which actually makes sense for every 1 ≤ p ≤ ∞. Equation (2.1) explains thewording “reproducing kernel” for kλ. We then have

P+f(z) = 〈f, kz〉,


and this is well-defined for every f ∈ Hp, 1 ≤ p ≤ ∞, z ∈ D. However, when f ∈L1(T) (or L∞(T)) then the function P+f may be in weak-L1(T) (or in BMO(T)).We will not discuss these issues here and concentrate mainly on p = 2 or on 1 <p <∞.

Pick now a bounded function ϕ ∈ L∞(T), then the Toeplitz operator with Tϕ

symbol ϕ is defined by

Tϕf = P+(ϕf).

When we consider Tϕ on H2, then via the identification H2 with 2 mentionedearlier this is exactly the form we had discussed in the introduction and which thusgeneralizes to Hp. Note that we can also write

Tϕf(z) = 〈ϕf, kz〉,which not only makes sense for 1 < p <∞ but also when p = 1 or p =∞.

A similar set-up can be given in the upper half plane.

2.2. Factorization and model spaces. Let us turn back to Hardy spaces.It is well-known that every function f ∈ Hp may be decomposed in three factors

f = BSF,

where B is the Blaschke product carrying the zeros Λ of f (counting multiplicities):

B =∏λ∈Λ

bλ.

Here b0(z) = z and, when λ = 0,

bλ(z) =λ

λ

λ− z

1− λz, z ∈ D,

(the normalization is chosen to make the Blaschke factors positive at zero). Notethat the product B converges if and only if Λ satisfies the Blaschke condition∑

λ∈Λ

(1− |λ|2) <∞.

The function S is the singular inner function determined by a positive singularmeasure μ on T:

S(z) = exp

(−∫ π

−π

eit + z

eit − zdμ(eit)

), z ∈ D.

Finally, F is an outer function which is uniquely determined by the modulus of fon T

F (z) = exp

(∫ π

−π

eit + z

eit − zlog |f(eit)|dt)

), z ∈ D.

Observe that |B| = |S| = 1 a.e. on T. Functions with |I| = 1 a.e. on T are calledinner, and any inner function is a product of Blaschke product and a singularinner function (possibly trivial). Inner functions play a central role in Beurling’scharacterization of shift invariant subspaces. Recall that

S : Hp −→ Hp, Sf(z) = zf(z),

is called the shift operator.


Theorem 2.2 (Beurling). Let M ⊂ H2 be a closed subspace, non-trivial (M = H2

and M = {0}). Then M is shift invariant if and only if there is a (unique) innerfunction I such that M = IH2.

We can also consider the adjoint shift S∗ = Tz which on D is given by

S∗f(z) =f(z)− f(0)

z, z ∈ D.

Then passing to orthogonal complements we deduce that a closed non-trivial sub-space M ⊂ H2 is S∗-invariant if and only if there exists a (unique) inner functionI such that

(2.3) M = K2I := (IH2)⊥ = H2 � IH2.

We will also use the notation KI = K2I . These are the so-called model spaces which

play a central role in the theory of function models for certain classes of Hilbertspace contractions (see e.g. [39, Chapter C]). By a result of Douglas, Shapiroand Shields [16], these spaces can also be defined by existence of so-called pseudo-continuations (which we won’t discuss here). The definition (2.3) is not adapted toa generalization to values p = 2. It is easy to check that denoting by H2

0 = zH2

the H2-functions vanishing at 0, then

K2 = H2 ∩ IH20 ,

where the equality has to be interpreted on the circle via non-tangential limits(the bar-sign here means complex conjugation). This definition immediately passesto any value of p (but these spaces behave strangely for p < 1 as discussed byAleksandrov, for which we refer to the monograph [12]):

Kp := Hp ∩ IHp0 .

The spaces Kp are widely studied spaces but are still far from being completelyunderstood. They constitute a central building block for kernels of Toeplitz oper-ators.

Let us recall three simple examples of model spaces.

• When I(z) = zn, then KpI = K2

I = Poln−1 the analytic polynomials ofdegree at most n− 1;

• When I(z) = ea(z+1)/(z−1), then KI is isometrically isomorphic (essen-tially via the Fourier transform) to the Paley-Wiener space PWa/2 ofentire functions of exponential type a/2 which are square integrable on R.

• When I = B is a Blaschke product with simple zeros then KpB = Lin(kλ :

λ ∈ Λ)− (closure in Hp, 1 < p < ∞). In the special situation when thezeros of B form an interpolating sequence, then every Kp

B-function is anp-sum of normalized reproducing kernels kλ/‖kλ‖p.

For 1 < p < ∞, the spaces KpI can also be understood as ranges of the following

projection (orthogonal for p = 2)

PI = IP−I, P− = Id− P+,


where I is an inner function, so that KpI = PIH

p, and

kIλ(z) = PIkλ(z) =1− I(λ)I(z)

1− λz

is the reproducing kernel.

2.3. Elementary properties of Toeplitz operators. As is well-known

(2.4) ‖Tϕ‖H2→H2 = ‖ϕ‖∞(for 1 < p <∞, p = 2, this equality is an equivalence);

(2.5) T ∗ϕ = Tϕ;

(2.6) TϕTψ = Tϕψ ⇐⇒ ϕ ∈ H∞ or ψ ∈ H∞;

(2.7) u inner =⇒ KerTu = Ku;

(2.8) ϕ ∈ H∞ =⇒ TϕKu ⊂ Ku.

Observe that (2.8) implies that the model spaces enjoy the so-called F -property: Ifw is an inner function then

(2.9) f ∈ Ku and f/w ∈ Hp =⇒ f/w ∈ Ku.

3. Invertibility and left-invertibility of Toeplitz operators

It is slightly surprising that the injectivity problem for Toeplitz operators receivedsignificantly less attention in the past in comparison with the invertibility problem.As part of the invertibility problem, the so-called injectivity with closed range (left-invertibility) of Toeplitz operators has been thoroughly studied. This is why webegin by giving a brief overview of the most well-known results about invertibilityand left-invertibility of Toeplitz operators which could be used as a guide in thestudy of the injectivity problem.

Recall that an operator A : X → Y between two Banach spaces is called left-invertible (bounded from below, injective with closed range) if there exists a con-stant ε > 0 such that for every x ∈ X, ‖Ax‖Y ≥ ε‖x‖. Left invertibility of Tϕ isequivalent to surjectivity of its adjoint Tϕ. The following well-known result gives anecessary condition for a Toeplitz operator to be left-invertible (we refer to [9, p.65],see also [38, Theorem B4.2.7])

Theorem 3.1 (Hartman-Wintner). If ϕ ∈ L∞ but not invertible in L∞, then Tϕ

is not bounded below.

Hence, if Tϕ is left-invertible or onto, then we can assume that the symbol ϕ isbounded away from zero, i.e., 0 < δ ≤ |ϕ| ≤ M < ∞. In that situation, log |ϕ| isobviously integrable and hence there is an outer function g ∈ H2 such that |g| = |ϕ|a.e. on T. As a matter of fact g is invertible in H∞ and thus Tg is invertible withinverse T1/g. Hence the Toeplitz operator Tϕ behaves like the Toeplitz operatorTϕ/g which has a unimodular symbol.

Invertibility of Toeplitz operators has been studied in the 1960’s by Devinatz andWidom in the Hilbert space situation. Their result can be stated as follows.


Theorem 3.2. Let ϕ ∈ L∞. Then the following are equivalent:

• Tϕ is invertible on H2.

• ϕ−1 ∈ L∞, and there exists h ∈ H2 such that

ϕ

|ϕ| = εh

|h| ,

and |h|2 satisfies the Helson-Szego condition (ε is a uni-modular constant).

Recall that a measurable function w on T is said to satisfy the Helson-Szego con-dition if

w = eu+v,

where u and v are real valued bounded functions with ‖v‖∞ < π/2. As a matterof fact the Helson-Szego condition turns out to be equivalent to the Muckenhoupt(A2) condition. We introduce here the general case of the (Ap) condition.

Definition 3.3. Let w be a positive measurable function on T. If

supI

(1

|I|

∫I

wp(eit)dt

)1/p

·(

1

|I|

∫I

w−p′(eit)dt

)1/p′

<∞,

where 1/p+1/p′ = 1 and I runs through all the subarcs of T, then we say that wp

satisfies the (Ap) condition.

Hunt, Muckenhoupt and Wheeden have shown that the already introduced Rieszprojection P+ is continuous on Lp(T, wpdm), 1 < p <∞, if and only if wp ∈ (Ap).

It is possible to give more equivalent conditions for invertibility of Toeplitz op-erators. We just mention here the relation with Hankel operators for which theNehari theorem gives ‖Hϕ‖ = dist(ϕ,H∞). Since Hankel and Toeplitz operatorscomplete each other: ϕf = Tϕf + Hϕf (the reader who does not know anythingabout Hankel operators might take this last identity as a definition of a Hankel op-erator), and when ϕ is uni-modular we get an equivalent criterion for invertibilityof Toeplitz operators via Nehari’s theorem. Indeed, it is necessary and sufficientthat dist(ϕ,H∞) < 1 and dist(ϕ,H∞) < 1. We shall not use this connection toomuch here.

Note also that in the special situation when ϕ is unimodular then the conditionϕ−1 ∈ L∞ is automatic and the condition on invertibility is essentially that of writ-ing ϕ = h/h where |h|2 satisfies the Helson-Szego or Muckenhoupt (A2) condition.

With the (Ap) condition in mind, it is natural to ask whether an invertibilitycriterion can be found for Toeplitz operators on Hp. As expected, and shown byRochberg [43], invertibility for Toeplitz operators on Hp is characterized by the(Ap)-condition. In case ϕ is unimodular this reads as: there is g ∈ Hp such that|g|p ∈ (Ap) and

ϕ =g

ga.e. on T.

In this situation, it is possible to define explicitely an inverse at least on a suitabledense set by

gT1/g.


Indeed, this operator makes sense on the dense set g Pol ⊂ Hp, and by directinspection it is seen to be the inverse to Tg/g on this dense set (observe that theinverse is in general not a Toeplitz operator).

4. Injectivity of Toeplitz operators and rigid functions

In this section we start discussing some general elementary injectivity properties ofToeplitz operators.

The following well-known lemma by Coburn provides a very interesting specialproperty possessed by the class of Toeplitz operators (see e.g. [39, Lemma B4.5.6]).

Lemma 4.1. Let ϕ ∈ L∞, then at least one of the kernels KerTϕ or KerT ∗ϕ = KerTϕ

has to be trivial.

Notice that there exist Toeplitz operators Tϕ such that both Tϕ and Tϕ are injective(pick for instance any bounded outer function or its conjugate). Understanding suchsymbols is in the heart of the injectivity problem.

For most applications one can concentrate on unimodular symbols. For such sym-bols we have

(4.2) g outer in H2 =⇒ Tg/g injective on H2.

This is easily seen by contradiction. Indeed, if f ∈ KerTg/g then (g/g)f = ψ for

some function ψ ∈ H2, ψ(0) = 0. Then H1 # gf = gψ ∈ H10 which is possible only

when gf = gψ = 0, i.e. f = 0.

Note that when I is inner, then Ig/g = (1 + I)g/((1 + I)g) also gives a symbol ofan injective Toeplitz operator.

If we reverse the numerator and the denominator, then the situation changes cru-cially. Answering the injectivity problem for such symbols requires the notion ofrigidity which we will define now.

Definition 4.3. A function f ∈ H1 is rigid if it is uniquely determined in H1 upto a positive constant multiplier by its argument: for every g ∈ H1, f/g > 0 a.e.T, implies f = λg, λ ≥ 0.

This parallels in a way the definition of outer functions which are determined bytheir moduli. Rigid functions are necessarily outer, since the argument of any innerfactor I is for instance also given by (1+ I)2 (which can be immediately seen fromelementary geometric considerations). It is known that f is rigid in H1 if and onlyif f/‖f‖2 is an exposed point in the unit ball of H1 ([15]). A simple sufficientcondition for rigidity is for instance f ∈ H1 and 1/f ∈ H1 but this condition is notnecessary. In [28] the authors show that every polynomial with no zeros in D andsimple zeros on T is rigid. Moreover for every such polynomial and f ∈ H∞ with‖f‖∞ ≤ 1, the function p ◦ f is also rigid. However, no useful characterization forrigid functions is known. It turns out, rigidity is the right notion for injectivity ofTg/g:

g2 rigid in H1 ⇐⇒ Tg/g injective on H2.


In fact this observation can also be used as an equivalent definition of rigid functions(see [46, X-2]).

Rigid functions arise naturally in the frame of completely non-deterministic gauss-ian processes. The idea is to relate the spectral measure dμ = wdm of the process,where w is log-integrable so that w = |f |2 for some outer function f ∈ H2, andconsider the map T : L2(wdm) −→ L2, Th = fh which maps isometrically the

future of the process to H20 and the past to (f/f)H2. The process is completely

non-deterministic if the future and the past have trivial intersection, which happensexactly when Tf/f is injective. We refer to [7] for more precise information on this

topic.

5. Kernels of Toeplitz operators and extremal functions

5.1. Nearly invariant subspaces. We start recalling a definition from theintroduction.

Definition 5.1. A closed non-trivial subspace M of H2 is called nearly invariant,if

f ∈M and f(0) = 0 =⇒ S∗f ∈M.

Any space of the form GK2I with G ∈ H2 and G(0) = 0 is automatically nearly

invariant as can be seen from the following simple argument: if f = Gh ∈ GK2I

and f(0) = 0, then necessarily h(0) = 0. Hence

(5.2) S∗f = S∗(Gh) = zGh = Gzh = GS∗h ∈ GK2I .

However, the space GK2I is in general neither contained in H2, nor closed even if

it were contained in H2. For this we would need that |G|2dm is a Carleson anda reverse Carleson measure for K2

I (the first condition guarantees that GK2I is a

subspace of H2 and the second one that it is closed; we refer to [13], [49], [6]).

In his work on invariant subspaces of H2 on an annulus, Hitt [26] described thenearly invariant subspaces in a precise way (he also wrote an unpublished paper forthe case 1 < p < ∞, [25]). Pick a nearly invariant subspace M and associate theextremal function which is the solution to the problem

sup{Re g(0) : g ∈M, ‖g‖2 ≤ 1}.This problem has a unique solution. Indeed, the existence of a solution follows froman argument based on normal families, and switching to the equivalent problem

inf{‖g‖2 : g(0) = 1, g ∈M},which is a closed convex set in the strictly convex space Hp, 1 < p < ∞, we seethat the solution has to be unique.

Hitt observed that in the case p = 2 it is possible to divide isometrically by theextremal function and that the resulting space is S∗-invariant:

Theorem 5.3 (Hitt). Let M be a nontrivial nearly S∗-invariant subspace of H2,with extremal function g. Then M = gK2

I , where I is some inner function vanishingat the origin. Furthermore, g is an isometric divisor on M : ‖f/g‖2 = ‖f‖2 for allf ∈M .


We will discuss the situation when p = 2 in Section 7.

The function g appearing in this result is not necessarily outer. Observe thatgiven any nearly invariant subspace with extremal function g and associated innerfunction I, then picking another inner function J with J(0) = 0, it follows from(5.2) that JgKI is nearly invariant.

On the contrary, when I(z) = z and g is any outer function, then the space M =gKI is nearly invariant (as a matter of fact, the only function in M vanishing at0 is the zero function, the backward shift of which is trivially in M). Picking aninner function J with J(0) = 0 leads to a space JgKI which is not nearly invariant.

Sarason described in a precise way the extremal function g using a de Branges-Rovnyak spaces approach. This also allowed him to actually characterize all theisometric multipliers on a given model space K2

I . For this we need to associatetwo parameters with g. First it is clear that the extremal function is of unit norm,‖g‖ = 1. Then the measure |g(eit)|2dt is a probability measure. Now the function

1

2π

∫ π

−π

eit + z

eit − z|g(eit)|2dt

has positive real part and hence is the Cayley transform of a function b in the unitball of H∞:

(5.4)1 + b(z)

1− b(z)= G(z) :=

1

2π

∫ π

−π

eit + z

eit − z|g(eit)|2dt.

This may be reinterpreted as saying that |g(eit)|2dt is the Aleksandrov-Clark mea-sure associated with the function b. Observe that |g(eit)|2dt is absolutely continuousso that b is not an inner function (we won’t go further here into de Branges-Rovnyakspaces). Now set [44]

a =2g

G+ 1,

and observe that

b =G− 1

G+ 1.

Then

|a|2 + |b|2 =4|g|2 + |G|2 − 2ReG+ 1

|G+ 1]2,

and since |G| = |g| a.e. on T, we obtain |a|2 + |b|2 = 1 a.e. T. As a result, everynormalized H2-function g can be written as

g =a

1− b,

where a and b are bounded analytic functions and |a|2 + |b|2 = 1 a.e. T.

Sarason’s achievement in this context was to observe that whenever I divides b,then setting b = Ib0, the function

g =a

1− Ib0

multiplies isometrically on KI . We have to assume I(0) = 0 which ensures thatthe extremal function is in the space as it should. Observe that if g multiplies


isometrically on KI , where I(0) = 0, then it is automatically the extremal functionas seen from the following equality (we can assume that Re(g(0)) > 0):

sup{|gh(0)| : gh ∈ gK2I , ‖gh‖ = 1}

= g(0) sup{|h(0)| : h ∈ K2I , ‖h‖ = 1} = g(0)

(the last observation follows from the fact that |h(0)| ≤ ‖h‖ ≤ 1 and 1 ∈ KI). Letus state this observation as a separate result [44, Theorem 2].

Theorem 5.5 (Sarason). Let I be any inner function, I(0) = 0. Then for everycouple a, b0 of bounded holomorphic functions on T with |a|2+|b0|2 = 1 the function

g =a

1− Ib0

multiplies isometrically on KI (and is thus an extremal function of gK2I ).

5.2. Kernels of Toeplitz operators. Let ϕ ∈ L∞(T), and consider theToeplitz operator Tϕ. Since a Toeplitz operator is also defined by the operatoridentity (1.2) we easily see that the kernel of a Toeplitz operator is nearly invariant.Let us formally check this observation already mentioned earlier. Suppose Tϕf = 0and f(0) = 0, then

Tϕ(S∗f) = (S∗TϕS)(S

∗f) = S∗Tϕf = 0,

since when f(0) = 0 then SS∗f = f .

This introduces immediately the next question: is it possible to identify the kernelsof Toeplitz operators among arbitrary backward invariant subspaces? The answerto this question was given by Hayashi [24]. It is here that we need again rigidfunctions. He gave the following classification.

Theorem 5.6 (Hayashi). Let M be a nearly invariant subspace with associatedextremal function g and inner function I (I(0) = 0). Then M is the kernel of aToeplitz operator if and only if the function

g0 =a

1− b0

is rigid. Here g is given by the Sarason parameters as

g =a

1− Ib0.

It is known that when g0 is rigid so will be g (see e.g. [45, Corollary to Proposition6]). However, the converse is false: we refer to [24, Section 3] for an example of afunction g = a/(1− Ib0) such that g2 is rigid but g20 = (a/(1− b0))

2 is not, and thenearly invariant subspace gK2

I is not the kernel of a Toeplitz operator.

Observe that in this situation g is automatically outer (since rigid functions areouter as we have seen earlier). Note however that it is not required that b0 is outer,it can actually have inner factors.

Moreover, with the functions g and I associated with the kernel of a Toeplitzoperators Tϕ, ϕ ∈ L∞, it is easy to check that

KerTϕ = gKI = KerTIg/g.


So, as already observed, when considering kernels of Toeplitz operators, we canalways assume that ϕ is a unimodular function represented as

ϕ =Ig

g.

Note that we do not claim that an arbitrary Toeplitz operator Tϕ can be representedby TIg/g.

5.3. Surjectivity. As it turns out, the extremal function g of the kernel ofa Toeplitz operator contains even more information about the operator itself. Wehave seen earlier that rigidity of g20 , where g0 = a/(1 − b0), guarantees that thenearly invariant subspace is the kernel of a Toeplitz operator which can be chosento be TIg/g. If moreover |g0|2 satisfies the (A2) condition, then the Toeplitz operator

TIg/g is onto. That is the result in [21].

Theorem 5.7 (Hartmann-Sarason-Seip). Let ϕ be a unimodular symbol. If KerTϕ

is non-trivial and KerTϕ = gKI , where g = a/(1 − Ib0), then Tϕ is onto if andonly if |g0|2 ∈ (A2).

This result also makes a connection with left invertibility since Tϕ is onto if and onlyif Tϕ is left-invertible. As discussed in the introduction, left-invertibility of Toeplitzoperators plays a central role for describing Riesz sequences of reproducing kernels(or interpolating sequences) in model spaces (see [39, Section 4.4]).

Note that rigidity of g20 is equivalent to injectivity of Tg0/g0 while |g0|2 ∈ (A2) isequivalent to invertibility of Tg0/g0 , so the (A2) condition is a stronger requirement.

In order to construct a function g0 such that g20 is rigid but |g20 | /∈ (A2), one can usethe fact that the (A2)-condition is open, which means that when a weight satisfies(Ap) then it also satisfies (Ar) for r sufficiently close to p. It is easily seen that ifg0 ∈ H2 and 1/g0 ∈ H2 but 1/g0 /∈ H2+ε for any ε > 0, then g20 is rigid but doesnot define an (A2)-weight. For such an example we refer to [21].

6. Bourgain factorization

Another way of writing the kernel of a Toeplitz operator was explored by Dyakonov[17]. Using the Bourgain factorization he was able to prove the result below. Weshall first introduce some notation. When Tϕ is considered on Hp, then we denoteby Kerp Tϕ the corresponding kernel. By a triple we mean three functions (B, b, g),where B and b are Blaschke products and g is an invertible function in H∞.

Theorem 6.1 (Dyakonov, 2000). (i) For any 0 = ϕ ∈ L∞, there exists a triple(B, b, g) such that

(6.2) Kerp Tϕ =g

b(Kp

B ∩ bHp), 1 ≤ p ≤ ∞.

(ii) Conversely, given a triple (B, b, g), one can find a ϕ ∈ L∞ for which (6.2) holdstrue. In fact, it suffices to pick ϕ = bBg/g.


The triple appearing in (i) of the above theorem is not unique.

The representation (6.2) is interesting in that it is naturally related to the injectivityproblem and hence to the completeness problem in Kp

I . Indeed, if KpI ∩ bHp is non

trivial, then there is a function in KpI vanishing on the zeros of b. We will address

the completeness problem in Section 8.

Putting both representations together, i.e. the Hitt-Hayashi representation and theDyakonov representation of the kernel of a Toeplitz operator, we obtain (in the casep = 2):

GKI = KerTIG/G =g

b(K2

B ∩ bH2),

where G = a/(1 − Ib0) is the extremal function of KerTIG/G and (B, b, g) is the

associated Dyakonov triple. The triple (I, 1, G) found from Hitt’s result is in generalnot suitable for the Dyakonov description since G is in general neither in H∞ nor1/G.

Starting from his result on kernels of Toeplitz operators, Dyakonov went on furtherconstructing symbols ϕ for which the dimension of the kernels Kerp Tϕ varies in aprescribed way depending on p. More precisely he has the following result.

Theorem 6.3. Given exponents 1 = p0 < p1 < · · · < pN = ∞, and integersn1 > n2 > · · · > nN = 0, there exist Blaschke products B and b satisfying

dim(KpB ∩ bHp) = nj , p ∈ [pj−1, pj),

for j = 1, . . . , N .

The idea of the construction may be illustrated for N = 3 by choosing

G(z) =

m1∏k=1

(z − ζk)−1/p1 ·

n2∏l=1

(z − ηl)−1/p2 ,

where ζk and ηl are different points on the circle and m1 = n1−n2. Then G ∈ Hp,p < p1 and 1/G ∈ H∞ (implying that G is rigid). It remains to put ϕ = zn1G/G.For more details we refer the reader to [17].

7. More on the case p = 2

Concerning the theory developed in Section 5, the situation is much less clear inthe non-Hilbert situation. For instance the extremal function does no longer havethe nice multiplier properties it had for p = 2. This will be discussed in Subsection7.1 below.

We will also discuss a more general notion of rigidity. This can actually be definedfor the case q > 0 as it was defined for q = 1 (see Definition 4.3, and replace H1

by Hq), i.e. a function f ∈ Hq is rigid if it is uniquely determined in Hq (up to apositive constant multiplier) by its argument. As in Definition 4.3 this is equivalentto say that for every function g ∈ Hq

g/f ≥ 0 a.e. T =⇒ g = λf for some λ ≥ 0.

Concerning injectivity of specific Toeplitz operators, the following result can beshown by a similar argument as in the case p = 2 (see also [11, Theorem 5.4]).


Theorem 7.1. The Toeplitz operator Tf/f is injective on Hp if and only if f2 is

rigid in Hp/2.

With this notion of rigidity in mind, we will present some extensions of Hayashi’sresult to p = 2 as discussed in [11] and [10] in Subsection 7.2. Those authors alsoobserved that the notion of near invariance can be replaced in the following way.Let M be a closed subspace of Hp, then M is nearly invariant if

f ∈M, fz ∈ Hp =⇒ f ∈M.

Since fz /∈ Hp when f(0) = 0 the above observation is immediate. The idea is toreplace the function z by more general functions, and call M nearly η-invariant if

f ∈M, fη ∈ Hp =⇒ f ∈M.

Note that by the F -property (2.9) kernels of Toeplitz operators are automaticallynearly θ-invariant whenever θ is inner. However, we will not discuss this matter inthis survey.

We end the section with some results on minimal kernels.

7.1. Extremal functions. Extremal functions for general p are defined inexactly the same way as in the case p = 2, i.e. if M = Kerp Tϕ, then G is theunique solution to the extremal problem

sup{ReG(0) : g ∈M, ‖G‖p ≤ 1}.Concerning the extremal function of the kernel of a Toeplitz operator, as soon asp = 2 we lose the nice isometric multiplier property. In general G is even not anisomorphic multiplier as illustrated by the following result ([22]), and which relieson a notion of variational identity for extremal problems valid for general p (as canbe found in [47]).

Theorem 7.2 (Hartmann-Seip). Let Tϕ be a Toeplitz operator on Hp, 1 < p <∞,and G the extremal function of Kerp(Tϕ).

(1) If p ≤ 2, then GK2I ⊂ Kerp(Tϕ) ⊂ GKp

I and ‖f/G‖p ≤ cp‖f‖p for everyfunction f ∈ Kerp(Tϕ).

(2) If p ≥ 2, then GKpI ⊂ Kerp(Tϕ) ⊂ GK2

I and ‖f‖p ≤ cq‖f/G‖p for everyfunction f ∈ Kerp(Tϕ) (1/p+ 1/q = 1).

In general, none of these norm estimates can be reversed unless p = 2.

We refer to [22] were explicit examples are constructed showing the failure of thereverse inequalities in general.

Still, we can relate the function G with the situation p = 2. Since G ∈ Kerp Tϕ, we

have ϕG = Iψ, with I an inner function vanishing at 0 and ψ an outer function inHp. The function I will be called the associated inner function.

Theorem 7.3 (Hartmann-Seip). Let Tϕ be a Toeplitz operator on Hp, 1 < p <∞,and G the extremal function of Kerp Tϕ with associated inner function I. Then g =

Gp/2 is the extremal function of a nearly S∗-invariant subspace of H2 expressibleas gK2

I .


Since g is extremal for gK2I , from Sarason’s description we know that

g =a

1− Ib,

where a and b are bounded analytic functions with |a|2 + |b|2 = 1 a.e. on T. Anatural question is to ask whether gK2

I is the kernel of a Toeplitz operator, or inother words if g20 is rigid in H1 where g0 = a/(1− b). An answer to this question,especially whenKp

I is not finite dimensional, would certainly give some more insightin the structure of kernels of Toeplitz operators for p = 2. There are some partialanswers due to Camara and Partington that will be discussed in the next subsection.

7.2. Finite dimensional kernels of Toeplitz operators. In this subsectionwe discuss the special situation when Kerp Tϕ is finite dimensional. Recall fromTheorem 7.2 that ifG is the extremal function of the kernel, then Kerp Tϕ is includedbetween GKp

I and GK2I (with the right order depending on whether p ≥ 2 or p ≤ 2).

From that observation, the only way of Kerp Tϕ to be finite dimensional is that Iis a finite Blaschke product of order n. In this situation, we get in particular thatKp

I = K2I which is just a space of rational functions:

KpI = K2

I =

{p(z)∏n

j=1(1− λjz): p ∈ Poln−1

},

where λj are the zeros of I repeated with multiplicity, and Poln−1 are the analyticpolynomials of order at most n− 1. In particular

Kerp Tϕ = GK2I = GKp

I = G

{p(z)∏n

j=1(1− λjz): p ∈ Poln−1

}.

Observing that h(z) =∏n

j=1(1 − λjz) is an outer function which is obviouslyinvertible in H∞, we can also write

Kerp Tϕ =G

hPoln−1 .

See also [10, Theorem 2.8]. Note that replacing the denominator∏n

j=1(1−λjz) by

any other denominator h :=∏n

j=1(1− μjz), where μ are zeros of some other finiteBlaschke product with same degree as I we get

Kerp Tϕ = GK2I = GK2

B,

where G = Gh/h, and h/h is invertible in H∞.

A more subtle question is to decide whether a given space of the form GPoln−1

can be the kernel of a Toeplitz operator. A partial answer to this question can befound in Dyakonov’s result Theorem 6.2 which states that when G is boundedlyinvertible, then GKI is the kernel of a Toeplitz operator (and in particular whenI(z) = zn which gives KI = Poln−1). So the interesting case appears when G isunbounded. Let us consider the case n = 1, then we need that CG = Kerp Tϕ. Thisimmediately gives that G is the extremal function (thus necessarily outer) of thekernel and the associated inner function is I(z) = z. Hence

Kerp Tϕ = GKI = CG.


Note that when p = 2, then by Hayashi’s and Sarason’s result we know that

G(z) =a(z)

1− zb(z)

where |a|2 + |b|2 = 1 a.e. on T, and G20 = (a/(1 − b))2 is rigid. Even in the

more general finite dimensional situation (and for arbitrary 1 < p < +∞) it turnsactually out that the rigidity assumption is not needed for g20 but only for g2 (thiscontrasts to the general situation discussed for p = 2).

In this connection we start citing [11, Theorem 5.4] here for the disk.

Theorem 7.4 (Camara-Partington). Let f ∈ Hp and M = Cf the space generatedby f . Then M is the kernel of a Toeplitz operator if and only if f2 is rigid in Hp/2.

Observe that replacing GKz by GKbλ as explained above conserves rigidity (G and

G are simultaneously rigid or not).

The above result generalizes to finite dimensional spaces (see [10, Theorem 3.4])which we again cite for the disk.

Theorem 7.5 (Camara-Partington). Let M ⊂ Hp (1 < p <∞) be a finite dimen-sional subspace dimM = N < ∞. Then M is the kernel of a Toeplitz operator ifand only if M = GKzN = GPolN−1 and G2 is rigid in Hp/2.

We mention that for a unimodular symbol ϕ [33, Lemma p.15] states thatdimKerp Tϕ = n + 1 if and only if dimKerp Tbni ϕ

= 1 where bi is the Blaschkefactor of the upper half-plane vanishing at i.

7.3. Minimal kernels. Another observation from the work of Camara andPartington concerns minimality of kernels. For a given function f ∈ Hp, the min-imal kernel associated with f is the kernel of a Toeplitz operator such that thekernel of any other Toeplitz operator which annihilates f contains this minimalkernel. The following, in a sense natural, result holds ([11, Theorem 5.1]).

Theorem 7.6 (Camara-Partington). Given f = Iu ∈ Hp, 1 < p < ∞, where I isinner and u is outer. Then the minimal kernel is given by

Kmin(f) = Kerp TIu/u.

In other words, for every ϕ ∈ L∞(T), if f ∈ Kerp Tϕ, then

Kmin(f) ⊂ Kerp Tϕ.

Note that it is clear that Kmin(f) contains f . If f2 were rigid, then we wouldhave Kmin(f) = Cf as seen in Theorem 7.4, and the above theorem would betrivial. So, the interesting situation is when f is not rigid. This still remainslargely unexploited territory. We refer to [11] where some examples are discussedand again some links with rigidity are established. Note that when p = 2, and u canbe represented as a/(1−Ib) with |a|2+ |b|2 = 1, and u2

0 = (a/(1−b))2 is rigid, thenKmin(f) = uK2

I and u is extremal for this kernel. In particular, the minimal kernelcan have arbitrary dimension (it can be finite or infinite dimensional). Theorem[11, Theorem 5.2] gives a general criterion for Kmin(f) to be finite dimensional.


Theorem 7.7. Let 0 = f = Iu ∈ Hp. Then Kmin(f) is finite dimensional if andonly if I is a finite Blaschke product and Kerp Tu/u is finite dimensional.

We refer to [10, Theorem 3.7] which gives a criterion for the latter kernel to befinite-dimensional in terms of suitable factorizations of the symbol. Still, it is notclear how to obtain these factorizations from arbitrary symbols. Note that forinstance if u = (1 + θ)u0, where θ is inner so that (1 + θ)u0 is outer, then it isalways possible to write

u

u=

(1 + θ)u0

(1 + θ)u0=

θu0

u0,

which defines a symbol containing at least u0Kθ.

8. Triviality of Toeplitz kernels and applications

In the preceding sections of this survey we have largely discussed the structure of(non-trivial) kernels of Toeplitz operators. It is now time to discuss the problem ofdeciding when the kernel of a Toeplitz operator is non-trivial. An answer to thatquestion was proposed in the work by Makarov and Poltoratski who realized that aresolution of several classical open problems in the area of complex and harmonicanalysis can be reformulated in terms of injectivity of Toeplitz kernels. As alreadymentioned earlier, these include the completeness problem for a wide class of modelspaces, but also the gap problem, the type problem, the determinacy problem, andseveral others. The goal of this section is to give an overview of these results and toexplain the general ideas behind their proofs. The accent will be of course placedon the role of the Toeplitz kernels.

This somewhat longer section will be organized in the following way. In the firstpart we will concentrate on the triviality problem for Toeplitz kernels. In the secondpart we will touch upon several of the above mentioned applications.

As already mentioned in the introduction, a more natural setting for the resultsthat will be discussed in this section is the continuous setting, so everything inthis section will be done exclusively in the upper half-plane and the real line. LetTU : H2 → H2 be a Toeplitz operator with a unimodular symbol U . The mostinteresting case in applications is the one when the symbol is a quotient of twomeromorphic inner functions, i.e., U = ΘΨ with Θ, Ψ meromorphic inner functions.Recall that an inner function is said to be meromorphic if it can be extended to ameromorphic function on C. It is not hard to see that those are exactly the innerfunctions that can be represented as SaBΛ, where S(z) = eiz and BΛ(z) is theBlaschke product whose zero set Λ has no accumulation points on the real line. Wewill keep using S(z) (as Makarov and Poltoratski do) to denote the singular innerfunction eiz. Every meromorphic inner function Θ can be represented on the realline by Θ = eiθ, where θ : R → R is some strictly increasing continuous branchof the argument of Θ. Note that the fact that Θ is a meromorphic inner functionimplies that θ is real-analytic. Therefore, the symbols that are of most interest canbe written as U = eiγ , with γ : R → R being a real-analytic function of boundedvariation (difference of two real-analytic increasing functions).

The fundamental problem that was solved by Makarov and Poltoratski was theinjectivity problem for Toeplitz operators with this type of symbols. As we said


earlier one would like to be able to say whether TU is injective or not just by lookingat the argument γ of the symbol U . Recall first the two extreme cases discussed inSubsection 1.2, namely, U = Θ and U = Ψ. In the first case clearly TU = TΘ is notinjective, whereas in the second case TU = TΨ is injective. If we look at this in termsof the arguments it suggests that if the argument function γ is decreasing, then wedon’t have injectivity, and if the argument function is increasing we have injectivity(actually more than injectivity). It turns out that this trivial guess is not that farfrom the truth, but making it precise requires a great deal of effort. The first steptowards the goal is given by the following simple (but fundamental) lemma. Beforestating this lemma we recall that each outer function H ∈ H2 can be represented

(on the real line) as H = eh+ih, where h = log |H| and satisfies h ∈ L1(dΠ) and

eh ∈ L2(R). Here dΠ(t) = dt1 + t2

. Conversely, each function h : R → R satisfying

the last two conditions defines an outer function H determined on the real line by

H = eh+ih. We use h in the above formulas (and throughout this section) to denotethe Hilbert transform of h ∈ L1(dΠ) defined by

h(x) =1

πv.p.

∫R

(1

x− t+

1

1 + t2

)h(t)dt.

Lemma 8.1 (Makarov, Poltoratski). A Toeplitz operator TU : H2 → H2 with uni-modular symbol U is non-injective if and only if there exists an inner function Φand an outer function H ∈ H2 such that

U = ΦH

H.

Alternatively, in terms of arguments, TU is non-injective if and only if the real-analytic increasing argument γ of U = eiγ can be represented in the form γ = −ϕ−h, where ϕ is the argument of some meromorphic inner function and h ∈ L1(dΠ),eh ∈ L1(R).

Part of this lemma has already been discussed in (4.2).

Roughly speaking this lemma says that a Toeplitz operator TU is not injective ifU is “close to being equal” to Φ for some inner function Φ which is exactly oneof the extreme cases discussed above. The phrase “close to being equal”, that weused in the previous sentence is admittedly very imprecise. However, as we willtry to explain below it is not as wrong as it looks. Indeed, one of the crucialparts of Makarov-Poltoratski’s proof consists in showing that the term H/H canbe discarded by paying only a very small penalty.

The lemma above tells us that we need to find a way to tell when the argument ofthe symbol γ can be represented in the form −ϕ − h, where ϕ is the (increasing)argument of some meromorphic inner function and h ∈ L1(dΠ), eh ∈ L2(R). To dothis Makarov and Poltoratski first considered the following simpler problem: Howto tell by looking at γ whether it can be represented as γ = d+h, for some decreasingd and h ∈ L1(dΠ). They solved this problem under the assumption γ(±∞) = ∓∞.They were able to make this assumption since they actually worked with γε(x) =γ(x) − εx, ε > 0 instead of γ (it is not very hard to show that γε(±∞) = ∓∞ isa necessary condition for this representation to hold). To make the presentationclearer we are not going to switch to γε and will make the assumption on γ directly.


This assumption allows one to apply the classical Riesz sunrise idea by considering,as usual, the portion of the real line Σ(γ) ⊂ R defined by

Σ(γ) =

{x ∈ R : γ(x) = sup

t∈[x,∞)

γ(t)

}.

It is not hard to show that this set is open and consequently can be representedas a countable disjoint union of open intervals Σ(γ) = ∪In. Furthermore, for eachconnected component In = (an, bn) we have γ(an) = γ(bn) and γ(x) < γ(an) =γ(bn) for all x ∈ (an, bn). As a side note we mention that this construction wasused by F. Riesz to provide one of the first proofs that the maximal operator is ofweak type (1, 1). It was later used as a basis for the famous Calderon-Zygmunddecomposition (see e.g. [20]). Let γ∗ : R→ R be the function defined by γ∗(x) :=sup{γ(t) : t ∈ [x,∞)}. Clearly γ∗ is a decreasing function which is constant oneach of the intervals In. Moreover, the difference δ := γ∗ − γ is a non-negativefunction supported on Σ(γ). This way we have a representation γ = γ∗− δ with γ∗

decreasing. It is left to examine when the difference function δ can be representedas h for some h ∈ L1(dΠ). It turns out that this is not always the case. However,under an appropriate assumption on Σ(γ) (that will be given momentarily) it isnever too far from being true. The idea is to compare δ to a similar but muchsimpler function β which is also supported on Σ(γ). We define β on each In bythe tent function Tn whose graph is an isosceles right triangle with a base equal toIn, i.e., Tn(x) = dist(x,R \ In). Define then β : R → R as a linear combination ofthe tent functions β =

∑n εnTn with a freedom to choose the coefficients later. It

is easy to see that a sufficient condition for β ∈ L1(dΠ) is the so-called shortnesscondition:

(8.2)∑n

|In|21 + dist(0, In)2

<∞,

With this condition in place it is not hard to show that for any ε > 0 one can choosethe coefficients 0 < εn < ε so that

(1) |β′| � ε, and

(2) δ − β can be represented as a sum of atoms, i.e., it belongs to the realHardy space H1

Re(dΠ).

Since H1Re(dΠ) ⊂ L1(dΠ) this gives the following representation

γ(x)− εx = γ∗(x) + (δ(x)− β(x)) + (β(x)− εx).

Obviously the last term can be written as a sum of a bounded and a decreasingfunction. Therefore, it can be absorbed by the first two terms. This way, underthe shortness assumption, we obtain the desired representation of γ(x) − εx forany ε > 0. This beautiful idea, to use atoms to show that a function is a Hilberttransform of an L1-function, can be traced back to Beurling-Malliavin [5].

It is remarkable that the shortness condition above is almost necessary for γ tohave such a representation. Namely, it can be shown that if the shortness condition


fails, i.e., if

(8.3)∑n

|In|21 + dist(0, In)2

=∞,

then for no ε > 0 the function γ(x)+εx can be represented as a sum of a decreasing

function d and h, with h ∈ L1(dΠ).

To summarize, we have the following result as the first step towards the solution ofthe injectivity problem for Toeplitz kernels. It is sometimes called “little multipliertheorem”.

Theorem 8.4 (Makarov-Poltoratski). Let γ : R→ R be a real-analytic function ofbounded variation (difference of two increasing functions). Assume also that γ′ isbounded from below.

(i) If Σ(γ) does not satisfy the shortness condition (8.2), then for every ε > 0

the function γ(x)+εx cannot be represented as a sum d+ h of a decreasingfunction d and some h ∈ L1(dΠ).

(ii) If Σ(γ) satisfies the shortness condition 8.2, then for every ε > 0 the

function γ(x)−εx can be represented as d+h for some decreasing functiond and some h ∈ L1(dΠ).

This result has an interesting function-theoretic interpretation. Namely, in analogywith usual Toeplitz kernels in H2 one can also define Smirnov-Nevanlinna Toeplitzkernels Ker+ TU as sets of locally integrable functions f ∈ N+ such that U f ∈ N+,where we use N+ as usual to denote the Smirnov class on the upper half-plane. Itcan be shown that for a unimodular symbol U = eiγ the Toepliz kernel Ker+ TU

is trivial if and only if the argument γ can be represented as a sum d + h of adecreasing function d and some h ∈ L1(dΠ). This way one can view the theoremabove as a solution to the injectivity problem in the Smirnov-Nevanlinna case.

To solve the injectivity problem in the Hardy case requires an even more ingeniousidea which also goes back to the work of Beurling and Malliavin [4]. It is sometimescalled the “big multiplier theorem”. It shows that under some rather mild regularityassumptions on γ the shortness condition above is enough (up to an arbitrary smallε gap) to determine whether a function γ : R → R can be represented in the form

γ = −ϕ− h, where ϕ is argument of some meromorphic inner function, h ∈ L1(dΠ),and morover eh ∈ L1(R). Thus, in view of Lemma 8.1, the shortness condition canbe used as an almost necessary and sufficient condition to test injectivity of Toeplitzoperators. To prove this one needs to address the following two problems:

(1) replace the decreasing function d with a stronger requirement that d = −ϕ,where ϕ is an argument of some meromorphic function,

(2) in addition to condition h ∈ L1(dΠ) we would also need the conditioneh ∈ L2(R).

The second problem is much harder and its solution lies much deeper.

We will outline here only the main idea behind the proof skipping many of thetechnical difficulties. For details we refer to [34]. It should be noted that a similaridea was also used in [14]. The crucial part of this problem can be formulated in


the following way: Given a non-negative real-analytic function h ∈ L1(dΠ), with

h′ � 1, and ε > 0, find a function m : R → R such that h ≤ m and εx − m(x) isessentially an increasing function. Indeed, assume that we can show that such afunction m exists. Let γ = d+ h, with d decreasing and h ∈ L1(dΠ). Then

γ(x)− εx = d(x)− (εx− m(x)) + (h(x)− m(x)).

This implies that there exists an argument of a meromorphic inner function (not a

finite Blaschke product) ϕ1 and a bounded function u such that γ = −ϕ1 + u+ h1,where h1 = h − m ≤ 0. It is then not hard to show that any such function canbe represented as −ψ + k, where ψ is an argument of some inner function andek ∈ Lp(R) with p < 1 (see [34] for details). Finally, one can improve the condition

ek ∈ Lp(R), p < 1 to ek1 ∈ L1(R) by moving an appropriate part of ψ to k. We nowreturn to the main problem; to find a function m ∈ L1(dΠ) such that h ≤ m andε − m′(x) ≥ o(1). The first step consists in showing that the a priori assumptions

on h imply that h0(x) = h(x)/|x| satisfies∫h0(x)h

′0(x)dx < ∞. The second step

is then to use the finiteness of the last integral to set up the following extremalproblem: Minimize

I(m0) =

∫h0(x)h

′0(x)dx+ ε

∫|x|h0(x)dΠ(x)

over all non-negative functions m0 ∈ L1(dΠ) satisfying m0 ≤ h0. Usual abstractarguments can be used to show that this extremal problem has a solution m0.Finally, the way the extremal problem is set up allows one to show that m(x) =|x|m0(x) satisfies our desired conditions.

We can now state the “almost characterization” of the injectivity of Toeplitz oper-ators in terms of the shortness condition.

Theorem 8.5 (Makarov-Poltoratski). Let TU : H2 → H2 be a Toeplitz operatorwith a unimodular symbol U = eiγ . Let γ : R→ R be a real-analytic function suchthat γ = ϕ− ψ where ϕ is an argument of a meromorphic inner function and ψ isan increasing function such that |ψ′| % 1.

(i) If Σ(γ) does not satisfy the shortness condition 8.2, then for every ε > 0the Toeplitz operator TV : H2 → H2 with symbol V = Ueiεψ is NOTinjective.

(ii) If Σ(γ) satisfies the shortness condition 8.2, then for every ε > 0 theToeplitz operator TV : H2 → H2 with symbol V = Ue−iεψ is injective.

In fact, Makarov and Poltoratski proved a more general result which also includesToeplitz operators for which the function ψ appearing in the above theorem isallowed to satisfy |ψ′(x)| � |x|κ as x→ ∞. This can be viewed as an extension ofthe classical Beurling-Malliavin theorem.

We now show how this result and its extensions were used in the recent solutionsof several long-standing classical problems. More details as well as several otherapplications can be found in [41].


8.1. Completeness problem. We have already mentioned in the introduc-tion how the completeness of a sequence of reproducing kernels {kIλ}λ∈Λ in a modelspace KI can be characterized by the injectivity condition of TIBΛ

. The classi-cal problem for completeness of non-harmonic complex exponentials on L2[0, 1] isequivalent to the completeness problem for the normalized reproducing kernels inthe model space KS where S(z) = eiz. Thus, in this classical case, Theorem 8.5can be reformulated in terms of the Beurling-Malliavin densities. Namely,

Proposition 8.6. Let Λ = {λn} ⊂ R be a discrete sequence of real numbers. LetΘ = eiθ be some meromorphic inner function whose increasing argument θ satisfiesθ(λn) = 2nπ for all n.

(i) If |θ′(x)| % |x|κ, then sup{a ≥ 0 : KerTΘSa = {0}} is equal to D−BM (Λ),

the interior Beurling-Malliavin density of Λ.

(ii) For general θ we have inf{a ≥ 0 : KerTSaΘ = {0}} is equal to D+BM (Λ),

the exterior Beurling-Malliavin density of Λ. In other words, the radiusof completeness for the sequence of complex exponentials {eiλnx}, i.e., thesupremum of all a > 0 for which the sequence is complete in L2[0, a] isequal to D+

BM (Λ).

The same approach can be utilized to solve the completeness problems for otherfamilies of special functions which naturally show up as eigenfunctions of someclassical singular Sturm-Liouville (Schrodinger) operators. Many of these operatorsare singular at the endpoints which prevents direct application of the classicalBeurling-Malliavin result (which corresponds to the case when the characteristicfunction Ψ for the operator satisfies the condition |Ψ′(x)| � 1). In these instancesthe general form of Theorem 8.5 which allows a polynomial growth of |Ψ′(x)| needsto be used.

8.2. Spectral gap and oscillation. Recall from the introduction that in theclassical gap problem one is interested in the gaps in the support of the measure andits Fourier transform. The heuristics says that these gaps cannot be simultaneouslytoo big. In other words, if the support of the measure has gaps that are too big thenthe support of its Fourier transform cannot have too large gaps. It is customaryto call the gap in the support of the Fourier transform a spectral gap. The mostrudimentary form of this principle is the well-known fact that a measure that issupported on a finite interval [−a, a] (so it is zero on a big portion of the real line)cannot have a spectral gap of positive measure. Indeed, the Fourier transform ofsuch a measure is an entire function which obviously cannot vanish on a set ofpositive measure on the real line. A more advanced version is the Riesz brotherstheorem saying that a measure that is supported on [0,∞) (so still being zero on abig portion of R) also cannot have any spectral gaps. The general problem — thegap problem — can be formulated in the following way. Given a closed set X ⊆ R,determine the largest spectral gap that a measure supported on X can have. Moreprecisely, we would like to find a way to compute the so-called gap characteristicsG(X) of X which is defined by

G(X) := sup{a ≥ 0 : ∃μ, suppμ ⊆ X, μ = 0 on [0, a]}.


There are several classical results that address this problem, especially the partwhen G(X) = 0. The well-known Beurling gap theorem [30] says that if a non-zero measure μ is supported on a closed set X whose complement Xc is long inthe sense of 8.3, then μ cannot vanish on any interval of positive length. Anotherwell-known result in this direction is due to Levinson [32] who showed that if thetail M(x) = |μ|(x,∞) of a non-zero measure μ satisfies

∫logM(x)dΠ(x) = −∞,

then again the measure μ cannot have a spectral gap of positive length. Moresophisticated results generalizing these two classical statements were proved byde Branges [14], and later Benedicks [3].

A first step towards the final solution of the gap problem was obtained by Poltoratskiand the second author in [36] where it was proved that for closed sets X whichare discrete and separated the gap characteristics of X is equal to the interiorBeurling-Malliavin density of X.

Theorem 8.7 (Mitkovski-Poltoratski). If X = {xn} is a separated discrete set,i.e., infm �=n |xn − xm| > 0, then G(X) = D−

BM (X).

This result was later extended by Poltoratski [40] to arbitrary closed sets X. Heproved that in the general case, besides the density ofX, an additional subtle energycondition enters into play. The shortest way to formalize the energy condition isthrough the notion of d-uniform sequences. A real sequence Λ = {λn} is d-uniformif 1) it is regular with density d, i.e., there exists a sequence of disjoint intervals{In} in R satisfying the shortness condition

∑n(|In|/(1 + dist(0, In)))

2 < ∞, and|In| → ∞ as n→ ±∞, such that

#(Λ ∩ In)− d|In| = o(|In|) as |n| → ∞;

and 2) it satisfies the following energy condition: there exists a short partition {In}such that ∑

n

#(Λ ∩ In)2 log+ |In| − EIn(dnΛ)

1 + dist(0, In)2<∞,

where EI(μ) =∫∫

I,Ilog |x − y|dμ(x)dμ(y), is the usual energy of the compactly

supported measure 1I(x)dμ(x). The measure dnΛ entering in the energy conditionabove is the counting measure on Λ. The solution of the gap problem is given bythe following theorem.

Theorem 8.8 (Poltoratski [40]). For any closed set X ⊂ R,

G(X) = π sup{d : ∃ d-uniform sequence {λn} ⊂ X}.

Very recently, the second author jointly with Poltoratski, refined these results evenfurther [37], and obtained a generalization of the Beurling spectral gap theoremthat strengthened this theorem by a factor of two. More precisely, in this paper,among other things, a metric description of the gap characteristic was obtainedwhen the positive and the negative parts of the measure are supported in certainprescribed parts of the set. The gap characteristic of a pair of disjoint closed subsetsA and B of R is defined by

G(A,B) = sup{a > 0 : ∃μ ≡ 0, suppμ− ⊆ A,

suppμ+ ⊆ B, μ ≡ 0 on (−a, a)}.


As in the case of G(X), the description of G(A,B) depends on two properties of Aand B: their density and their energy.

Theorem 8.9 (Mitkovski-Poltoratski [37]). For any disjoint closed sets A,B ⊂ R,

G(A,B) = π sup{d : ∃ d-uniform sequence {λn},{λ2n} ⊂ A, {λ2n+1} ⊂ B}.

As a simple consequence of this oscillation theorem one obtains a sharpening of thefamous oscillation theorem of Eremenko and Novikov [18] (their theorem solved anold problem of Grinevich from 1964 that was included in Arnold’s list of problems).

Theorem 8.10 (Mitkovski-Poltoratski [37]). If σ is a nonzero signed measure withspectral gap (−a, a) then there exists an a/π-uniform sequence {λn} such that σhas at least one sign change in every (λn, λn+1).

The crucial step towards the proof of both the gap theorem and the oscillationtheorem above is the following result about annihilating measures for a very largeclass of de Branges spaces. This is where Toeplitz kernels enter into play.

Theorem 8.11 (Mitkovski-Poltoratski [37]). Suppose that BE is a regular

de Branges space and let Φ(z) = E(z)/E(z) be the corresponding meromorphicinner function. If μ is a non-zero measure that annihilates BE then there exists ameromorphic inner function Θ such that {Θ = 1} ⊂ suppμ+ for which the Smirnov-Nevanlinna kernel Ker+ TΦΘ is non-trivial. The same holds for the support of thenegative part of μ as well.

8.3. Polya’s problem. As discussed in the introduction, a Polya sequence isa separated real sequence with the property that there is no non-constant entirefunction of exponential type zero (entire functions that grow slower than exponen-tially in each direction) which is bounded on this sequence. Historically, first resultson Polya sequences were obtained in the work of Valiron [48], where it was provedthat the set of integers Z is a Polya sequence. Later, in ignorance of the work ofValiron, this problem was popularized by Polya, who posted it as an open problem.Subsequently many different proofs and generalizations were given (see for examplesection 21.2 of [31] or chapter 10 of [8] and references therein).

In his famous monograph [32] Levinson showed that if |λn − n| ≤ p(n), where p(t)satisfies

∫p(t) log |t/p(t)|dt/(1 + t2) < ∞ and some smoothness conditions, then

Λ = {λn} is a Polya sequence. In the same time for each such p(t) satisfying∫p(t)dt/(1 + t2) = ∞ he was able to construct a sequence Λ = {λn} that is not a

Polya sequence. As it often happens in problems from this area, the constructiontook considerable effort (see [32], pp. 153-185). Closing the gap between Levinson’ssufficient condition and the counterexample remained an open problem for almost25 years until de Branges [14] solved it assuming extra regularity of the sequence.These results have remained strongest for a very long time.

Jointly with A. Poltoratski [36] the second author recently derived the followingcomplete characterization of Polya sequences.

Theorem 8.12 (Mitkovski-Poltoratski [36]). A separated real sequence Λ = (λn)n∈Z

is a Polya sequence if and only if its interior Beurling-Malliavin density D−BM (Λ)

is positive.


Here are some details of how Toeplitz kernels enter in the solution of this problem.Let F be an entire function of exponential type 0 which is bounded (say by M)on Λ. Choose a meromorphic inner function Θ such that {Θ = 1} = Λ. Thisshould be done carefully, but we won’t go into technicalities here. The choice of Θand the description of D−

BM (Λ) in terms of Toeplitz kernels imply that KerTΘSc isnontrivial for all c > 0. As before S denotes the inner function S(z) = eiz. Picksome h ∈ KerTΘSc with L2-norm 1. It can be shown that one has the followingClark-type representation for the function hF :

(8.13) h(z)F (z) =1−Θ(z)

2πi

∫F (t)h(t)

t− zdσ(t),

where σ is the Clark measure associated to the inner function Θ. Furthermore forany n ∈ N, Fn is still an entire function of exponential type 0 (bounded by Mn onΛ) so that one has the same Clark-type representation for hFn as well. A simpleestimate then provides a bound for h(x)F (x)n for all x ∈ R. Taking the n-th rootand using that n is arbitrary help to get rid of h. This way we obtain that F mustbe bounded on R which combined with the fact that F is of zero type implies thatF is constant function.

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[45] Donald Sarason, Kernels of Toeplitz operators, Toeplitz operators and related topics (SantaCruz, CA, 1992), Oper. Theory Adv. Appl., vol. 71, Birkhauser, Basel, 1994, pp. 153–164.MR1300218

[46] Donald Sarason, Sub-Hardy Hilbert spaces in the unit disk, University of Arkansas LectureNotes in the Mathematical Sciences, 10, John Wiley & Sons, Inc., New York, 1994. A Wiley-Interscience Publication. MR1289670

[47] Harold S. Shapiro, Topics in approximation theory, Springer-Verlag, Berlin-New York, 1971.With appendices by Jan Boman and Torbjorn Hedberg; Lecture Notes in Math., Vol. 187.MR0437981

[48] G. Valiron. Sur la formule d’interpolation de Lagrange. Bull. Sci. Math, 49(2):181–192, 1925.[49] A. L. Vol′berg and S. R. Treil′, Embedding theorems for invariant subspaces of the inverse

shift operator (Russian, with English summary), Zap. Nauchn. Sem. Leningrad. Otdel. Mat.Inst. Steklov. (LOMI) 149 (1986), no. Issled. Linein. Teor. Funktsii. XV, 38–51, 186–187, DOI10.1007/BF01665042; English transl., J. Soviet Math. 42 (1988), no. 2, 1562–1572. MR849293

Institut de Mathematiques de Bordeaux, Universite de Bordeaux, 351 cours de la

Liberation 33405 Talence Cedex, France


Department of Mathematical Sciences, Clemson University, O-110 Martin Hall, Box

340975, Clemson, South Carolina 29634


URL: http://people.clemson.edu/~mmitkov/


Some open questions in analysis for Dirichlet series

Eero Saksman and Kristian Seip

Abstract. We present some open problems and describe briefly some possi-ble research directions in the emerging theory of Hardy spaces of Dirichlet se-ries and their intimate counterparts, Hardy spaces on the infinite-dimensionaltorus. Links to number theory are emphasized throughout the paper.

1. Introduction

We have in recent years seen a notable growth of interest in certain functionalanalytic aspects of the theory of ordinary Dirichlet series

∞∑n=1

ann−s.

Contemporary research in this field owes much to the following fundamental obser-vation of H. Bohr [20]: By the transformation zj = p−s

j (here pj is the jth prime

number) and the fundamental theorem of arithmetic, an ordinary Dirichlet seriesmay be thought of as a function of infinitely many complex variables z1, z2, .... Moreprecisely, in the Bohr correspondence,

(1) F (s) :=∞∑

n=1

ann−s ∼ f(s) :=

∑ν∈N∞

fin

aνzν ,

where n = pν11 · · · p

νk

k and we identify aν with the corresponding coefficient an, andN∞

fin stands for the finite sequences of positive indices. By a classical approximationtheorem of Kronecker, this is much more than just a formal transformation: If,say, only a finite number of the coefficients an are nonzero (so that questions aboutconvergence of the series are avoided), the supremum of the Dirichlet polynomial∑

ann−s in the half-plane Re s > 0 equals the supremum of the corresponding

polynomial on the infinite-dimensional polydisc D∞. In a groundbreaking work ofBohnenblust and Hille [19], it was later shown that homogeneous polynomials—thebasic building blocks of functions analytic on polydiscs—may, via the method ofpolarization, be transformed into symmetric multilinear forms. Bohnenblust andHille used this insight to solve a long-standing problem in the field: Bohr had shown

2010 Mathematics Subject Classification. Primary 11C08, 11C20, 11M06, 11N60, 32A05,30B50, 42B30, 46B09, 46G25, 47B35, 60G15, 60G70.

The first author was supported by the Finnish CoE in Analysis and Dynamics Research andby a Knut and Alice Wallenberg Grant.

The second author was supported by Grant 227768 of the Research Council of Norway.


179

180 EERO SAKSMAN AND KRISTIAN SEIP

that the width of the strip in which a Dirichlet series converges uniformly but notabsolutely is ≤ 1/2, but Bohnenblust and Hille were able to prove that this upperestimate is in fact optimal.

In retrospect, one may in the work of Bohr and Bohnenblust–Hille see theseeds of a theory of Hardy Hp spaces of Dirichlet series. However, this researchtook place before the modern interplay between function theory and functionalanalysis, as well as the advent of the field of several complex variables, and thearea was in many ways dormant until the late 1990s. One of the main goals of the1997 paper of Hedenmalm, Lindqvist, and Seip [48] was to initiate a systematicstudy of Dirichlet series from the point of view of modern operator-related functiontheory and harmonic analysis. Independently, at the same time, a paper of Boasand Khavinson [18] attracted renewed attention, in the context of several complexvariables, to the original work of Bohr.

The main object of study in [48] is the Hilbert space of Dirichlet series∑

n ann−s

with square summable coefficients an. This Hilbert space H2 consists of functionsanalytic in the half-plane Re s > 1/2. Its reproducing kernel at s is ks(w) = ζ(s+w),where ζ is the Riemann zeta function. By the Bohr correspondence, H2 may bethought of as the Hardy space H2 on the infinite-dimensional torus T∞. Bayart[10] extended the definition to any p > 0 by defining Hp as the closure of Dirichlet

polynomials F (s) =∑N

n=1 ann−s under the norm

‖F‖Hp :=(

limT→∞

1

2T

∫ T

−T

|F (it)|pdt)1/p

.

By ergodicity (or see [80] for an elementary argument), the Bohr correspondenceyields the identity

(2) ‖F‖Hp = ‖f‖Hp(T∞) :=(∫

T∞|f(z)|pdm∞(z)

)1/p,

where m∞ stands for the Haar measure on the distinguished boundary T∞, i.e.,for the product of countably many copies of normalized Lebesgue measure on thecircle T. Since the Hardy spaces on the infinite dimensional torus Hp(T∞) may bedefined as the closure of analytic polynomials in the Lp-norm on T∞, it follows thatthe Bohr correspondence provides an isomorphism between the spaces Hp(T∞) andHp. This linear isomorphism is both isometric and multiplicative.

The classical theory of Hardy spaces and the operators that act on them servesas an important source of incitement for the field of Dirichlet series that has evolvedafter 1997. Two distinct features should however be noted. First, a number ofnew phenomena, typically crossing existing disciplines, appear that are not presentin the classical situation. Second, many of the classical objects change radicallyand require new viewpoints and methods in order to be properly understood andanalyzed.

In the following sections, we sketch briefly some research directions and listseveral open problems (thus updating [47]). In our selection of problems, we havefollowed our own interests and made no effort to compile a comprehensive list. Asa consequence, several interesting recent developments such as for instance [13] or[68] will not be accounted for and discussed. The reader should also notice that thedifficulty of the problems may vary considerably. It seems likely that for some of theproblems mentioned below, further progress will require novel and unconventional

SOME OPEN QUESTIONS IN ANALYSIS FOR DIRICHLET SERIES 181

combinations of tools from harmonic, functional, and complex analysis, as well asfrom analytic number theory.

2. Basic properties of the spaces Hp and Hp(T∞)

The study of the boundary limit functions in the spaces Hp has a number ofinteresting features. Several central points have been clarified, such as questionsconcerning convergence of the Dirichlet series [49], to what extent ergodicity ex-tends to the boundary [80], properties of the boundary limit functions for Dirichletseries in H2 [72], and zeros of functions in H2 and, at least partially, in Hp for p > 2[83]. The diversity of techniques involved is considerable, ranging from functiontheory in polydiscs and ergodic theory to classical harmonic analysis, Hardy spacetechniques, Fourier frames, estimates for solutions of the ∂ equation, and Ramanu-jan’s estimates for the divisor function. Still, a very natural problem first consideredin [10] (see [80] for further discussion on it) remains unsolved and represents oneof the main obstacles to further progress:

Problem 2.1 (The embedding problem). Is the Lp integral of a Dirichlet poly-

nomial∑N

n=1 ann−s over any segment of fixed length on the vertical line Re s = 1/2

bounded by a universal constant times ‖∑N

n=1 ann−s‖pHp?

This is known to hold for p = 2 and thus trivially for p an even integer. Onemay notice a curious resemblance with Montgomery’s conjectures concerning norminequalities for Dirichlet polynomials (see [66, pp. 129, 146] or [56, p. 232–235]).It remains to be clarified if there is a link between this question and Montgomery’sconjectures.

An affirmative answer to Problem 2.1 for p < 2 would have immediate func-tion theoretic consequences regarding for instance zero sets and boundary limits.Namely, following [72], we would be able to answer

Problem 2.2. Characterize Carleson measures for Hp on {Re s > 1/2} forp < 2.

More modest but nontrivial open questions are:

Problem 2.3. Do the zero sets of functions inHp for p < 2 satisfy the Blaschkecondition in the half-plane Re s > 1/2?

Problem 2.4. Are elements of Hp for p < 2 locally in the Nevanlinna class?

There are similar problems of a dual flavor regarding interpolating sequencesfor Hp. Indeed, it follows from [73] that the Shapiro–Shields version of Carleson’sclassical theorem in the half-plane Re s > 1/2 remains valid when 1/p is an eveninteger. We would like to know if this result extends to other values of p.

By a theorem of Helson [50], the partial sum operator [50] is uniformly boundedon Hp for 1 < p <∞ (see [4] for an alternative treatment), and hence the functionsn−s for n ≥ 1 form a basis for Hp for these exponents p. The following questionsstated in [4] seem to be open:

Problem 2.5. Does Hp have an unconditional basis if p ∈ (1,∞) and p = 2?

Problem 2.6. Does H1 have a basis? Does it have an unconditional basis?


The last two problems are equivalent to corresponding statements for Hp(T∞).There are also natural and interesting questions that are specific for function theoryin infinite dimensions. In [5] (see [80] for the first steps in this direction), it wasshown that Fatou or Marcin- kiewicz–Zygmund-type theorems on boundary limitsremain true for all classes Hp(T∞) or for their harmonic counterparts hp(T∞),assuming fairly regular radial approach to the distinguished boundary T∞; thesimplest example of such approach is of the form (reiθ1 , r2eiθ2 , r3eiθ3 , . . .) withr ↑ 1−. However, [5] also constructs an example of an element f in H∞(T∞) suchthat at almost every boundary point, f fails to have a radial limit under a certainradial approach that is independent of the boundary point.

Problem 2.7. Give general conditions for a radial (or non-tangential) approachin D∞ to T∞ such that Fatou’s theorem holds for elements in Hp(T∞).

The Hp spaces are well defined (via density of polynomials) also in the range0 < p < 1. Again, one may consider the analogue of the embedding problem (nowstated in term of local Hardy spaces on Re s = 1/2). For all values other than p = 2,even partial non-trivial results pertaining to the following widely open question (see[80]) would be interesting.

Problem 2.8. Describe the dual spaces of Hp.

3. Operator theory and harmonic analysis

Viewing our Hardy spaces as closed subspaces of the ambient Lp spaces onthe infinite-dimensional torus T∞, we are led to consider classical operators likethe Riesz projection (orthogonal projection from L2 to H2), Hankel operators, andFourier multiplier operators. The paper [4] contains some results on multipliersand Littlewood–Paley decompositions. It has become clear, however, that most ofthe classical methods are either not relevant or at least insufficient for the infinite-dimensional situation. For example, the classical Nehari theorem for Hankel forms(or small Hankel operators) does not carry over to T∞, see [74]. This leads us to askif a reasonable replacement can be found and, more generally, how the different rolesand interpretations of BMO (the space of functions of bounded mean oscillation)manifest themselves in our infinite-dimensional setting.

Problem 3.1. What is the counterpart to Nehari’s theorem on T∞? In partic-ular, what can be said about the Riesz projection of L∞(T∞) and other BMO-typespaces on T∞?

This and similar operator theoretic problems may be approached along severaldifferent paths. In [29], a natural analogue of the classical Hilbert matrix wasidentified and studied. This matrix was referred to as the multiplicative Hilbertmatrix because its entries am,n := (

√mn log(mn))−1 depend on the product m · n.

This matrix represents a bounded Hankel form on H20×H2

0 with spectral problemssimilar to those of the classical Hilbert matrix. (Here H2

0 denotes the subspace ofH2 consisting of functions that vanish at +∞.) Its analytic symbol ϕ0 is a primitiveof −ζ(s + 1/2) + 1, and by analogy with the classical situation, we are led to thefollowing problem.

Problem 3.2. Is the symbol ϕ0(s) = 1 +∑∞

n=2(log n)−1n−1/2−s the Riesz

projection of a function in L∞(T∞)?


It is interesting to notice that a positive answer to Problem 2.1 for p = 1would yield a positive answer to this question, via an argument involving Carlesonmeasures. We refer to [29] for details.

The beautiful pioneering contribution of Gordon and Hedenmalm [42] and agrowing number of other papers have established the study of composition operatorson Hardy spaces of Dirichlet series as an active research area in the interface of oneand several complex variables. In the series of papers [12, 16, 78], quantitativeand functional analytic tools have been developed in this context, for examplenorm estimates for linear combinations of reproducing kernels, Littlewood–Paleyformulas, and (soft) functional analytic remedies for the fact that Hp fails to becomplemented when 1 ≤ p <∞ and p = 2.

Problem 3.3. Characterize the compact composition operators on H2.

4. Moments of sums of random multiplicative functions

There has during the last few years been an interesting interplay between thestudy of sums of random multiplicative functions and problems and methods comingfrom Hardy spaces. This topic has a long history, beginning with an importantpaper of Wintner [85]. One of the links to Hardy spaces comes from

Problem 4.1 (Helson’s problem [52]). Is it true that ‖∑N

n=1 n−s‖H1 = o(

√N)

when N →∞.

This intriguing open problem arose from Helson’s study of Hankel forms and acomparison with the one-dimensional Dirichlet kernel. However, it seems to be morefruitful to think of the problem in probabilistic terms, viewing the functions p−s

j asindependent Steinhaus variables. Resorting to a decomposition into homogeneouspolynomials and using well known estimates for the arithmetic function Ω(n), it was

shown in [26] that ‖∑N

n=1 n−s‖H1 (

√N(logN)−0.05616. This was later improved

by Harper, Nikeghbali, and Radziwi�l�l [45] who, using methods from [44], found the

lower bound√N(log logN)−3+o(1). In a recent preprint [46], Heap and Lindqvist

made a prediction based on random matrix theory that Helson’s conjecture is false.The paper [26] also gave a precise answer to the question of for which m the

homogeneous Dirichlet polynomials∑

Ω(n)=m,n≤N n−s have comparable L4 and L2

norms. Indeed, this happens if and only m is, in a precise sense, strictly smallerthat 1

2 log logN . An interesting problem coming from analytic number theory andthe work of Hough [54], is to extend this result to higher moments.

Problem 4.2. Assume k is an integer larger than 1. For which m (dependingon N) will the L2k norms of m-homogeneous Dirichlet polynomials of length N becomparable to their L2 norms?

Cancellations in the partial sums of the Riemann zeta function on the criticalline can be studied through a similar problem concerning Hp norms.

Problem 4.3. Determine the asymptotic behavior of the norms∥∥∑Nn=1 n

−1/2−s∥∥Hp when N →∞ for 0 < p ≤ 1.

An interesting modification of this problem is the following.

Problem 4.4. Determine the precise asymptotic growth of the norms

‖∑N

n=1[d(n)]γn−1/2−s‖Hp when N →∞ for p ≤ 1.


A more general problem is to do the same for polynomials with coefficients rep-resented by multiplicative functions satisfying appropriate growth conditions.[22]established the inequality

(3)

(N∑

n=1

|μ(n)||an|2[d(n)]log plog 2−1

)1/2

≤ ‖f‖Hp ,

valid for f(s) =∑N

n=1 ann−s and 0 < p ≤ 2, where μ(n) is the Mobius function.

This inequality, which should be recognized as an Lp-analogue of an inequality ofHelson [50], yields the lower bound

(4)∥∥∥ N∑

n=1

n−1/2−s∥∥∥Hp( (logN)p/4

for all 0 < p < ∞. An estimate in the opposite direction in the range 1 < p < ∞follows by applying Helson’s theorem on the Lp boundedness of the partial sumoperator [50] on suitably truncated Euler product. When p = 1 the same methodyields that an additional factor log logN appears on the right-hand side when (is replaced by ) in (4), and thus Problems 4.3 and 4.4 remain open exactly in therange p ≤ 1. Some results for Problem 4.4 are contained in the manuscript [24].

A closely related and more general problem concerns the natural partial sumoperator of the Dirichlet series whose Lp norm can be estimated by Helson’s theorem[50] for finite p and a result from [8] for p =∞.

Problem 4.5. Determine the precise asymptotic growth of the norm of the

partial sum operator SN :∑∞

n=1 ann−s �→

∑Nn=1 ann

−s when N → ∞ for p = 1(or more generally, for p ≤ 1 or p =∞).

In the case p = 1, a trivial one dimensional estimate yields a lower bound oforder log logN , whereas [24] gives an upper bound of order logN/ log logN , sothat presently there is a large gap between the known bounds.

We finish this section by recalling a pointwise version of the analogue of Helson’sproblem on the torus. Thus, for primes p let χ(p) : we i.i.d random variables with

uniform distribution on T and define χ(n) =∏�

k=1 χ(pk)�l for n = pk1

1 . . . pk

� .

Problem 4.6. Determine the almost sure growth rate (in N) of the charactersum

N∑n=1

χ(n).

This problem stems from Wintner, and is listed by Erdos, although in the orig-inal version instead χ(p):s are Rademacher variables. Deep results on the problemwere provided by Halasz [43] in the 1980s, and the recent papers [44] of Harperand [62] of Lau, Tenenbaum, and Wu contain remarkable improvements.

5. Estimates for GCD sums and the Riemann zeta function

The study of greatest common divisor (GCD) sums of the form

(5)

N∑k,�=1

(gcd(nk, n�))2α

(nkn�)α


for α > 0 was initiated by Erdos who inspired Gal [40] to solve a prize problem ofthe Wiskundig Genootschap in Amsterdam concerning the case α = 1. Gal provedthat when α = 1, the optimal upper bound for (5) is CN(log logN)2, with C anabsolute constant independent of N and the distinct positive integers n1, ..., nN .The problem solved by Gal had been posed by Koksma in the 1930s, based on theobservation that such bounds would have implications for the uniform distributionof sequences (nkx) mod 1 for almost all x.

Using the several complex variables perspective of Bohr and seeds found in[61], Aistleitner, Berkes and Seip [2] proved sharp upper bounds for (5) in therange 1/2 < α < 1 and a much improved estimate for α = 1/2, solving in par-ticular a problem of Dyer and Harman [34]. The method of proof was based onidentifying (5) as a certain Poisson integral on D∞. The acquired bounds werealso used to establish a Carleson–Hunt-type inequality for systems of dilated func-tions of bounded variation or belonging to Lip1/2, a result that in turn settled twolongstanding problems on the almost everywhere behavior of systems of dilatedfunctions. The Carleson–Hunt inequality and the original inequality of Gal (see (6)below) were later optimized by Lewko and Radziwi�l�l [60].

Additional techniques were introduced by Bondarenko and Seip [25,26] to dealwith the limiting case α = 1/2, and finally the range 0 < α < 1/2 was clarified in[22]. Writing

Γα(N) :=1

Nsup

1≤n1<n2<···<nN

N∑k,�=1

(gcd(nk, n�))2α

(nkn�)α,

we may summarize the state of affairs as follows:

Γ1(N) ∼ 6e2γ

π2log logN(6)

log Γα(N) *α(logN)(1−α)

(log logN)α, 1/2 < α < 1

log Γ1/2(N) *√

logN log log logN

log logN

log Γα(N)− (1− 2α) logN *α log logN, 0 < α < 1/2,

where in (6), γ denotes Euler’s constant; these estimates remain the same if wereplace Γα(N) by the possibly larger quantity

Λα(N) := sup1≤n1<n2<···<nN ,‖c‖=1

N∑k,�=1

ckc�(gcd(nk, n�))

2α

(nkn�)α,

where the vector c = (c1, c2, ..., cN ) consists of nonnegative numbers and ‖c‖2 :=c21 + c22 + · · · c2N .

Aistleitner [1] made the important observation that such estimates can be usedto obtain Ω-results for the Riemann zeta function. Indeed, using Hilberdink’s ver-sion of the resonance method [44], he found a new proof of Montgomery’s Ω-resultsfor ζ(α+ it) in the range 1/2 < α < 1 [65]. In turn, Bondarenko and Seip appliedthe particular set {n1, n2, ..., nN} yielding the lower bound for Λ1/2(N) in combina-tion with the resonance method of Soundararajan [84] to obtain (unconditionally)

the following: given c < 1/√2, there exists a β, 0 < β < 1, such that for every


sufficiently large T

(7) supt∈(Tβ ,T )

|ζ(1/2 + it)| ≥ exp

(c

√log T log log log T

log log T

).

This gives an improvement by a power of√log log log T compared with previously

known estimates [7,84].We list two rather general questions pertaining to these recent developments.

Problem 5.1. Link the estimates for GCD sums to the function and operatortheory of the spaces Hp.

Problem 5.2. Develop further applications to and links with the Riemannzeta function.

Problem 5.1 originates in the observation from [2] that GCD sums can beinterpreted as Poisson integrals on polydiscs. Taking into account the prominentrole played by Poisson integrals and the Poisson kernel in the classical setting (forinstance in connection with functions of bounded mean oscillation), we are led toask for potential function and operator theoretic interpretations or applications ofour estimates for GCD sums.

Finally, we would like to give an example related to the rather vague and generalProblem 5.2. It concerns estimates relating the size of the coefficients to the Hp

norm of a Dirichlet series, which can be traced back to Bohr’s problem of computingthe maximal distance between the abscissas of absolute and uniform convergence.Bohnenblust and Hille’s solution to this problem [19] relied on a revolutionarymethod of polarization for estimating the size of the coefficients of homogeneouspolynomials. There was a revival of interest in Bohnenblust and Hille’s work afterthe 1997 paper of Boas and Khavinson [18] on so-called Bohr inequalities. Itwas gradually recognized that the original estimate of order mm for the constantC(m) in the Bohnenblust–Hille inequality was not sufficiently accurate to reach thedesired level of precision in various applications. Based on a re-examination of theoriginal proof, a sophisticated version of Holder’s inequality due to Blei [17], anda Khinchin-type inequality of Bayart [10], Defant, Frerick, Ounaıes, Ortega-Cerda,and Seip established in [31] that C(m) grows at most exponentially in m. This wasrecently improved further by Bayart, Pellegrino, and Seoane-Sepulveda [15] whowere able to show, by taking a new approach to Blei’s inequality, that C(m) growsat most as exp

(c√m logm

)for some constant c.

The most important application of the improved version of the Bohnenblust–Hille inequality was to the compute the Sidon constant S(N) which is defined as thesupremum of the ratio between |a1|+· · ·+|aN | and supt∈R

∣∣a1+a22it+· · ·+aNN it

∣∣,with the supremum taken over all possible choices of nonzero vectors (a1, ..., aN ) inCN . The following remarkably precise asymptotic result holds [31]:

S(N) =√N exp

((− 1√

2+ o(1)

)√logN log logN

)when N → ∞. This formula has a long history and relies on the contribution ofmany researchers, most notably Queffelec and Konyagin [58] and de la Breteche[28]. The proof involves an unconventional blend of techniques from function the-ory on polydiscs (the Bohnenblust–Hille inequality), analytic number theory (theDickman function), and probability (the Salem–Zygmund inequality).


There is a striking resemblance between the formula for the Sidon constantS(N) and the following conjecture from [35], based on arguments from randommatrix theory, conjectures for moments of L-functions, and also by assuming arandom model for the primes [35]:

max0≤t≤T

∣∣ζ(1/2 + it)∣∣ = exp

((− 1√

2+ o(1)

)√log T log log T

)when T →∞. It is natural to ask if this resemblance is more than just a coincidence.

6. Random Dirichlet series

A classical result due to Selberg states that the distribution of the Riemannzeta function on the critical line is asymptotically Gaussian, after suitable renor-malisation. More precisely, the distribution of{(

1

2log log(T )

)−1/2

log |ζ(1/2 + it)| : t ∈ [0, T ]

}tends to that of a standard normal variable N (0, 1) as T → ∞. Recently, Fyo-dorov, Keating and Hiary computed heuristically the covariance of the translationsof the zeta function and observed that in the first approximation a logarithmiccorrelation structure emerges. Similar covariance structure is exhibited by (theone-dimensional) restriction of the Gaussian free field (GFF), a fundamental prob-abilistic object that figures prominently in e.g. Liouville quantum gravity, SLE andrandom matrix theory. Based on the classical (after Montgomery) heuristic con-nection between ζ(1/2 + it) and random matrices, and the conjectured behaviourof random matrices they proposed the following

Problem 6.1. [39] Consider [0, T ] as a probability space, with normalisedLebesgue measure, and denote the corresponding variable by ω ∈ [0, T ]. Then, asT →∞, one has

maxh∈[0,1]

log |ζ(1/2 + ih+ iω)| = log log T − 3

4log log log T + E,

where the error term E is bounded in probability as T →∞.

Very recently Arguin, Belius and Harper [6] established the analogue of the aboveconjecture for a natural model that is derived from the Euler product of the zeta-function, i.e. for partial sums of random Dirichlet series of the type

X(x) =∑p

1√p(cos(x log p) cos θp + sin(x log p) sin θp) ,

where θp : s are i.i.d. and unifrom on [0, 2π] and indexed by prime numbers.The GFF heuristics of the zeta-function over the critical line has been useful

also in connection with the Helson conjecture [45]. Many fascinating questionsremain to be studied in this general domain of probabilistic behaviour of the zetafunction and related models. For many random Gaussian fields (taking values ingeneralised functions) one may construct the corresponding multiplicative Gaussianchaos measure see e.g. [57], [32], [9]). Naturally, after Selberg’s result one mayinquire if one could produce a gaussian chaos as a suitable scaling limit of theRiemann zeta function on the critical line. An easier task would be to consider


Problem 6.2 (6.1). Let the field X be defined as in (8). Study the propertiesnon-Gaussian chaos “exp(βX(x))”. 1

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Department of Mathematics and Statistics, University of Helsinki, FI-00170

Helsinki, Finland


Department of Mathematical Sciences, Norwegian University of Science and

Technology, NO-7491 Trondheim, Norway



Some problems on optimal approximants

Daniel Seco

Abstract. We present an account of different problems that arise in relationwith cyclicity problems in Dirichlet-type spaces, in particular with polynomialsp that minimize the norm ‖pf − 1‖.

1. Introduction

Denote by D, the unit disk of the complex plane and let α ∈ R. The Dirichlet-type space, Dα, is the space of analytic functions f : D → C, given by a Taylorexpansion f(z) =

∑∞k=0 akz

k, for which

(1.1) ‖f‖2α =

∞∑k=0

|ak|2(k + 1)α <∞.

This one-parameter family joins together three classical examples: Bergman (α =−1), Hardy (α = 0) and Dirichlet (α = 1) spaces. Since α will be fixed in eachproblem, whenever there is no ambiguity, we will use the notation ‖ · ‖ and 〈·, ·〉for the norm and inner product in Dα. All the Dα spaces are reproducing kernelHilbert spaces, meaning that for each ω ∈ D there exists a function kω,α such thatfor any f ∈ Dα we have

(1.2) 〈f, kω,α〉 = f(ω).

They are obviously nested, in the sense that Dα ⊂ Dβ for α > β, and the differen-tiation operator sends Dα to Dα−2 in such a way that ‖f‖2α ≈ |f(0)|2 + ‖f ′‖2α−2.When α < 2, an equivalent norm to that in (1.1) is

(1.3) ‖f‖2α,∗ = |f(0)|2 +∫D

|f ′(z)|2(1− |z|2)1−αdA(z),

where dA(z) = dxdyπ . For α > 1, the spaces are closed under multiplication and

the functions in these spaces are continuous to the boundary, whereas this is nottrue for any space for which α ≤ 1, giving the Dirichlet space a critical situationfor many problems. However, the operator Mp, of multiplication by an analytic

2010 Mathematics Subject Classification. Primary 47A16; Secondary 46E22, 30C15.Key words and phrases. Cyclicity, Dirichlet-type spaces, optimal approximation.The author was supported by ERC Grant Agreement n.291497 of the EU’s Seventh Frame-

work Programme (FP/2007-2013), and by Ministerio de Economıa y Competitividad ProjectsMTM2011-24606 and MTM2014-51824-P.


193

194 D. SECO

polynomial p, is always a bounded operator in Dα. For more details, we refer thereader to [11,12,14,15].

In this work, we will concentrate on problems related to the (forward) shiftoperator S : Dα → Dα, given by

(1.4) Sf(z) = zf(z).

In terms of multiplication operators, S = Mz, but in terms of Taylor series, theoperator shifts each coefficient to the next position (and hence its name). It is adeeply studied operator (see, for instance, [18]). A function f ∈ Dα is called cyclic(in Dα and with respect to the shift operator) if the polynomial multiples of fspan a dense subspace. Denote by [f ] the smallest closed subspace of Dα which isinvariant under the shift and contains f . Then cyclic functions are exactly those forwhich [f ] = Dα. From this definition, and since polynomials are themselves densein Dα, it becomes clear that g ≡ 1 is always a cyclic function and that a necessaryand sufficient condition for cyclicity of f is that there exists a cyclic function g ∈ [f ].It is therefore sufficient to study whether there exist some family of polynomials{pn}n∈N such that

(1.5) ‖pnf − 1‖2 n→∞→ 0.

From now on, we will denote by Pn the space of polynomials of degree less or equalto n.

Definition 1.1. A polynomial p that minimizes the norm ‖pf − 1‖2 amongthose p ∈ Pn is called an optimal approximant (to 1/f of degree n in Dα).

The objective of this work is to present several problems related with opti-mal approximants in this sense. It is clear that their behavior characterizes cyclicfunctions, and they were introduced in [3].

The boundedness of point evaluations (a direct consequence of the reproducingkernel property) guarantees that a function f is not cyclic if f(ω) = 0 at somepoint ω ∈ D, since ‖pf − 1‖ will be controlled from below by a constant dependingon ω times |p(ω)f(ω) − 1|. On the other hand, a function which is holomorphicin a disk of radius larger than 1, without zeros on the closed disk will be cyclic inany Dα since the Taylor polynomials of 1/f will make the norm in (1.5) tend to0. Hence, the behavior towards the boundary of the disk is often relevant in thestudy of cyclicity. Also, from (1.5) and the definitions of the norms it is clear thatthere exists a hierarchy with respect to the cyclicity in Dirichlet-type spaces, i.e.,for α > β, cyclic in Dα implies cyclic in Dβ .

In any case, a characterization for cyclicity in general is not available althoughseveral steps have been taken in this direction. A celebrated theorem of Beurlingstates that a function is cyclic in the Hardy space if and only if it is an outer function.Outer functions are those for which the logarithm satisfies a Mean Value Propertyover the unit circle. A previous result by Smirnov and reproved by Beurling allowsto factor f ∈ D0 as the product of an outer θ and an inner function I, that is, aholomorphic function such that |I(z)| ≤ 1 for all z ∈ D and |I(eiθ)| = 1 for almostevery θ ∈ [0, 2π). In this sense, Beurling showed that a function is cyclic in Hardyif and only if its inner factor is trivial.

In general Dirichlet-type spaces, the question was studied in detail by Brownand Shields in [10]. They showed that in the case when the space is an algebra(α > 1), a function f is cyclic if and only if 1/f is in the space H∞ of bounded

OPTIMAL APPROXIMANTS 195

analytic functions, that is, if f has no zeros in the closed unit disk. Brown andShields also showed that cyclic functions in the Dirichlet space are different fromthose in Hardy and different from those in Dα for α > 1. They are different fromthe Hardy space case in that their zero sets on the boundary need to be small(sets of zero logarithmic capacity), and different from α > 1 since polynomialswith zeros on the boundary are cyclic in Dirichlet space (for example, the functionf(z) = 1 − z). For information on logarithmic capacity, we refer the reader to[12,14,19]. In their article, the authors proposed the following question:

Conjecture 1.2 (Brown-Shields). A function f ∈ D1 is cyclic in D1 if andonly if it is outer and it has a set of boundary zeros of logarithmic capacity equalto zero.

This problem stands open today and it has attracted attention from a diversityof authors. Several of the problems we will present are motivated by this conjecture.

Several attempts at solving Conjecture 1.2 have dealt with the concept ofBergman-Smirnov Exceptional (BSE) sets, a family of subsets of T for which therelative part of Conjecture 1.2 holds and that includes all countable and many un-countable closed sets. An excellent reference for this topic is the book [12] andwe will not deal with it in here, since the sets for which the BSE condition is notknown have complicated expressions and therefore, it seems out of the question bynow to work with the optimal approximants for functions that would be of interestin this context.

2. Polynomial proofs of cyclicity theorems

2.1. The classical theorems. The concept of optimal approximant as inDefinition 1.1 was first introduced in [3]. See also [13]. There it was shown that foreach α ∈ R, n ∈ N, and f ∈ Dα (not identically zero) there exists a unique optimalapproximant to 1/f of degree n in Dα, p

∗n. Moreover, the authors showed that the

coefficients (ck)nk=0 of p∗n are given as the only solution to a linear system, of the

form

(2.1) Mc = b

where M = (⟨zjf, zkf

⟩)nj,k=0, and b = (

⟨1, zjf

⟩)nj=0.

In the particular case of the Hardy space, it is easy to obtain, as a corol-lary, a new proof of part of Beurling’s Theorem, namely, that there can only beone cyclic function in D0 with a given outer part: Indeed, let f ∈ D0. FromParseval identity, the elements of the matrix M , Mj,k =

⟨zjf, zkf

⟩are equal to

limr→1

∫ 2π

0rj−ke(j−k)iθ|f(reiθ)|2dθ, which does not depend on the inner part of the

function f .In fact, it is easy to see that the property < zjf, f >= 0 for all j ≥ 1 character-

izes inner functions among Hardy space functions (not identically zero). From nowon, we will refer to < zjf, f > as the moments of f . The simplicity of the proofencourages one to wonder whether it is possible to complete a proof of Beurling’sTheorem in this way:

Problem 2.1. Show that outer functions are cyclic in D0, using only informa-tion on the optimal approximants and the moments of f .

Two different and simple characterizations of cyclicity in Hardy that use onlyinformation on the approximants were already given in [5]. The proof of existence

196 D. SECO

and uniqueness of p∗n lies in the fact that Vn = Pnf is a finite dimensional subspaceof a Hilbert space, and therefore there is a unique orthogonal projection. In fact,p∗nf will be the orthogonal projection of 1 onto Vn and this tells us that ‖p∗nf−1‖2 =1−p∗n(0)f(0). In particular, a function is cyclic if and only if p∗n(0) tends to 1/f(0)as n tends to ∞. This holds in any of the Dα spaces. From now on, Z(f) willdenote the zero set of a function f . The following was recently shown in [5]:

Theorem 2.2. Let f ∈ D0. Then f is cyclic if and only if

(2.2)

∞∏n=0

⎛⎝1−∏

zk∈Z(p∗n)

|zk|−2

⎞⎠ =|f(0)|2‖f‖2 ,

where {p∗n}n∈N is the sequence of optimal approximants to 1/f .

It would be desirable to complete the proof of Beurling’s Theorem with a directproof that any of the known characterizations actually match the definition of anouter function.

Brown-Shields Theorem on capacity of the zero sets can be indeed easily provedfrom the definition of cyclicity in terms of polynomials, as a corollary to the follow-ing result (which can be found as Theorem 3.3.1 in [12]):

Denote by f∗ the function defined on the boundary by nontangential limits off , and by Cap(E), the logarithmic capacity of a set E.

Theorem 2.3 (Weak-type inequality for capacity). There exists an absoluteconstant C such that for f ∈ D1 and t > 0 we have

(2.3) Cap({|f∗| > t}) ≤ C

t2‖f‖2.

From here, to prove Brown and Shields Theorem one only needs to see that

Z(f) ⊂ Z(p∗nf) ⊂ {|p∗nf − 1| > 1− ε}.A strengthening of the other result of Brown and Shields (on simple functions

with zeros on the boundary being cyclic) using only the optimal approximants wasalready given in [3], and so, this theory could represent a unified approach to severalresults on cyclic functions.

2.2. Matrices and algorithms. A Grammian is a matrix given by the innerproducts of a sequence of functions with themselves, a matrix with entries Gj,k =〈fj , fk〉. Grammians form a family of matrices that have been studied for more thana hundred years, for their relations with the orthogonal projection. The matrix Mappearing in (2.1) is a Hermitian Grammian but in some cases it will have additionalstructure. Continuing with the case α = 0, we may notice that Mj,k = Mj−k,0.A matrix with this property is called a Toeplitz matrix. The Toeplitz structurewas exploited in [5] in order to show that we can characterize cyclicity in Hardyin terms of the zeros of p∗n exclusively. In order to show this, it was relevant tostudy the Levinson algorithm, which is an efficient algorithm for the inversion ofa Toeplitz matrix. See [17]. Toeplitz inversion algorithms are typically based oneither the Schur (see [20]) or the Levinson algorithms.

The reason why the matrices appearing in Hardy are Toeplitz matrices is thatthe shift is an isometry in the Hardy space (onto its image). Unfortunately, this isclearly not true in any other of the Dα spaces: for α > 0, the shift increases thenorm of a function, whereas for α < 0, it reduces this norm. However, the shift in


the Dirichlet space (α = 1) does have a special property: it is a 2-isometry, meaningthat

(2.4) ‖f‖21 − 2‖Sf‖21 + ‖S2f‖21 = 0.

In other words,

(2.5) Mj,k −Mj+1,k+1 = Mj+1,k+1 −Mj+2,k+2.

Therefore we can propose the following problem:

Problem 2.4. Develop an analogous of the Levinson or Schur algorithms thatexploits the structure of a Hermitian matrix which satisfies (2.5), in order to com-pute its inverse.

Ideally, a recursive formula for the inverse matrix could lead to a similar con-dition to that in (2.2).

3. Examples and the role of the logarithmic potential

3.1. Brown and Cohn’s examples. There are several sources of positiveresults for the Brown-Shields Conjecture. For instance, right after Brown andShields paper appeared, it was shown in [9] that the conjecture is sharp:

Theorem 3.1 (Brown-Cohn). Let E ⊂ T be a closed set of logarithmic capacityzero. Then there exists a cyclic function f ∈ D1 such that E ⊂ Z(f).

The example functions constructed in this paper satisfy additional regularityproperties: they are functions continuous to the boundary, and they have logarithmsthat are also in D1. Although the proof of cyclicity of these functions is left for thereader, this is easily derived from a more general later statement by Aleman ([2]),showing as a particular case that any f ∈ D1 such that log f ∈ D1 must be cyclic.A sufficient condition for a function f to be outer is log f ∈ H1 (where H1 denotesthe space of holomorphic functions on the disk, integrable on the boundary), orequivalently (log f)1/2 ∈ D0. At the same time, if α ∈ (0, 1] and there exists anumber t > 0 such that (log f)t ∈ Dα then the α-capacity of the zero set of f willsatisfy the corresponding necessary condition for cyclicity, that Capα(Z(f)) = 0.Therefore, it seems natural to ask the following:

Problem 3.2. Fix α ∈ (0, 1]. What are the values of t > 0 such that for anyf ∈ Dα, if (log f)

t ∈ Dα then f is cyclic in Dα?

We know 1/2 works (and is optimal) for α = 0 and 1 does for α > 0. It seemsnatural to think that 1/2 may work in all cases. This seems connected with anotherproblem that is closely related with Brown-Shields Conjecture:

Problem 3.3. Fix α ∈ (0, 1]. What are the values of (t, β) ∈ R2 such that forany f ∈ Dα, whenever (log f)t ∈ Dβ and the α-capacity of Z(f) ∩ T is zero, thenf is cyclic in Dα?

Again, clearly, t ≥ 1/2, β = 0 work for α = 0 and so do t ≥ 1 when β = α. Infact, all the examples of cyclic functions for Dirichlet we know of satisfy some suchcondition with β = 1: cyclic polynomials satisfy it whenever t < 1/2 or β < 1, andthe Brown-Cohn examples satisfy it with t = 1 = β.

198 D. SECO

3.2. An unresolved case. Another result in [10] states that for outer func-tions in D2 their cyclicity in D1 depends only on their zero sets, so it seems naturalto think that the Brown-Shields conjecture will be true for the particular case off ∈ D2, although this is yet to be shown. It would be enough to show that Brownand Cohn’s result in [9] can be improved in terms of the regularity of the functions,although the zero sets for D2 form a smaller class: so called Carleson thin sets.

Problem 3.4. Is it true that for each E ⊂ T, Carleson thin set of logarithmiccapacity 0, there is an f ∈ D2 that is cyclic in D1 and such that E ⊂ Z(f)?

It seems reasonable to expect that theD2 condition does help the function to becyclic and that the answer to Problem 3.4 is positive. After several candidates fora level of regularity that would improve the cyclicity of the function, the questionarises of finding a function for which optimal approximants can be numericallycomputed, but satisfying none of the unnecessary regularity conditions.

Problem 3.5. Is there an outer function f ∈ D1\H∞ satisfying all of thefollowing:

(1) Z(f) ∩ T has zero logarithmic capacity.(2) For all t > 0, (log f)t /∈ D1.(3) The elements of the matrix Mj,k =

⟨zjf, zkf

⟩can be computed from

existing information.

The assumption that f /∈ H∞ guarantees that f does not belong to any of themultiplicative algebras (Dα with α > 1). The last requirement could be replacedby any other that allows to work towards proving or disproving the cyclicity of thefunction.

3.3. The minimization of logarithmic energies. A classical problem inanalysis is that of finding sets of n points that minimize the energy generated bya given potential with certain restrictions. In the plane such potentials are usuallyrelated with the logarithmic potential and this is connected with the problem ofdetermining the orthogonal polynomials for a particular measure over the unitcircle. A good summary of such situations can be found in [19].

In [5], the authors show a correspondence between orthogonal polynomials forsome such measures and optimal approximants for a function in Dα. In fact, inthe Hardy space, the zeros of orthogonal polynomials are reflections of the zeros ofoptimal approximants (with respect to the unit circle), and hence it may happenthat sets minimizing energies tied to some logarithmic potentials describe the zerosets of the optimal approximants. A very ambitious program could be based on thefollowing problem:

Problem 3.6. Given f ∈ Dα. Determine whether there exists and describea potential for an energy which is minimized, for all n ∈ N, at the zero set of theoptimal approximant p∗n to 1/f in Dα.

A plausible reduction of this problem is that it could be enough to study only2 points in the zero set: on one hand, the interaction between any two points ofthe zero set will minimize some energy described by the rest of the points and thefunction f ; on the other, any two zeros z0, z1 of an optimal approximant p of degreen ≥ 2 for a function f determine also the optimal approximant of degree 2 to the


function pf/(z − z0)(z − z1). Hence it could be enough to solve for polynomials ofdegree 2 for all functions. This may still be a large problem.

As an illustration of how to find a closed formula for the optimal approximantsto a small collection of functions, we can look at the functions fa = (1 − z)a,a ∈ N and a ≥ 2, which has a root of multiplicity a at z = 1. The optimalapproximants to 1/fa may be computed explicitly with a closed formula in thecase of the Hardy space. In the present paper, we denote by B the beta function,

B(x, y) =∫ 1

0tx−1(1− t)y−1dt.

Proposition 3.7. Let a ∈ N. The nth-order optimal approximant to 1/(1−z)a

in D0 is given by

(3.1) pn(z) =n∑

k=0

((k + a− 1

k

)B(n+ a+ 1, a)

B(n− k + 1, a)

)zk.

Since the first version of this article, this formula has been generalized to a ∈ C

with positive real part. This result will appear in the forthcoming paper [6].

Proof. Let us first compute the elements Mj,k of the matrix M in (2.1) as-sociated with fa = (1 − z)a. Since the matrix in question is Hermitian, we can,without loss of generality, take j ≥ k, and since multiplication by zk is an isometry,we have that

Mj,k =⟨zj−k(1− z)a, (1− z)a

⟩.

We will use the standard notation that(nk

)is null whenever k < 0 or k > n.

Substituting the Taylor coefficients of fa, we see that

Mj,k =a∑

l=0

a∑s=0

(a

l

)(a

s

)(−1)l+s

⟨zl+j−k, zs

⟩.

By the orthogonality of the system of monomials, only the term in s = l+ j−kis non-zero, and in view of basic properties of binomial coefficients,

Mj,k = (−1)j−k

a+k−j∑l=0

(a

l

)(a

a− l + k − j

).

Now, applying the Chu-Vandermonde identity, we obtain

(3.2) Mj,k = (−1)j−k

(2a

a+ k − j

).

We can, from now on, take this to be the definition of Mj,k, extending its domainto all integers j and k. This will simplify notation.

It will be shown that the solution ck,n of the system (2.1) is of the form

(3.3) ck,n = tn

(a−1∏s=1

(k + s)

)(a∏

r=1

(n+ r − k)

),

for some tn ∈ C depending only on n. Note that, for all n, this is a polynomial in kof degree 2a−1 and that ck,n = 0 for k = 1−a, 2−a, ...,−1 and for k = n+1, ..., n+a.

200 D. SECO

If we substitute the values of Mj,k from (3.2) into the linear equations (2.1),we obtain, for j = 0, . . . , n,

(3.4)

n∑k=0

Mj,kck,n =

n∑k=0

(−1)j−k

(2a

a+ k − j

)ck,n =: Aj .

Suppose, firstly, that j ∈ {a, . . . , n−a} (and hence, that n ≥ 2a). Set qn,a(s) =(−1)acj−a+s,n, which is a polynomial in s of the same degree as ck,n in terms of k(that is, less or equal than 2a− 1). Then Aj may be rewritten as

(3.5) Aj =

2a∑s=0

(−1)s(2a

s

)qn,a(s).

For any polynomial of degree 2a− 1 or less, the result of (3.5) is equal to 0 byNewton’s theory of finite differences. Now we know that, for j = a, . . . , n − a, wehave

(3.6) Aj = 0.

Define E := {1, . . . , a − 1} ∪ {n − a + 1, . . . , n}, and suppose that we chose anypolynomial on k, ck,n, of degree less or equal to 2a − 1 = #E such that ck,n = 0for all k ∈ {1− a, . . . ,−1} ∪ {n+ 1, . . . , n+ a}. Then for all j ∈ E, (3.4) can stillbe completed, by adding 0-terms, to the formula (3.5). We have seen that if ck,n isdefined as in (3.4) then (3.6) holds for all j = 1, . . . , n.

That is, the system (2.1) is satisfied, provided that A0 = 1, and we are stillfree to choose tn. Clearly, since cs,n = 0 for s = 1− a, . . . ,−1, we know that

A0 =a∑

k=0

M0,kck,n =a∑

k=1−a

M0,kck,n.

Newton differences tell us thata∑

k=−a

M0,kck,n = 0,

and, hence, by the symmetry of the binomial coefficients,

A0 = −M0,ac−a,n = (−1)a+1c−a,n.

Therefore, it is enough to choose tn so that c−a,n = (−1)a+1. Evaluating c−a,n

in (3.3) gives

c−a,n = tn(−1)a−1Γ(a)Γ(n+ 2a+ 1)

Γ(n+ a+ 1).

Therefore, choosing

tn =Γ(n+ a+ 1)

Γ(a)Γ(n+ 2a+ 1),

we have the optimal approximants.Multiplying all the different factors together and expressing everything in terms

of the gamma function, we find that

(3.7) ck,n =Γ(k + a)Γ(n+ a+ 1− k)Γ(n+ a+ 1)

Γ(k + 1)Γ(n− k + 1)Γ(a)Γ(n+ 2a+ 1).


A simple expression for the same quantity in terms of binomial coefficients andthe beta function B is

(3.8) ck,n =

(k + a− 1

k

)B(n+ a+ 1, a)

B(n− k + 1, a).

To see that (3.7) and (3.8) are equivalent, just substitute(k + a− 1

k

)=

Γ(k + a)

Γ(k + 1)Γ(a)and B(x, y) =

Γ(x)Γ(y)

Γ(x+ y).

�

The beginning of the previous method can be used in a general Dα space. Forthe particular case of the Dirichlet space, we can go as far as in (3.2) and show thatthe elements Mj,k of the matrix M are given by

Mj,k = (−1)j−k

(2a

a+ k − j

)k + j + a+ 2

2.

For the functions fa in Proposition 3.7, it is in fact possible to check explicitlythat pn(0) converges to 1 = 1/f(0), and although we knew a priori that the functionfa is cyclic, this method may be of interest in itself. By a ≈ b we will mean thereexist universal nonzero constants C1 and C2 such that C1b ≤ a ≤ C2b.

Proposition 3.8. For the functions fa = (1 − z)a, and the optimal approxi-mants p∗n of degree n to 1/f in D0,

(3.9) ‖p∗nfa − 1‖2 ≈ a2/(n+ a+ 1).

Proof. First, from the expression (3.7) it is easy to see that

(3.10) pn(0) =Γ(n+ a+ 1)2

Γ(n+ 1)Γ(n+ 2a+ 1).

Now we will use Euler’s formula for the gamma function:

Γ(t) =1

t

∞∏k=1

(1 + 1k )

t

1 + tk

Applied to (3.10), we arrive at

pn(0) =∞∏

k=n+1

k(k + 2a)

(k + a)2

That is

pn(0) =

∞∏k=n+1

(1− a2

(k + a)2

)=: eCn

which tends to 1 as n goes to infinity since

Cn =

∞∑k=n+1

log

(1− a2

(k + a)2

)≈ −a2

∞∑k=n+1

1

(k + a)2= −a2

∞∑t=n+a+1

t−2.

An elementary computation shows then that Cn is comparable to −a2/(n+a+1).

Now we apply that pnf − 1 is orthogonal to Pnf , to see that

d2n = 〈pnf − 1, pnf − 1〉 = 〈1− pnf, 1〉 = 1− pn(0),

202 D. SECO

and, hence, the distance dn is approximated (in terms of absolute constants) as

d2n ≈ 1− e−a2/(n+a+1) ≈ a2/(n+ a+ 1).

�With not much work we can solve the quadratic equation p2 = 0, to obtain

that

(3.11) Z(p2) := {z0, z1} = {−1± i√2/a}.

Therefore, one can obtain the distances between the zeros (2√2/a), the dis-

tances between the zeros and the significant point z = 1, which is√4 + 2/a, or

the modulus of the zeros (√

1 + 2/a). It would be a starting step to identify acorresponding family of potentials whose energies are minimized at these distances.This is an inverse problem from that of identifying the points of minimal energy,given the functional.

When we take a = 1, the optimal approximants may be given in more generalspaces than D0 (see [3, 5, 13]). Other natural quantities that may influence thedescription of the potential are the distances between two zeros of the function forwhich we compute the optimal approximants and the multiplicities of these zeros.Adding a few degrees of generality, we expect the problem to stay tractable:

Problem 3.9. Find a closed formula for logarithmic potentials with externalfields whose energy is minimized among sets of 2 points by Z(p∗2) where p∗2 is theoptimal approximant of degree 2 to 1/f , and

(3.12) f(z) = (1− z)β[(z − eiθ)(z − e−iθ)

]γfor α ∈ R, β, γ ≥ 0, θ ∈ (0, π].

3.4. An extremal problem in Bergman spaces. Zeros of optimal approx-imants are restricted as to their positions. The following result was proved in [5]:

Theorem 3.10. Let f ∈ Dα not identically zero, p∗n the corresponding optimalapproximant, and z0 ∈ Z(p∗n). Then

(3.13) |z0| > min(1, 2α/2).

Moreover, 1 is sharp for all α ≥ 0 and for all α < 0, there exists a function f ∈ Dα

such that z0 ∈ D.

The proof is based on the fact that every zero of an optimal approximant ofdegree n to some function 1/f is the zero of an optimal approximant of degree 1to a different function. This reduces the problem to approximants of degree 1, towhich the solution of the linear problem (2.1) becomes trivial. The solution z0 is

(3.14) z0 =‖zf‖2〈f, zf〉 .

Applying Cauchy-Schwartz inequality and computing the norm of the shift operatoryields then the result.

Naturally, one can ask what is the sharp constant for each α < 0 (these are oftencalled Bergman spaces). A way to deal with this problem may be to reformulateit in terms of an extremal problem. The theory of extremal problems in Bergmanspaces has been often fruitful (see, for instance, [1]) and the variety of techniquesmay help solve the problem. We will concentrate on the case α = −1. In order to


find the sharp constant for Theorem 3.10, we would like to find the infimum of theabsolute values of the right-hand side in (3.14), or equivalently,

(3.15) sup| 〈g, zg〉 |‖zg‖2 ,

where the supremum is taken over all the functions g ∈ D−1.Renaming f = zg/‖zg‖, we obtain any function in the unit sphere of D−1

with f(0) = 0. Using the integral expression of the norm of D−1, we arrive to thefollowing problem:

Problem 3.11. Compute

sup

{∣∣∣∣∫D

|f(z)|2z

dA(z)

∣∣∣∣ : f(0) = 0, ‖f‖2−1 ≤ 1

}.

By all of the above, the solution should be a number in the interval (1,√2].

In fact, in [5], the authors showed that for the lower optimal constant must beat least 121/119, since this is the corresponding value obtained for the functiong(z) = (1 + z)/(1− z)4/5.

Since the first version of this article, this problem has been completely solved.The solution will appear in the forthcoming paper [6].

4. Higher dimensional phenomena

Several articles have dealt already with cyclicity in more than 1 complex vari-able. In the case of the bidisk, D2 = D×D, Dirichlet-type spaces are usually definedwith a product norm:

Definition 4.1. The Dirichlet-type space Dα(D2), of parameter α over the

bidisk is defined as the space of functions f of two variables that are holomorphicon each variable at each point of the bidisk, defined by a Taylor series f(z1, z2) =∑

j,k∈Naj,kz

j1z

k2 that satisfies

(4.1) ‖f‖2α,D2 =∑j,k∈N

|aj,k|2((j + 1)(k + 1))α <∞.

Some problems on cyclicity in this family of spaces have been tackled. See [4,7]and the references therein for background information on this topic. A difficultythat arises when increasing the dimension to 2 is the lack of a Fundamental Theoremof Algebra: the structure of irreducible polynomials in 2 variables is much richer.

However, the approach in terms of optimal approximants follows the sameprinciples as in 1 variable: for each finite set of monomials X, one can find theorthogonal projection of the constant function 1 onto Y = f spanX and that willyield the optimal approximant within Y to 1/f . If we choose a sequence of sets

{Xn} with Xn ⊂ Xn+1 and⋃Xn = {zj1zk2 , (j, k) ∈ N2}, a function will be cyclic

depending only on the behavior of these optimal approximants. Natural choicesfor Xn are the monomials of degree less or equal to n where the definition of thedegree can be taken to be the maximum or the sum of the degrees in each variable.In algebraic geometry, it is more often the latter while the first one is commonlyused in analysis. Here we will use the algebraic version.

Definition 4.2. By the optimal approximant of degree n to 1/f we denote the

optimal approximant to 1/f within f span{zj1zk2 : (j, k) ∈ N2, j + k ≤ n}.

204 D. SECO

Discrete sets of points that minimize some energy are often studied in higherdimensions (for example, in sampling theory), but algebraic varieties, of dimensiongreater or equal to 1, that minimize a functional are pointing in a completelydifferent direction. The typical pathologies of minimal currents may occur onlywhen taking limits.

Let us explore an example. Choose f(z1, z2) = 1 − z1+z22 . We can compute

the optimal approximant of degree 1, p∗1: from the symmetry of the coefficientsand the uniqueness of the orthogonal projection, one can see that p∗1 will be of theform p∗1(z1, z2) = a0(a1 + (z1 + z2)). The constants a0 and a1 will depend on theparameter α of the space, but a1 can be shown to be a real number larger than 2.In particular, Z(p∗1) does not intersect the bidisk.

This example is in concordance with what happens in one dimension, at leastfor α ≥ 0, although an analogous to Theorem 3.10 is not known yet. A possiblerestriction could be that zero sets of optimal approximants can’t intersect the bidiskwhen α ≥ 0. But this wouldn’t tell which irreducible polynomials are feasible asoptimal approximants (observe this is answered by Theorem 3.10). When α <0, the question retains some uncertainty equivalent to solving the Problem 3.11.Describing all the irreducible polynomials that appear as optimal approximantsseems a difficult task, but many subproblems may be of interest. We propose thefollowing:

Problem 4.3. For each value of α ≥ 0, determine which algebraic curves arezero sets of optimal approximants of degree 2 or less to 1/f , for some f ∈ Dα(D

2).

References

[1] Dov Aharonov, Catherine Beneteau, Dmitry Khavinson, and Harold Shapiro, Extremal prob-lems for nonvanishing functions in Bergman spaces, Selected topics in complex analysis,Oper. Theory Adv. Appl., vol. 158, Birkhauser, Basel, 2005, pp. 59–86, DOI 10.1007/3-7643-7340-7 5. MR2147588

[2] A. Aleman, The multiplication operator on Hilbert spaces of analytic functions, Habilitation-sschrift, Fernuniversitat Hagen, 1993.

[3] Catherine Beneteau, Alberto A. Condori, Constanze Liaw, Daniel Seco, and Alan A. Sola,Cyclicity in Dirichlet-type spaces and extremal polynomials, J. Anal. Math. 126 (2015), 259–286, DOI 10.1007/s11854-015-0017-1. MR3358033

[4] Catherine Beneteau, Alberto A. Condori, Constanze Liaw, Daniel Seco, and Alan A. Sola,Cyclicity in Dirichlet-type spaces and extremal polynomials II: functions on the bidisk, PacificJ. Math. 276 (2015), no. 1, 35–58, DOI 10.2140/pjm.2015.276.35. MR3366027

[5] C. Beneteau, D. Khavinson, C. Liaw, D. Seco, and A. A. Sola, Orthogonal polynomials,reproducing kernels, and zeros of optimal approximants, to appear on Journal Lond. Math.Soc.

[6] C. Beneteau, D. Khavinson, C. Liaw, D. Seco, and B. Simanek, Zeros of optimal polynomialapproximants: Jacobi matrices and Jentzsch-type theorems, preprint

[7] C. Beneteau, G. Knese, �L. Kosinski, C. Liaw, D. Seco, and A. A. Sola, Cyclic polynomials intwo variables, to appear on Trans. Amer. Math. Soc.

[8] Arne Beurling, On two problems concerning linear transformations in Hilbert space, ActaMath. 81 (1948), 17. MR0027954

[9] Leon Brown and William Cohn, Some examples of cyclic vectors in the Dirichlet space, Proc.Amer. Math. Soc. 95 (1985), no. 1, 42–46, DOI 10.2307/2045570. MR796443

[10] Leon Brown and Allen L. Shields, Cyclic vectors in the Dirichlet space, Trans. Amer. Math.Soc. 285 (1984), no. 1, 269–303, DOI 10.2307/1999483. MR748841



[12] Omar El-Fallah, Karim Kellay, Javad Mashreghi, and Thomas Ransford, A primer on theDirichlet space, Cambridge Tracts in Mathematics, vol. 203, Cambridge University Press,Cambridge, 2014. MR3185375

[13] Emmanuel Fricain, Javad Mashreghi, and Daniel Seco, Cyclicity in reproducing kernel Hilbertspaces of analytic functions, Comput. Methods Funct. Theory 14 (2014), no. 4, 665–680, DOI10.1007/s40315-014-0073-z. MR3274894

[14] John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics,

vol. 236, Springer, New York, 2007. MR2261424[15] Haakan Hedenmalm, Boris Korenblum, and Kehe Zhu, Theory of Bergman spaces, Graduate

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[17] Norman Levinson, The Wiener RMS (root mean square) error criterion in filter design andprediction, J. Math. Phys. Mass. Inst. Tech. 25 (1947), 261–278. MR0019257


[19] Edward B. Saff and Vilmos Totik, Logarithmic potentials with external fields, Grundlehren derMathematischenWissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316,Springer-Verlag, Berlin, 1997. Appendix B by Thomas Bloom. MR1485778

[20] J. Schur, Uber Potenzreihen, die im Innern des Einheitskreises beschrankt sind (German),J. Reine Angew. Math. 147 (1917), 205–232, DOI 10.1515/crll.1917.147.205. MR1580948

Departament de Matematica Aplicada i Analisi, Facultat de Matematiques, Univer-

sitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain.



Some open problems in complex and harmonic analysis:Report on problem session held during the conference

Completeness problems, Carleson measures, and spaces of analyticfunctions

Catherine Beneteau, Alberto A. Condori, Constanze Liaw,William T. Ross, and Alan A. Sola

Abstract. The following article reports on the contributions of AlexandruAleman, Emmanuel Fricain, Alexei Poltoratski, and Rishika Rupam to theproblem session at the Mittag-Leffer conference on Completeness problems,Carleson measures, and spaces of analytic functions, in July 2015. Prob-lems considered included approximation in de Branges-Rovnyak spaces, inversespectral problems for the Schrodinger operator, completeness of exponentials,meromorphic inner functions, and the analysis of evolution equations gener-ated by quadratic differential operators on L2(Rn).

1. Overview

This article reports on a problem session that took place on July 1, 2015, atthe conference “Completeness problems, Carleson measures, and spaces of analyticfunctions” at the Mittag-Leffler Institute. A wide variety of interesting problemswere presented by the following contributors: Emmanuel Fricain discussed prob-lems regarding approximation in de Branges-Rovnyak spaces; Alexei Poltoratskireported on questions about inverse spectral problems for the Schrodinger operatorand their relationship to completeness of exponentials on sets, related to these in-verse spectral problems; Rishika Rupam explored meromorphic inner functions andquestions about what conditions on real data guarantee the existence of a mero-morphic inner function matching that data; Alexandru Aleman addressed issuesrelated to quadratic differential operators on L2(Rn) and the Weyl-quantization ofquadratic forms. The contributors generously gave us their notes on these problems,and this article is a slightly expanded and edited version of those notes.

2. Approximation in de Branges-Rovnyak spaces

2.1. Definitions and background. Emmanuel Fricain discussed two prob-lems concerning approximation in de Branges–Rovnyak spaces H(b), with b ∈b(H∞). HereH∞ denotes the space of bounded analytic functions on the open unit

2010 Mathematics Subject Classification. Primary 35-06, 30-06.Key words and phrases. Approximation in de Branges-Rovnyak spaces, inverse spectral prob-

lems, meromorphic inner functions, evolution equations, FBI transform.


207

208 BENETEAU, CONDORI, LIAW, ROSS, AND SOLA

disk D normed by ‖f‖∞ := supz∈D |f(z)|, and b(H∞) := {g ∈ H∞ : ‖g‖∞ � 1} isthe closed unit ball in H∞. For b ∈ b(H∞), the de Branges-Rovnyak space H(b) isthe reproducing kernel Hilbert space of analytic functions on D whose kernel is

kbλ(z) :=1− b(λ)b(z)

1− λz, λ, z ∈ D.

It turns out that this Hilbert space can also be defined using Toeplitz operators.Recall that for ϕ ∈ L∞ = L∞(T), the associated Toeplitz operator Tϕ is a boundedoperator on the Hardy space H2 of D, defined by the formula

Tϕ(f) = P+(ϕf),

where P+ stands for the Riesz (orthogonal) projection from L2 onto H2. Then,for b ∈ b(H∞), the operator I − TbTb = I − TbT

∗b is a positive operator. We can

therefore consider its square root, and the associated de Branges–Rovnyak spaceH(b) is defined by

H(b) = (I − TbTb)1/2 H2,

equipped with the norm

‖(I − TbTb)1/2f‖b = ‖f‖2, f ∈ H2 ∩ (Ker(I − TbTb)

1/2)⊥.

That is to say, H(b) is normed to make (I − TbTb)1/2 a partial isometry of H2

onto H(b). When ‖b‖∞ < 1, the operator I − TbTb is an isomorphism on H2 andH(b) = H2 with an equivalent norm. At the other extreme, when b is an innerfunction Θ, meaning |Θ| = 1 almost everywhere on T, then TΘTΘ is the orthogonalprojection of H2 onto ΘH2 and thus H(Θ) turns out to be the so-called modelspaces KΘ = H2 ∩ (ΘH2)⊥.

These spaces (and, more precisely, their general vector-valued versions) wereintroduced by de Branges and Rovnyak [14, 15] as universal model spaces forHilbert space contractions. Thanks to pioneering work of Sarason, we know that deBranges–Rovnyak spaces play an important role in numerous questions of complexanalysis and operator theory (e.g. [6,26,42,44]). For the general theory of H(b)spaces, we refer to [21,42]. We just note that the theory often bifurcates into twodirections depending on whether or not b is an extreme point of the closed unit ballof H∞. Note that the function b is an extreme point of b(H∞) if and only if∫

T

log(1− |b|) dm = −∞.

2.2. Density of continuous functions. We are interested here in the densityof C(D−), the algebra of continuous functions on the closed unit disk D−, in H(b).The first positive result in this direction is due to Sarason [43].

Theorem 1 (Sarason, 1986). Let b ∈ b(H∞). The (analytic) polynomialsbelong to H(b) if and only if b is a non-extreme point of the closed unit ball of H∞.Moreover, in that case, the polynomials are dense in H(b).

Note that since the polynomials are continuous functions on the closed unit diskD−, we immediately get that the set H(b)∩C(D−) is a dense set in H(b) when b isnon-extreme. The proof given by Sarason [42] uses duality and is non-constructive.In a recent work [19], the authors gave a constructive way to get approximation bypolynomials but it can be showed that, contrary to the H2 case, one cannot take

OPEN PROBLEMS IN COMPLEX AND HARMONIC ANALYSIS 209

the Taylor polynomials. The solution takes the form of a Toeplitz-type summationmethod.

At the other extreme, when b = Θ is inner, Aleksandrov [3] proved the follow-ing.

Theorem 2 (Aleksandrov, 1981). Let Θ be an inner function. The spaceKΘ ∩ C(D−) is dense in KΘ.

We also refer to [10, Theorem 8.5.1] for a beautiful presentation of this deepresult. The proof is highly non-constructive and uses several of results on theCauchy transforms such as Aleksandrov’s characterization of Cauchy transforms,the F-property (a certain property of division by inner function) for Cauchy trans-forms, and the distribution theorem for Cauchy transforms. It is remarkable thatfor instance when Θ is a singular inner function, it is is even not obvious why thereare any non-trivial functions in KΘ ∩ C(D−). Yet the result of Aleksandrov saysthat they are dense! Note that the paper [18] contains further information as towhether or not KΘ contains functions from various smoothness classes.

These two results naturally leads to the following question.

Problem 1. Let b ∈ b(H∞). Is it true that H(b) ∩ C(D−) is dense in H(b)?Theorems 1 and 2 say that the answer is positive when b is non-extreme and

when b is inner. What can be said in the remaining cases?

2.3. Cyclicity. Recall that if T is a bounded linear operator on a separablecomplex Hilbert space H, then we say that T is cyclic if there exists an x ∈ H suchthat the orbits of x under T spans all the space, that is,

(1) span(Tnx : n ≥ 0) = H.

Here span(. . . ) denotes the closed linear subspace generated by (. . . ). A vector xwhich satisfies (1) is called a cyclic vector.

The problem of characterizing the cyclic vectors for a given cyclic operatorshas been shown to be a challenging problem. It is remarkable that there exists acomplete description for the forward and backward shift operator on H2. Recallthat

(Sf)(z) = zf(z), f ∈ H2, z ∈ D

is the forward shift, while

(S∗f)(z) =f(z)− f(0)

z, f ∈ H2, z ∈ D,

is the backward shift. Smirnov [45] proved that a function f ∈ H2 is a cyclic vectorfor S if and only if f is outer, which means that f can be written as

f(z) = λ exp

(∫T

ζ + z

ζ − zlog |f(ζ)| dm(ζ)

), z ∈ D,

with λ ∈ T and log |f | ∈ L1.It turns out that when b is a non-extreme point of the closed unit ball of H∞,

then H(b) is S-invariant and we set Yb := S|H(b).

Problem 2. Describe the cyclic vectors for Yb.


It should be noted that since polynomials are dense in H(b) when b is non-extreme, then the constant function f = 1 is cyclic for Yb. Moreover, the densityof polynomials in H(b) also shows that f is cyclic for Yb if and only if there existsa sequence of polynomials (pn)n such that

limn→+∞

‖1− pnf‖b = 0.

Since the spaceH(b) is contractively embedded intoH2, if f is cyclic for the operatorYb, then it is cyclic for S and thus f is necessarily an outer function. This conditionis also sufficient in a particular situation. In the theory of H(b) spaces, when b isnon-extreme a particular role is played by the unique outer function a such thata(0) > 0 and |a|2 + |b|2 = 1 almost everywhere on T. The function a is called thePythagorean mate associated to b. Now define the space M(a) = aH2 equippedwith the range norm ‖ag‖M(a) = ‖g‖2, g ∈ H2. It turns out that the space M(a)is contractively contained in H(b).

In [24], Fricain, Mashreghi, and Seco proved the following result.

Theorem 3. Let b be a non-extreme point of b(H∞) and let a be its Pythagoreanmate. Then

(1) If a−2 ∈ L1(T) and f ∈M(a) is outer, then f is cyclic for Yb.(2) If |a|2 ∈ (A2), then f ∈ H(b) is cyclic for Yb if and only if f is an outer

function.

Recall that |a|2 ∈ (A2) means that

supI

(1

|I|

∫I

|a|2 dm)(

1

|I|

∫I

|a|−2 dm

)< +∞,

where the supremum is taken over all sub arcs I ⊂ T. A result of Sarason [42]shows that when |a|2 ∈ (A2), then H(b) =M(a) with equivalent norms.

It appears that when |a|2 /∈ (A2), the situation is more complicated as thefollowing example suggests. Let b(z) = (1 + z)/2 and f ∈ H(b). Then, it is shownin [24] that f is cyclic for Yb if and only if f is an outer function and f/a /∈ H2. Theproof of this equivalence is based on a complete description of invariant subspacesfor Yb obtained by Sarason [43] in the case when b(z) = (1+ z)/2. The descriptionof these invariant subspaces in the general case is also an open problem.

As this example suggests, the condition for cyclicity should involve the zerosof the function on the boundary. This idea is also justified because of the relationbetween Problem 2 and the Brown–Shields conjecture on cyclic vectors for theshift operator on the Dirichlet space D. Indeed, due to works of Sarason [41]and Costara–Ransford [12], we know that de Branges–Rovnyak spaces coincidein certain cases with some D(μ) spaces. However, it should be noted that thisequivalence only occurs when the measure μ is singular which excludes the classicalDirichlet space D.

3. Inverse spectral problems, completeness of exponentials, andmeromorphic inner functions

Alexei Poltoratski examined inverse spectral problems for the Schrodinger op-erator and their relationship with problems about completeness of exponentials onsets, and Rishika Rupam examined the connection with these problems and ques-tions about meromorphic inner functions.


3.1. The Schrodinger operator and inverse spectral problems. We candefine a time-independent 1-dimensional Schrodinger equation in the following way:

(2) −u′′ + qu = λu

on some interval (a, b) and assume that the potential q(t) is locally integrable anda is a regular point i.e., a is finite and q is in L1 at a. Let us fix the followingstandard self-adjoint boundary conditions:

cos(α)u(a) + sin(α)u′(a) = 0

cos(β)u(b) + sin(β)u′(b) = 0.

We can consider the corresponding operator L : L2(a, b)→ L2(a, b) as

(3) u �→ −u′′ + qu.

Here we also take into account the above boundary conditions. Thus, L will reallymean L(q, α, β).

Let us consider this Schrodinger operator on the interval (0, 1). Suppose thatwe know the spectrum σ of this operator for the self-adjoint boundary conditions atthe endpoints, as above. It is known ([28]) that given σ and q on (0, 1/2) one canrecover the operator uniquely. The next natural question becomes, if q is known ona bigger interval, (0, c), c > 1/2, say, then what part of σ is sufficient to recover theoperator? In [29], M. Horvath solved this problem by establishing its equivalencewith the Beurling-Malliavin problem on completeness of exponentials in L2(0, 1−c).

The next question is, what if instead (0, c) the potential q is known on a unionof two intervals (0, c) and (d, 1)? On an arbitrary subset of (0, 1)? This problemremains open. An analogous problem of completeness of exponentials on sets, otherthan an interval is also open in general. Is there a Horvath-type result connectingthese two problems?

3.2. Meromorphic inner functions. A slightly different inverse problem(see [33] for details) is related to so-called meromorphic inner functions. Onedefines meromorphic inner functions (MIFs) in the following way. Let Θ : C+ → C

be bounded and holomorphic, with the properties that |Θ| = 1 on R and Θ can becontinued meromorphically to C. Then Θ is called a meromorphic inner functionon the upper half plane. These functions are ubiquitous in the study of certain2nd order differential equations. One often encounters its twin, the well knownm-function, first studied by Weyl and later Titchmarsh [20] who employed it tostudy the properties of the differential equation. On the other hand, the relatedMIF has received comparatively less attention in the past. However, over the lastfew years, these functions have been used extensively by Makarov and Poltoratski([32, 33, 37]) in the study of Toeplitz kernels and model spaces to provide newconnections between inverse spectral problems and complex function theory.

One inverse spectral problem related to MIFs is the following: suppose, a priori,we are not given q and we only know one of the boundary conditions, let’s say α atthe point a. Then, it is known that there is a corresponding MIF Θ (called a Weylinner function) such that

(4) σ(L(q, α,D)) = σ(Θ),


where D refers to the Dirichlet boundary condition at b, and the spectrum σ(Θ) ofthe MIF Θ is defined as the set

σ(Θ) = {x ∈ R : Θ(x) = 1}.In fact, Marchenko proved [34] that the MIF, coming from a Schrodinger operatoruniquely defines the operator. Thus, this problem reduces to recovering MIFs fromparts of their spectrum.

Problem 3. Suppose we are given some data about the spectrum of an operator.Does it

(1) correspond to the spectrum of a 2nd order differential operator (or moregenerally a canonical system)?

(2) correspond to a 2nd order differential operator of a specific kind, for ex-ample a Schrodinger or Dirac operator?

In terms of MIFs, these questions can be rephrased as follows.

Problem 4. Given some data on R, does there

(1) exist an MIF that corresponds to this data?(2) exist an MIF, arising out of a Schrodinger or Dirac operator that corre-

sponds to this data?

There are some other natural questions that arise in these studies. For example:Suppose we are given a Weyl inner function Θ corresponding to a Schrodinger (orDirac) operator on L2(a, b). Let the argument function of Θ be θ. One may ask the

question that if we perturb θ slightly, to a new function θ, will there exist anotherMIF Θ, also corresponding to a Schrodinger (or Dirac) operator on L2(a, b) such

that arg(Θ) = θ on R? In fact, in the case of the Schrodinger operator, we knowwhat the argument function should look like ([9,29]). One still does not know whathappens in the general case, or in other words, if one restricts attention to a generalclass of operators.

One may ask questions related to the above about what the argument of aMIF looks like without referring to operators. To get a feel for these questions, itis helpful to consider some examples of MIFs, such as Blaschke factors z−ω

z−ω (with

ω ∈ C+) and singular inner functions of the form eiaz, where a ≥ 0. There is alsoa useful factorization of MIFs due to Riesz and Smirnov [33]:

Theorem 4. If Θ is a MIF, then it has the form

(5) Θ(z) = BΛ(z)eiaz

where a ≥ 0 and BΛ is the Blaschke product of the zeros of the function given byΛ = {λn ∈ C+}n, where |λn| → ∞ and satisfy the convergence criterion,

(6)∑λn∈Λ

�λn

1 + |λn|2<∞.

Using this theorem, it is not difficult to prove that on the real line, Θ has aspecial property

Observation 1. If Θ is a MIF then,

(7) Θ = eiθ on R,

where θ : R→ R is an increasing, real-analytic function.


Using the factorization (5), we see that if θ is the argument of an MIF on R,then

(8) θ(t) = at+∑λn∈Λ

arctan

(2(xn − t)yn

(xn − t)2 − (yn)2

),

where λn = xn + iyn are the zeros of Θ. Here it is possible to define a continuousbranch of the argument function. It is natural to ask the following question. Givenan increasing, real analytic function, is it possible to determine whether it is theargument of an MIF? The answer in general is no! Let us consider the followingexamples.

Example 1. Consider the following function defined on R. Let

(9) θ(x) = πx

for all x ∈ R. Then θ is the argument of the MIF Θ(z) = eπiz.

Example 2. In the above example, we notice that σ(Θ) = {2n : n ∈ Z} andσ(−Θ) = {2m + 1 : m ∈ Z}. It is natural to wonder which other MIFs have thisso-called two-spectra property. The answer is given by Krein’s proof that gives aformula for all such MIFs [33]. We state his result precisely here. If Θ is an MIF

such that σ(Θ) = {2n : n ∈ Z} and σ(−Θ) = {2m+ 1 : m ∈ Z}, then

(10) Θ =Θ− c

1− cΘ, c ∈ (−1, 1),

where Θ(z) = eπiz. From this formula, it is clear that if we know the value of Θ atany one point besides the integers, then it is completely determined. This allows usto construct the following counterexample to the question above. Let us constructφ : R → R as an increasing, real analytic function with the following conditions:φ(n) = πn, for all integers n, φ(1/2) = π/3 and φ(3/2) = 5π/4. Then, φ does notcorrespond to an MIF, i.e., there does not exist an MIF Φ such that Φ = eiφ on R.

Example 3. Let θ : R → R be an increasing, real analytic function such thatθ(n) = πn for all n ∈ N ∪ {0}, θ(−∞) > −π and sup

x∈R

θ′(x) < ∞, then it has

been shown that ([2,40]) there does not exist an MIF Θ such that Θ = eiθ on R.We want to emphasize that it is easy to construct an increasing, real analytic θthat satisfies the conditions above. Yet, it is impossible for any such θ to be theargument of an MIF. This example arises from the studies of de Branges spaces ofanalytic functions. In his book [13], Louis de Branges made the following claim:Given any separated sequence Λ = {λn}n, there always exists a MIF Θ such that

σ(Θ) = Λ and supx∈R

|Θ′(x)| <∞.

De Branges had used this result in the proof of the gap theorem. In 2011, thiswas discovered to be false by Anton Baranov [2]. In fact, Baranov provided acounterexample to this claim, which is the same as our example. Thus, Baranovproved that if Λ = N ∪ {0}, then there does not exists any MIF Θ such thatσ(Θ) = Λ and sup

x∈R

|Θ′(x)| <∞. For more such counterexamples and details on this

lemma of de Branges’, we refer the reader to [40].

Thus, the interesting question that remains is, what conditions on a functionθ : R→ R will ensure that it is the argument of a MIF?


4. Quadratic differential operators and Weyl-quantization of quadraticforms

Alexandru Aleman proposed two problems that are inspired by the study ofquadratic differential operators on L2(Rn), not necessarily elliptic, and evolutionequations associated with such operators.

4.1. Evolution equations associated with quadratic differential oper-ators. Consider the evolution equation{

∂tu(t, x) +Qu(t, x) = 0u(0, x) = 0

The differential operators of interest here are defined via the Weyl quantizationof quadratic forms. More precisely, given a quadratic form q on Rn × Rn, writtenas

q(x, y) =

n∑i=1

n∑j=1

aijxiyj ,

its Weyl quantization is obtained by replacing the variables yj by partial differen-tiation with respect to xj denoted by ∂xj

. Thus the quadratic differential operatorcorresponding to q we obtain in this way is

(11) Q(x,D) =∑i,j

aijxi∂xj.

Example 4. In the simplest case of R×R and aij = δij , this procedure simplyyields the (rotated) harmonic oscillator Q = d2/dx2 + x2. This operator is elliptic,but an important feature of the analysis developed in [4,5] is that it applies also inthe non-elliptic setting. Fokker-Planck operators also fit into this framework, see[4,5].

Operators of the form (11) are studied extensively by practitioners of microlocalanalysis, but in certain cases they can also be modelled by operators acting onHilbert spaces of analytic functions, which ties in with the theme of the conference.

This correspondence with operators on Hilbert function spaces is set up viathe so-called Fourier-Bros-Iagolnitzer (FBI) transform. The FBI transform in thissetting is a unitary operator U from L2(Rn) onto the weighted Fock space Hφ

consisting of entire functions f : Cn → C endowed with the norm

‖f‖2φ =

∫Cn

|f(z)|2e−φ(z)dV <∞.

Here V denotes Lebesgue measure on Cn, z = (z1, . . . , zn) is a point in Cn, and φis a function of the special form

(12) φ(z) = |z|2 −Re(Az · z),where | · | denotes the Euclidean norm in Cn, A is a constant n × n matrix, andz ·w =

∑j zjwj . In the case of the classic Fock space, corresponding to A = 0, the

transformation reduces to the Bargmann transform.For a large class of quadratic form this unitary map U transforms the quadratic

differential operator Q(x,D) into an (unbounded) operator of the form

(13) (Tf)(z) = (Mz ·D)f =∑i,j

mijzj∂zif,


where the n × n matrix M = (mij) is constant. Despite a relatively simple form,these operators are in general far from symmetric or normal, meaning that estimat-ing their resolvent is a subtle task. However, the eigenspaces of the operator T arespanned by monomials.

Example 5. It is known that the Hermite functions diagonalize the harmonicoscillator. One can check (see [5, Section 3] and [4]) that the appropriate FBItransform (or Bargmann transform) maps the subspace of spanned by Hermitepolynomials associated with energy level α to the span of homogeneous polynomialsof degree α.

One can ask to what extent orthogonality carries over.

Problem 5. For which functions φ of the form (12) do monomials form aRiesz basis in the Fock space Hφ?

Problem 6. Estimate the norm of the resolvent (λI − T )−1, where T is anoperator of the form (13) on a Fock space Hφ as above.

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Department of Mathematics, University of South Florida, 4202 E Fowler Ave,

CMC342, Tampa, Florida 33620


Department of Mathematics, Florida Gulf Coast University, Fort Myers, Florida

33965


CASPER and Department of Mathematics, Baylor University, Waco, Texas 76798

E-mail address: Constanze [email protected]

Department of Mathematics and Computer Science, University of Richmond, Rich-

mond, Virginia 23173


Department of Mathematics, University of South Florida, 4202 E Fowler Ave,

CMC342, Tampa, Florida 33620


SELECTED PUBLISHED TITLES IN THIS SERIES

679 Catherine Beneteau, Alberto A. Condori, Constanze Liaw, William T. Ross,and Alan A. Sola, Editors, Recent Progress on Operator Theory and Approximation inSpaces of Analytic Functions, 2016

673 Alex Martsinkovsky, Gordana Todorov, and Kiyoshi Igusa, Editors, RecentDevelopments in Representation Theory, 2016

672 Bernard Russo, Asuman Guven Aksoy, Ravshan Ashurov, and ShavkatAyupov, Editors, Topics in Functional Analysis and Algebra, 2016

671 Robert S. Doran and Efton Park, Editors, Operator Algebras and TheirApplications, 2016

669 Sergiı Kolyada, Martin Moller, Pieter Moree, and Thomas Ward, Editors,Dynamics and Numbers, 2016

668 Gregory Budzban, Harry Randolph Hughes, and Henri Schurz, Editors,Probability on Algebraic and Geometric Structures, 2016

667 Mark L. Agranovsky, Matania Ben-Artzi, Greg Galloway, Lavi Karp, Dmitry

Khavinson, Simeon Reich, Gilbert Weinstein, and Lawrence Zalcman, Editors,Complex Analysis and Dynamical Systems VI: Part 2: Complex Analysis, QuasiconformalMappings, Complex Dynamics, 2016

666 Vicentiu D. Radulescu, Adelia Sequeira, and Vsevolod A. Solonnikov, Editors,Recent Advances in Partial Differential Equations and Applications, 2016

665 Helge Glockner, Alain Escassut, and Khodr Shamseddine, Editors, Advances inNon-Archimedean Analysis, 2016

664 Dihua Jiang, Freydoon Shahidi, and David Soudry, Editors, Advances in theTheory of Automorphic Forms and Their L-functions, 2016

663 David Kohel and Igor Shparlinski, Editors, Frobenius Distributions: Lang-Trotterand Sato-Tate Conjectures, 2016

662 Zair Ibragimov, Norman Levenberg, Sergey Pinchuk, and Azimbay Sadullaev,Editors, Topics in Several Complex Variables, 2016

661 Douglas P. Hardin, Doron S. Lubinsky, and Brian Z. Simanek, Editors, ModernTrends in Constructive Function Theory, 2016

660 Habib Ammari, Yves Capdeboscq, Hyeonbae Kang, and Imbo Sim, Editors,Imaging, Multi-scale and High Contrast Partial Differential Equations, 2016

659 Boris S. Mordukhovich, Simeon Reich, and Alexander J. Zaslavski, Editors,Nonlinear Analysis and Optimization, 2016

658 Carlos M. da Fonseca, Dinh Van Huynh, Steve Kirkland, and Vu Kim Tuan,Editors, A Panorama of Mathematics: Pure and Applied, 2016

657 Noe Barcenas, Fernando Galaz-Garcıa, and Monica Moreno Rocha, Editors,Mexican Mathematicians Abroad, 2016

656 Jose A. de la Pena, J. Alfredo Lopez-Mimbela, Miguel Nakamura, and JimmyPetean, Editors, Mathematical Congress of the Americas, 2016

655 A. C. Cojocaru, C. David, and F. Pappalardi, Editors, SCHOLAR—a ScientificCelebration Highlighting Open Lines of Arithmetic Research, 2015

654 Carlo Gasbarri, Steven Lu, Mike Roth, and Yuri Tschinkel, Editors, RationalPoints, Rational Curves, and Entire Holomorphic Curves on Projective Varieties, 2015

653 Mark L. Agranovsky, Matania Ben-Artzi, Greg Galloway, Lavi Karp, DmitryKhavinson, Simeon Reich, Gilbert Weinstein, and Lawrence Zalcman, Editors,Complex Analysis and Dynamical Systems VI: Part 1: PDE, Differential Geometry, RadonTransform, 2015

652 Marina Avitabile, Jorg Feldvoss, and Thomas Weigel, Editors, Lie Algebras andRelated Topics, 2015

For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/conmseries/.

This volume contains the Proceedings of the Conference on Completeness Problems, Car-

leson Measures, and Spaces of Analytic Functions, held from June 29–July 3, 2015, at the

Institut Mittag-Leffler, Djursholm, Sweden.

The conference brought together experienced researchers and promising young mathe-

maticians from many countries to discuss recent progress made in function theory, model

spaces, completeness problems, and Carleson measures.

This volume contains articles covering cutting-edge research questions, as well as longer

survey papers and a report on the problem session that contains a collection of attractive

open problems in complex and harmonic analysis.

ISBN978-1-4704-2305-6

9 781470 423056

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