A study of Schur multipliers and some Banach …ltu.diva-portal.org/smash/get/diva2:991334/FULLTEXT01.pdfDOCTORAL THESIS Department of Mathematics A study of Schur Multipliers and

DOCTORA L T H E S I S

Department of Mathematics

A study of Schur Multipliers and

some Banach Spaces of Infinite Matrices

Liviu-Gabriel Marcoci

ISSN: 1402-1544 ISBN 978-91-7439-092-6

Luleå University of Technology 2010

Liviu Gabriel M

arcoci A study of Schur M

ultipliers and some B

anach Spaces of Infinite Matrices

ISSN: 1402-1544 ISBN 978-91-7439-XXX-X Se i listan och fyll i siffror där kryssen är

A study of Schur Multipliers and some Banach Spaces of Infinite Matrices

Liviu-Gabriel Marcoci

Luleå University of Technology Department of Mathematics

Printed by Universitetstryckeriet, Luleå 2010

ISSN: 1402-1544 ISBN 978-91-7439-092-6

Luleå 2010

www.ltu.se

Abstract

This PhD thesis consists of an introduction and five papers, which deal withsome spaces of infinite matrices and Schur multipliers.

In the introduction we give an overview of the area that serves as a frame forthe rest of the Thesis.

In Paper 1 we introduce the space Bw(�2) of linear (unbounded) operators on

�2 which map decreasing sequences from �2 into sequences from �2 and we findsome classes of operators belonging either to Bw(�

2) or to the space of all Schurmultipliers on Bw(�

2).In Paper 2 we further continue the study of the space Bw(�

p) in the range1 < p < ∞. In particular, we characterize the upper triangular positive matricesfrom Bw(�

p).In Paper 3 we prove a new characterization of the Bergman-Schatten spaces

Lpa(D, �2), the space of all upper triangular matrices such that ‖A(·)‖Lp(D,�2) < ∞,

where

‖A(r)‖Lp(D,�2) =

(2

∫ 1

0

‖A(r)‖pCprdr

) 1p

.

This characterization is similar to that for the classical Bergman spaces. We alsoprove a duality between the little Bloch space and the Bergman-Schatten classesin the case of infinite matrices.

In Paper 4 we prove a duality result between Bp(�2) and Bq(�

2), 1 < p < ∞,1p + 1

q = 1, where by Bp(�p) we denote the Besov-Schatten space of all upper

triangular matrices A such that

‖A‖Bp(�2) =

[∫ 1

0

(1− r2)2p‖A′′(r)‖pCpdλ(r)

] 1p

< ∞.

In Paper 5 we introduce and discuss a new class of linear operators on quasi-monotone sequences in �2. We give some characterizations for such a class, forinstance we characterize the diagonal matrices.

i

Preface

This PhD thesis contains the following papers:

1. Updated and slightly revised version (2010) of the paper* A. N. Marcoci and L. G. Marcoci, A new class of linear operators on �2 and

Schur multipliers for them, J. Funct. Spaces Appl. 5(2007), No. 2, 151-165.2. A. N. Marcoci, L. G. Marcoci, L. E. Persson and N. Popa, Schur multiplier

characterization of a class of infinite matrices, Czechoslovak Math. J. 60(2010),No. 1, 183–193.

3. L. G. Marcoci, L. E. Persson, I. Popa and N. Popa, A new characteri-zation for Bergman-Schatten spaces and a duality result, J. Math. Anal. Appl.360(2009), No. 1, 67–80.

4. A. N. Marcoci, L. G. Marcoci, L. E. Persson and N. Popa, Besov-Schattenspaces and Hankel operators, Research Report No. 9, Department of Mathematics,Lulea University of Technology, 2009 (submitted).

5. L. G. Marcoci, A class of linear operators on quasi-monotone sequences in�2 and its Schur multipliers, Research Report No. 3, Department of Mathematics,Lulea University of Technology, 2010 (submitted).

These papers are put to a more general frame in an introduction, which alsoserves as a basic overview of the field.

iii

Acknowledgements

First of all I would like to express my deep gratitude to my main supervi-sors Professor Lars-Erik Persson (Department of Mathematics, Lulea Universityof Technology, Sweden) and Professor Nicolae Popa (Institute of Mathematics”Simion Stoilow” of the Romanian Academy, Romania) for their support and con-stant encouragement during my studies. I also thank my co-supervisors DocentSorina Barza (Sweden) and Docent Thomas Stromberg (Sweden) for supportingme and helping me in various ways.

Moreover, I thank Professor Maria-Jesus Carro for her generous advices, pa-tience and help during my visits in Barcelona. I also thank the Institute of Math-ematics ”Simion Stoilow”, of the Romanian Academy for financial support, whichmade it possible for me to attend in some conferences.

I also would like to thank Lulea University of Technology for their financialsupport. Finally, I want to thank everyone at the Department of Mathematics forhelping me in different ways and for the warm and friendly atmosphere. I warmlythank John Fabricius for his permanent willingness to help me.

v

Introduction

This PhD Thesis is dedicated to the study of some Banach spaces of infinitematrices. An important role is played by the upper triangular matrices calledanalytic matrices as well as some special operators acting on them, for instanceSchur multipliers. The first research component of the thesis is strongly relatedto a classical mathematical object having deep implications in the development ofmathematics in the last 150 years: infinite matrices.

The Bloch space has been studied for a long time in complex analysis, forthe first time in 1920 by A. Bloch (general references include K. Zhu’s book [45])regarding the boundary behavior of normal functions.

There are a few good sources for results and references about Bloch functionson the open unit disk. We mention here the paper of J. M. Anderson, J. Clunieand Ch. Pommerenke [2] and the survey papers of J. M. Anderson [1] and J. Cima[31].

The theory of Bergman spaces has evolved from several sources. A primarymodel is the related theory of Hardy spaces. For 0 < p < ∞, a function f ana-lytic in the unit disk D is said to belong to the Hardy space Hp if the integrals∫ 2π0

|f(reiθ)|pdθ remain bounded as r → 1. It belongs to the Bergman space Ap if

the area integral∫D|f(z)|pdσ is finite. It is clear that Hp ⊂ Ap.

The structural properties of invidual functions in Hp were studied actively inthe period 1915-1930, beginning with some classical papers of G. H. Hardy. Withthe emergence of functional analysis in the 1930’s, Hp spaces began to be viewedas examples of Banach spaces, for 1 ≤ p ≤ ∞. This point of view suggested avariety of new problems and provided effective methods for the solution of oldproblems. We mention here only a few mathematicians who early dealt with thesespaces: A. Beurling, H. Shapiro, L. Carleson and A. L. Shields. More generally,we mention here Triebel-Lizorkin spaces, which are very important in the theoryof function spaces. Details about these spaces can be found in H. Triebel’s books[42], [43] and [44]. Meanwhile, S. Bergman developed an elegant theory of Hilbertspaces of analytic functions in planar domains and in higher-dimensional complexspace, relying heavily on a reproducing kernel that became known as the Bergman

1

2 INTRODUCTION

kernel function [13]. Bergman’s work focused on spaces of analytic functions thatare square-integrable over the domain with respect to Lebesgue area or volumemeasure. When attention was later directed to the spaces Ap over the unit disk, itwas natural to call them Bergman spaces.

In the last twenty years the interest concerning these spaces has increased. Thepointwise multipliers of the Bloch space and the little Bloch space are characterizedby J. Arazy [4] in the case of open disk and by K. Zhu [45] in the case of the openunit ball. Coefficient multipliers of Bloch functions are described by J. M. Andersonand A. L. Shields in their paper [3]. J. Arazy remarked in [5] for the first timea similarity between functions and infinite matrices. This similarity has been thesource of many results and conjectures, one of the main results obtained for Schurmultipliers is due to G. Bennett in his paper [12].

Theorem 1 (Bennett’s Theorem). The Toeplitz matrix M is a multiplier ifand only if there exists a bounded and complex Borel measure μ on (the circlegroup) T with Fourier coefficients

μ(n) = cn for n = 0,±1,±2, · · · .Moreover, we then have

‖M‖(2,2) = ‖μ‖ = ‖M‖(∞,1).

An infinite matrix M is a Toeplitz matrix if it is on the form

mjk = cj−k (j, k = 0, 1, 2, · · · ).We call a matrix M a (p, q)-multiplier, 1 ≤ p, q ≤ ∞, if M ∗A maps �p into �q

whenever A does. The set M(p, q) of all such multipliers becomes a Banach spacewhen it is endowed with the norm

‖M‖(p,q) = sup{‖M ∗A‖p,q : ‖A‖ ≤ 1}.Another direction of research was that to study vector valued analytic func-

tions, but considered from a Banach space point of view.In this way appeared a series of papers e.g. by O. Blasco, J. L. Arregui, J. G.

Cuerva, A. Pelczynski, Q. H. Xu dedicated to Hardy, Bergman, Bloch and BMOvector valued spaces (see e.g. [6], [7], [14]-[25], [26], [27] and [28]). These paperscontinue the previous work of D. L. Burkholder [29] and J. Garcia Cuerva and J.L. Rubio de Francia [32] concerning singular operators and vector valued Hardyspaces.

We define the matriceal analogue of these spaces according to the correspondingdefinitions for analytic functions and we use the powerful device of Schur multipliersand its characterization in the case of Toeplitz matrices. Different spaces of infinitematrices, analytic matrices and Schur multipliers can be found in [8], [9], [10], [12],[36], [37], [39], [40] and [41].

INTRODUCTION 3

One alternative to study the proprieties of some Banach spaces of infinitematrices was to understand better the properties of sequence spaces and to findanother characterizations for them. Of course Hardy type inequalities plays animportant role here. There is a huge literature in this field but we mention hereonly the books [11], [30], [34] and [35] and the references given there.

In this PhD thesis we complement the theory and prove some new results inthis fascinating field in five papers:

Paper 1In the first paper of this PhD Thesis we introduce the space of infinite matrices

Bw(�2). This Banach space of infinite matrices can be regarded as a weaker version

of the space B(�2), the space of all bounded operators from �2 into �2. It is a weakerversion because it consists of those operators which maps the sequences which aredecreasing in modulus from �2 into �2. This space actually appeared in the study ofmatriceal analogue of classical function spaces like C(T) (the continuous functionson the torus), the Wiener algebra A(T) and the Lebesgue space L1(T).

It is easy to see that B(�2) ⊂ Bw(�2) and that the inclusion is proper. It is

interesting that these two spaces coincides on the subspace consisting of Toeplitzmatrices. More precisely, we prove that the Toeplitz matrix A belongs to B(�2) ifand only if A belongs to Bw(�

2). Using this result we prove a theorem which issimilar to G. Bennett’ s Theorem 1 above.

One main theorem of this paper is the following, which is in fact a new char-acterization of the positive Toeplitz upper triangular infinite matrices from B(�2):

Theorem 2. Let A =

⎛⎜⎜⎜⎜⎝a0 a1 a2 a3 ...0 a0 a1 a2 ...0 0 a0 a1 ...0 0 0 a0 ...... ... ... ... ...

⎞⎟⎟⎟⎟⎠ be an upper triangular Toeplitz

matrix. Then A ∈ B(�2)if and only if the sublinear operator TA is bounded from

�2 into �2 where

TA (b) (j) =1

j

j∑m=0

|(a ∗ b) (m)| , TA (b) (0) = |a0b0|

and (a ∗ b)(m) =∑

k+l=m

akbl, a = (ak)k≥0, b = (bk)k≥0 ∈ �2.

4 INTRODUCTION

Paper 2In the second paper we deal with the Banach space of infinite matrices Bw(�

p),1 < p < ∞. In the study of this space an important role is played by the followingold result that we obtain with a completely new proof: Let

d(q)× = ces(p)

where d(p) = {x = {xk}∞k=1 :(∑∞

k=1 supn≥k |xn|p) 1

p < ∞} and ces(p) = {x =

{xk}∞k=1 with∑∞

n=1

(1n

∑nk=1 |xk|

)p< ∞}.

Here d(q)× is the associate space of d(q), that is

d(q)× = {a = (an)n; such that

∞∑n=1

|anxn| < ∞ for all (xn)n ∈ d(q)}.

This result which gives us the Kothe dual of d(p) was obtained also by G.Bennett in [11] by using more technical methods, like factorization of some classicalinequalities. This problem was first investigated by A. A. Jagers in 1974 in thepaper [33].

In particular, in one important theorem from this paper we characterize someupper triangular operators from �p into another sequence spaces like d(p) and

g(p) ={x = {xk}∞k=1 : supn≥1

(1n

∑nk=1 |xk|p

)1/p< ∞

}in terms of some special

multipliers:

Theorem 3. Let 1 < p < ∞, 1p + 1

q = 1 and B be an upper triangular matrix.

Then

(1) B ∈ B(�p, d(p)) if and only if B ∗ [c] ∈ B(�p, �1) for all c ∈ ces(q).(2) B ∈ B(�p, �p) if and only if B ∗ [c] ∈ B(�p, �1) for all c ∈ �q.(3) B ∈ B(�p, g(p)) if and only if B ∗ [c] ∈ B(�p, �1) for all c ∈ �q · d(p).

Paper 3In the third paper we deal with analytic matrices. We continue the study

of Bergman-Schatten spaces and we give a characterization in terms of Taylorcoefficients namely:

Theorem 4. Let A be an analytic matrix. Then A ∈ Lpa(D, �2) if and only if

∞∑n=0

1

(n+ 1)2‖σn(A)‖pp < ∞.

INTRODUCTION 5

This characterization is similar with that of M. Mateljevic and M. Pavlovic in[38] for the classical Bergman spaces. Another important result we prove in thispaper is a duality between the Bergman-Schatten space and the little Bloch space.In the proofs we use very often the powerful tools of Schur multipliers and sometechniques from the theory analytic functions theory as well.

Paper 4In Paper 4 we continue the study of infinite matrix valued functions, namely

we consider the Besov-Schatten spaces in the framework of matrices. We define theBesov-Schatten matrix space Bp(�

2) to be the space of all upper triangular infinitematrices A such that

‖A‖Bp(�2) :=

[∫ 1

0


] 1p

< ∞,

where dλ is a positive measure on [0, 1) given by

dλ(r) :=2rdr

(1− r2)2.

We develop this theory by using in particular the powerfull device of Schur mul-tipliers and its characterizations in the case of Toeplitz matrices and prove somenew results. One of the main results in this paper is the following duality result:

Theorem 5. Under the pairing

< A,B >=

∫ 1

0

tr(V (A)[V (B)]∗)dλ(r)

we have the following dualities:(1) Bp(�

2)∗ ≈ Bq(�2) if 1 < p < ∞ and 1

p + 1q = 1;

(2) B0,c(D, �2)∗ ≈ B1(�2) and B1(�

2)∗ ≈ B(D, �2).

Paper 5In Paper 5 we consider a class of linear operators on quasi-monotone sequences

in �2 and its Schur multipliers. We define

Bαw(�

2) = {A infinite matrix ;Ax ∈ �2 for every

x = (xn)n ∈ �2 with|xn|nα

↘ 0, α ≥ 0}.

6 INTRODUCTION

In particular, for α = 0 we have that B0w(�

2) = Bw(�2) which was studied in

[36] and [37]. First we recall some definitions concerning infinite matrices andSchur multipliers mainly from [12] and [10] and then we give some preliminariesregarding Sawyer’s duality Theorem. We also include some additional definitionsthat we need in the sequel. We give a characterization for diagonal matrices tobelong in Bα

w(�2), α ≥ 0. Moreover, we consider the Schur product of matrices and

remark that for α ≥ 0, Bαw(�

2) is not closed under this product. One of the mainresults is that linear and bounded operators on �2 are Schur multipliers on Bα

w(�2)

with α > 0, a result which is not obvious since Bαw(�

2) is not a Schur algebra. Forα = 0 it is known that this result holds (see e.g. [36]). In what follows we presenttwo of the main Theorems in this paper.

Theorem 6. Let α ≥ 0, A = A0 be given by the sequence a = (an)n ThenA ∈ Bα

w(�2) if and only if

supn≥1

(∑nk=1 |ak|2k2α∑n

k=1 k2α

) 12

< ∞.

Moreover, the norm

‖A‖Bαw(�2) = sup

n≥1

(∑nk=1 |ak|2k2α∑n

k=1 k2α

) 12

.

Another main result is that linear and bounded operators on �2 are Schurmultipliers on Bα

w(�2) when α > 0.

Theorem 7. The space B(�2) is included in the space of all Schur multipliersfrom Bα

w(�2) to Bα

w(�2) where α ≥ 0.

Short description of the main contributions by L. G. Marcoci : This introduc-tion and Paper 5 are written by me. In all other papers I have substantiallycontributed to all parts, both concerning ideas and proofs. I mean that I am themain author of Papers 2 and 3. Moreover, for example, both the proof and idea ofTheorem 10 in Paper 1 (see Theorem 2) and Theorem 3.2 in Paper 4 (see Theorem5) are mainly due to me.

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[12] G. Bennett, Schur multipliers, Duke Math. J. 44(1977), 603–639.[13] S. Bergman, The kernel function and conformal mapping (second, revised edition), Mathe-

matical Surveys and Monographs, vol. 5, American Mathematical Society, Providence, RI,1970.

[14] O. Blasco, Boundary values of vector-valued functions in Orlicz-Hardy classes, Arch. Math.

(Basel)49(1987), no. 5, 434–439.[15] O. Blasco, Hardy spaces of vector-valued functions: duality, Trans. Amer. Math. Soc.

308(1988), no. 2, 495–507.

[16] O. Blasco, Boundary values of functions in vector-valued Hardy spaces and geometry onBanach spaces, J. Funct. Anal. 78(1988), no. 2, 346–364.

[17] O. Blasco, Spaces of vector valued analytic functions and applications. Geometry of Banach

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[20] O. Blasco, Vector-valued analytic functions of bounded mean oscillation and geometry ofBanach spaces, Illinois J. Math. 41(1997), no. 4, 532–558.

[21] O. Blasco, Vector-valued Hardy iequalities and B-convexity, Ark. Math. 38(2000), no. 1,

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[27] O. Blasco and A. Pelczynski, Theorems of Hardy and Paley for vector-valued analytic func-tions and related classes of Banach spaces, Trans. Amer. Math. Soc. 323(1991), no. 1,

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[28] O. Blasco and Q. Xu, Interpolation between vector-valued Hardy spaces, J. Funct. Anal.102(1991), no. 2, 331–359.

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and Weighted Inequalities, Memoirs of the American Mathematical Society, Number 877,2007.

[31] J. Cima, The basic properties of Bloch functions, International J. Math. Sci. 2(1979), 369–413.

[32] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted norm ienqualities and related topics,

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124.

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for them, J. Funct. Spaces Appl. 5(2007), no. 2, 151–165.

[37] A. N. Marcoci, L. G. Marcoci, L. E. Persson and N. Popa, Schur multiplier characterizationof a class of infinite matrices, Czechoslovak Math. J. 60(2010), No. 1, 183–193.

[38] M. Mateljevic and M. Pavlovic, Lp-behaviour of the integral means of analytic functions,Studia Math. 77(1984), 219–237.

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[40] V. V. Peller, Hankel operators of class Cp and their applications (rational approximation,

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hann Ambrosius Barth, Heidelberg, 1995.

[43] H. Triebel, Theory of function spaces, Basel, Birkhauser, 1983.[44] H. Triebel, Theory of function spaces II, Basel, Birkhauser, 1997.

[45] K. H. Zhu, Operator Theory in Function Spaces, Marcel Dekker, 1990.

Paper 1

A NEW CLASS OF LINEAR OPERATORS ON �2 ANDITS SCHUR MULTIPLIERS

A.-N. MARCOCI AND L.-G. MARCOCI

Abstract. We introduce the space Bw(�2) of linear (unbounded)

operators on �2 which map decreasing sequences from �2 into se-quences from �2 and we find some classes of operators belongingeither to Bw(�

2) or to the space of all Schur multipliers on Bw(�2).

For instance we show that the space B(�2) of all bounded oper-ators on �2 is contained in the space of all Schur multipliers onBw(�

2).

1. Introduction

Let A = (aij)i,j≥1 be an infinite matrix. We define Bw (�2) =

={A infinite matrix : Ax ∈ �2 for every x ∈ �2 with |xk| ↘ 0

},

where

�2 = {x = (xk)k≥1 :∞∑k=1

|xk|2 < ∞}

is the classical space of sequences.The above space of matrices has appeared in the study of the ma-

triceal analogue of some well-known Banach spaces as C (T), M (T),L1 (T).A similarity between the functions defined on the unit circle T and

the infinite matrices was remarked for the first time in 1978 by J. Arazy

2000 Mathematics Subject Classification. 47B35.Key words and phrases. Schur multiplier, Toeplitz matrices, matricial Wiener

algebra.1

2 A.-N. MARCOCI AND L.-G. MARCOCI

in [1]. Later on, in 1983, A. Shields has exploited further this similaritystarting with a few constructs used in harmonic analysis together withtheir matriceal analogues [12].Recently, in [4], the Fejer’s theory developed for Fourier series was

extended in the framework of matrices.The analogy is as follows: we identify a function f with the Toeplitz

matrix A = (aij)i,j≥1 ,

aij = ai−j for all i, j ∈ N∗

where (ak)k∈Z is the sequence of Fourier coefficients of the function f.The Schur product of two matrices is defined by

A ∗B = (aij · bij)i,j≥1 ,

where A = (aij)i,j≥1 , B = (bij)i,j≥1 . We denote by

M(�2)={M : M ∗ A ∈ B

(�2)for every A ∈ B

(�2)}

the space of all Schur multipliers equipped with the following norm

‖M‖ = sup‖A‖B(�2)≤1

‖M ∗ A‖B(�2) .

Let us define Ak = (a′ij)i,j≥1, k ∈ Z to be the matrix with the

elements

a′ij =

{aij if j − i = k,0 otherwise.

Ak is called the Fourier coefficient of k-order of the matrix A (seee.g. [4]). We have now a similarity between the expansion in Fourierseries of a periodical function f on T

f =∑k

akeikt

and the decomposition of matrix in diagonal matrices

A =∑k∈Z

Ak.

A NEW CLASS OF LINEAR OPERATORS 3

In this way we can say that B(�2) represents the matriceal analogueof L∞(T) and M(�2) is the analogue of M(T). Moreover in [4] wasintroduced the space of continuous matrices denoted by C(�2). A is acontinuous matrix if lim

n→∞σn(A) = A, where the limit is taken in the

norm of B(�2) and σn(A) is the Cesaro sum associated to Sn(A) :=n∑

k=−n

Ak.

This space is a Banach space with the following norm

‖A‖C(�2) = max(supn

‖σn(A)‖B(�2) , ‖A‖B(�2)).

In the way described earlier this space represents the matriceal ana-logue of the space C(T).Also N. Popa has defined

A(�2) =

{A infinite matrix : sup

l∈Z+

∑k∈Z

∣∣alk∣∣ < ∞}

where

alk =

{al l+k if k ≥ 0 and l ≥ 1

al−k l if k < 0 and l ≥ 1

and

A =

⎛⎝ a10 a11 ...a1−1 a20 ...... ... ...

⎞⎠ (see e.g. [3]).

This was the first attempt to define the matriceal analogue of theWiener algebra A(T) and we may call it matriceal Wiener algebra.

It is easy to see that the subspace of all Toeplitz matrices from A(�2)can be identified with A(T) but since A(�2) � C(�2) it is necessary tofind a larger space than B(�2). In order to see that A(�2) � C(�2) let


us consider

A =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 0 0 ...0 0 1 0 0 0 ...0 0 1 0 0 0 ...0 0 0 0 0 1 ...0 0 0 0 0 1 ...0 0 0 0 0 1 ...... ... ... ... ... ... ...

⎞⎟⎟⎟⎟⎟⎟⎟⎠.

Then supl∈Z+

∑k∈Z

∣∣alk∣∣ = 1 and A ∈ A (�2) implies∥∥∥Aen(n+1)

2

∥∥∥2=

√n, where

ek = (0, ..., 0, 1, 0, ...).One solution would be to choose a larger space than B (�2), and, for

this reason we introduce Bw (�2). Clearly Bw (�2) is a Banach spacewith the norm

‖A‖Bw(�2) = sup‖x‖2≤1|xk|↘0

‖Ax‖2 .

The following result due to E. Sawyer [11] will be used in the sequelas it appeared in [8]. For alternative simple proof see also [13].

If v = (v(n))n is a weight on N∗ = Z+, we put v =∞∑n=0

v(n)χ[n,n+1)

and V (t) =t∫0

v(s)ds.

Let us mention moreover that the relation f ≈ g means that thereare two positive constants a and b such that af ≤ g ≤ bf.

Then we have:

Theorem 1. Let w = (w(n))n, v = (v(n))n be weights on N∗ and let

S = supf↘

∞∑n=0

f(n)v(n)( ∞∑n=0

f(n)pw(n)

) 1p

.

Then


(i) If 0 < p ≤ 1

S = supn≥0

V (n)

W1p (n)

,

with W defined by W (n) =n∑

k=0

w(k) and similarly for V .

(ii) If 1 < p < ∞,

S ≈

⎛⎝∞∫0

(V (t)

W (t)

)p′−1

v(t)dt

⎞⎠ 1p′

≈

⎛⎝∞∫0

(V (t)

W (t)

)p′

w(t)dt

⎞⎠ 1p′

+V (∞)

W1p (∞)

where 1p+ 1

p′ = 1.

The paper is organized as follows: in Section 2 we show that thematriceal Wiener algebra A(�2) is a subset of Bw(�

2) (Proposition 2)and we give some criteria for diagonal matrices to belong to Bw(�

2).Moreover we consider the Schur product of matrices and remark thatBw(�

2) is not closed under this product. We prove in Section 3 the mainresult of the paper, namely that linear and bounded operators on �2

are Schur multipliers on Bw(�2), a result which is not obvious, since

Bw(�2) is not Schur algebra. Finally we collected in Section 4 all results

concerning the Toeplitz matrices. For instance there is no differencebetween Toeplitz matrices from B(�2) and those from Bw(�

2).

2. Preliminary results

Proposition 2. If A is an upper triangular matrix then ‖A‖Bw(�2) ≤‖A‖A(�2).


Proof. Let A ∈ A (�2) be an upper triangular matrix. Then for every(xl)l≥0 ∈ �2 with |xl| ↘ 0 and ‖x‖2 ≤ 1 we have

∞∑k=1

∣∣∣∣∣∞∑

l=k−1

akl−k+1xl

∣∣∣∣∣2

≤∞∑k=1

( ∞∑l=k−1

∣∣akl−k+1

∣∣ |xl|)2

≤∞∑k=1

( ∞∑l=k−1

∣∣akl−k+1

∣∣)2

supl≥k−1

|xl|2

≤∞∑k=1

(supk

∞∑l=k−1

∣∣akl−k+1

∣∣)2

supl≥k−1

|xl|2

≤ ‖A‖2A(�2)

∞∑k=1

supl≥k−1

|xl|2

= ‖A‖2A(�2) ‖x‖22 ≤ ‖A‖2A(�2)

which implies that ‖A‖Bw(�2) ≤ ‖A‖A(�2). �

Proposition 3. Let A = A0 given by (ak)∞k=1. Then A ∈ Bw (�2) if

and only if

supn∈N∗

⎛⎜⎜⎝n∑

k=1

|ak|2

n

⎞⎟⎟⎠12

< ∞.

Moreover

‖A‖Bw(�2) = supn∈N∗

⎛⎜⎜⎝n∑

k=1

|ak|2

n

⎞⎟⎟⎠12

.

Proof. The sufficiency follows immediately from the factorization

�2 = d (1, 2) · g (1, 2)


where

d (1, 2) =

{x :

∞∑n=1

supk≥n

|xk|2 < ∞}and

g (1, 2) =

{x :

n∑k=1

|xk|2 = O (n)

}(see e.g. [5], p. 9).

For the necessity take x(n) =(

1√n, 1√

n, ..., 1√

n, 0, ...

),∥∥x(n)

∥∥2

= 1,∣∣∣x(n)k

∣∣∣k↘ 0, n ≥ 1

∥∥Ax(n)∥∥2=

(1

n

n∑k=1

|ak|2) 1

2

≤ ‖A‖Bw(�2)

and supn

⎛⎝ n∑k=1

|ak|2

n

⎞⎠ 12

< ∞. �

If we translate this sequence (ak)∞k=1 above or below of the main

diagonal we obtain the following similar result.

Corollary 4. a) Let k > 0 and A = Ak given by (al)∞l=1. Then A ∈

Bw(�2) if and only if sup

n∈N∗

(1

n+k

n∑l=1

|al|2) 1

2

< ∞ and

‖A‖Bw(�2) = supn∈N∗

(1

n+ k

n∑l=1

|al|2) 1

2

.

b) Let k < 0 and A = Ak given by (al)∞l=1.Then A ∈ Bw(�

2) if and onlyif

supn∈N∗

(1

n

n∑l=1

|al|2) 1

2

< ∞

and ‖A‖Bw(�2) = supn∈N∗

(1n

n∑l=1

|al|2) 1

2

.


Remark 5. 1. Clearly B(�2) ⊆ Bw(�2) and using Proposition 3 for

the matrix A = A0 given by (ak)∞k=1,

ak =

{ √2n if k = 2n

0 if k �= 2nn � 1

we can easily show that the inclusion is proper.2. While B(�2) is closed under Schur multiplication Bw(�

2) is not.For example it is easy to see that A ∗ A /∈ Bw(�

2) where A is thematrix defined previously.3. The space Bw(�

2) cannot be compared with M(�2) meaning that

(1) Bw(�2) � M(�2) and

(2) M(�2) � Bw(�2).

For (1) there is A = A0 given by (ak)∞k=1 such that (ak)k /∈ �∞,

A ∈ Bw(�2) and A /∈ M(�2).

For (2) we take A =

⎛⎜⎜⎜⎜⎝1 1 ... 1 ...0 0 ... 0 ...0 0 ... 0 ...0 0 ... 0 ...... ... ... ... ...

⎞⎟⎟⎟⎟⎠ ∈ M(�2) but A /∈ Bw(�2).

4. Let A = A0 given by (ak)∞k=1.Then

A ∈ M(Bw(�2), Bw(�

2)) ={M : M ∗B ∈ Bw(�

2) for every B ∈ Bw(�2)}

if and only if (ak)k ∈ �∞.

A ∈ M(Bw(�2), Bw(�

2)) then

A ∗B ∈ Bw(�2) for B = B0 ∈ Bw(�

2) given by (bk)∞k=1.

From Proposition 3,

(bk)∞k=1 ∈ g(1, 2), ‖Ax‖2�2 =

∞∑k=1

|akbkxk|2 < ∞

for every b = (bk)k ∈ g(1, 2), and x ∈ d(1, 2).


Using the factorization �2 = d(1, 2) · g(1, 2), provided in [6] we getthat (an)n ∈ �∞.

Now let (an)n ∈ �∞. It is easy to see that

‖B0‖Bw(�2) ≤ ‖B‖Bw(�2) .

3. Main results

Lemma 6.

sup|xn|↘0

∣∣∣∣ ∞∑n=1

anxn

∣∣∣∣( ∞∑n=1

|xn|2) 1

2

= sup|xn|↘0

∞∑n=1

|an| |xn|( ∞∑n=1

|xn|2) 1

2

≈

⎛⎝ ∞∑n=1

(1

n

n∑k=1

|ak|)2⎞⎠ 1

2

,

where (an)n and (xn)n are sequences of complex numbers.

Proof. We denote S = sup|xn|↘0

∞∑n=1

|an||xn|( ∞∑

n=1|xn|2

) 12. From Theorem 1 we have

S ≈

⎛⎝∞∫0

(V (t)

W (t)

)2

w(t)dt

⎞⎠ 12

+V (∞)

W1p (∞)

,

where v(n) = |an| , w(n) = 1, f(n) = |xn| for every n nonnegativeinteger.In this case

v =∞∑n=0

v(n)χ[n,n+1) =∞∑n=0

|an|χ[n,n+1)(s), where a0 = 0,


therefore, for t ∈ (j, j + 1), we have

V (t) =

t∫0

v (s) ds =

j∫0

v (s) ds+

t∫j

v (s) ds

=

j−1∑m=0

m+1∫m

v (s) ds+

t∫j

v (s) ds

=

j−1∑m=0

|am|+ |aj| (t− j) ,

V (∞) =∞∫0

v (s) ds =∞∫0

∞∑n=0

|an|χ[n,n+1) (s) ds =∞∑n=1

|an| and

W (∞) =

∞∫0

w (s) ds = ∞,

since w (s) =∞∑n=0

χ[n,n+1) (s) .

Clearly, letting vM =M∑n=0

|an|χ[n,n+1) we have that

VM =

∞∫0

vM (s) ds

=

∞∫0

M∑n=0

|an|χ[n,n+1) (s) ds

=M∑n=1

|an| < ∞,


we get

S =supM

⎛⎜⎜⎜⎝ sup|xn|↘

M∑n=1

|xn| |an|( ∞∑n=1

|xn|2) 1

2

⎞⎟⎟⎟⎠

≈ supM

⎡⎢⎣⎛⎝∞∫

0

(VM (t)

t

)2

dt

⎞⎠ 12

+VM (∞)

W12 (∞)

⎤⎥⎦≈

⎛⎝∞∫0

(V (t)

t

)2

dt

⎞⎠ 12

.

But

∞∫0

(V (t)

t

)2

dt =∞∑j=1

j+1∫j

⎛⎜⎜⎜⎝j−1∑m=0

|am|+ |aj| (t− j)

t

⎞⎟⎟⎟⎠2

dt

≤∞∑j=1

⎛⎝j+1∫j

1

t2dt

⎞⎠( j∑m=1

|am|)2

≤∞∑j=1

(1

j

j∑m=1

|am|)2

.

On the other hand

∞∫0

(V (t)

t

)2

dt ≥∞∑j=1

j+1∫j

⎛⎜⎜⎜⎝j−1∑m=1

|am|

t

⎞⎟⎟⎟⎠2

dt+∞∑j=1

j+1∫j

|aj|2(t− j)2

t2dt ≥

≥∞∑j=1

(1

j

j∑m=1

|am|)2

,


which implies that S ≈(

∞∑n=1

(1n

n∑k=1

|ak|)2) 1

2

. �

Theorem 7. Let M(Bw(�2), Bw(�

2)) the space of Schur multipliersfrom Bw(�

2) to Bw(�2). Then

B(�2) ⊂ M(Bw(�2), Bw(�

2)).

Proof. Let us take A ∈ B (�2), B ∈ Bw (�2) and x = (xk)k≥1 ∈ �2 with|xk| ↘ 0. Using Holder inequality twice we have that

∑j

∣∣∣∣∣∑k

ajkbjkxk

∣∣∣∣∣2

≤∑j

(∑k

|ajk| |bjk| |xk|)2

≤∑j

(∑k

|ajk|2)(∑

k

|bjk|2 |xk|2)

≤ supj

(∑k

|ajk|2)∑

j

(∑k

|bjk|2 |xk|2).

Thus

‖(A ∗B)x‖2 ≤ ‖A‖2,∞(∑

j

(∑k

|bjkxk|2)) 1

2

(1)

≤ ‖A‖B(�2)

(∑j

(∑k

|bjkxk|2)) 1

2


To estimate the second term in (1) we will use Rademacher functionson [0, 1] denoted with rk, k ≥ 1 (see e.g. [9], p. 126).∑

j

(∑k

|bjkxk|2)

=∑j

∫ 1

0

∣∣∣∣∣∑k

bjkxkrk(t)

∣∣∣∣∣2

dt

≤ ess supt∈[0,1]∑j

∣∣∣∣∣∑k

bjkxkrk(t)

∣∣∣∣∣2

≤ ‖B‖2Bw(�2)‖x‖2�2 .The proof is complete.

�We give now a necessary condition in order that an upper triangular

Toeplitz matrix belong to B(�2) :

Proposition 8. Let A =

⎛⎜⎜⎜⎜⎜⎝a0 a1 a2 a3 a4 ...0 a0 a1 a2 a3 ...0 0 a0 a1 a2 ...0 0 0 a0 a1 ...0 0 0 0 a0 ...... ... ... ... ... ...

⎞⎟⎟⎟⎟⎟⎠. If A ∈ B(�2)

then A ∈ Bw(�2), where A is the diagonal matrix A = A0 given by the

sequence (am)∞m=0 and am =

m∑j=0

aj.

Proof. Let A ∈ B(�2). Then

∞∑k=0

∣∣∣∣∣∞∑j=0

ajxj+k

∣∣∣∣∣2

< ∞

for every (xj)j ∈ �2.

But ‖e1 + e2 + ...+ eN‖22 = N , where ek = (0, 0, ..., 0, 1, 0, ...).


Thus

A

(1√N(e1 + e2 + ...+ eN)

)=

1√N(aN,aN−1, ..., a1, a0, 0, ...)

which implies that supN

N∑m=0

|am|2

N< ∞ and, by Proposition 3 it follows

that A ∈ Bw(�2). �

We remark here that if we translate this result to the case of func-tions we shall get a necessary condition for a function to belong to H∞

i.e. if f ∈ H∞ then supN

N∑k=0

∣∣∣∣∣k∑

m=0f(m)

∣∣∣∣∣2

N< ∞, where f(k) is the Fourier

coefficient of k-order.

Theorem 9. Bw (�2)∩T = B (�2)∩T , where T is the set of all Toeplitzmatrices.

Proof. Let A be a Toeplitz matrix. Clearly, if A ∈ B (�2) it followsthat A ∈ Bw (�2). It is well known that a Toeplitz matrix A = (aij),where aij = ai−j for all i, j ∈ N∗, maps �2 into �2 precisely when thereexists a measurable function essentially bounded on [0, 2π] with Fourier

coefficients f (n) = an (n = 0,±1,±2, ...) and ‖A‖B(�2) = ‖f‖∞ see e.g.

[14].

Let f (t) =∞∑

k=−∞ake

2πikt, (xn)∞n=0 ∈ �2 with |xn| ↘ 0 and h (t) =

∞∑k=0

xke2πikt.

Then

‖Ax‖2 =

⎛⎝ ∞∑k=0

∣∣∣∣∣∞∑

j=−k

ajxk+j

∣∣∣∣∣2⎞⎠ 1

2

=

⎛⎜⎝ ∞∑k=0

∣∣∣∣∣∣1∫0

f (t) e2πikth (−t) dt

∣∣∣∣∣∣2⎞⎟⎠

12


= sup‖g‖2≤1

∣∣∣∣∣∣∞∑k=0

gk

1∫0

f (t) e2πikth (−t) dt

∣∣∣∣∣∣ = sup‖g‖2≤1

∣∣∣∣∣∣1∫0

f (t) g (t)h (−t) dt

∣∣∣∣∣∣ ,where g (t) =

∞∑k=0

gke2πikt.

Hence

‖A‖Bw(�2) = sup{|∫ 1

0

f(t)g(t)h(−t)dt| : ‖g‖2 ≤ 1,

‖h‖2 ≤ 1, h ∈ L2([0, 1])}.

If we take g (t) = 1√n

n−1∑j=0

e2πij(t−t0) and h (−t) = 1√n

n−1∑k=0

e2πik(t0−t) for

n = 2p+ 1, p ∈ N∗ and Kn-the Fejer kernel we obtain

‖A‖Bw(�2) ≥

∣∣∣∣∣∣1∫0

f (t)1

2p+ 1

∣∣∣∣∣p∑

j=−p

e2πij(t−t0)

∣∣∣∣∣2

dt

∣∣∣∣∣∣=

∣∣∣∣∣∣1∫0

f (t)K2p+1 (t− t0) dt

∣∣∣∣∣∣ a.e.→ |f(t0)|

which implies that ‖A‖Bw(�2) ≥ ‖f‖∞ = ‖A‖B(�2) . �

Theorem 10. Let A =

⎛⎜⎜⎜⎜⎝a0 a1 a2 a3 ...0 a0 a1 a2 ...0 0 a0 a1 ...0 0 0 a0 ...... ... ... ... ...

⎞⎟⎟⎟⎟⎠ be a positive upper tri-

angular Toeplitz matrix. Then A ∈ B (�2) if and only if the sublinearoperator TA is bounded from �2 into �2 where

TA (b) (j) =1

j

j∑m=0

|(a ∗ b) (m)| , TA (b) (0) = |a0b0|

and (a ∗ b)(m) =∑

k+l=m

akbl, a = (ak)k≥0, b = (bk)k≥0 ∈ �2.


Proof. From Theorem 9, A ∈ B(�2) if and only if A ∈ Bw (�2) if and

only if

(∞∑j=0

ajxj+k

)k≥0

∈ �2 for any x ∈ �2 with |xj| ↘ 0. This is

equivalent with

sup||b||�2≤1

∣∣∣∣∣∞∑k=0

( ∞∑j=0

ajxj+k

)bk

∣∣∣∣∣ < ∞ for all x ∈ �2, |xj| ↘ 0.

∞∑k=0

(∞∑j=0

ajxj+k

)bk =

∞∑l=0

xlcl, where cl =l∑

k=0

al−kbk. From lemma 6

sup|xl|↘0

∣∣∣∣∞∑l=0

xlcl

∣∣∣∣(∑l

|xl|2) 1

2

= sup|xl|↘0

∞∑l=0

|xl| |cl|(∑l

|xl|2) 1

2

≈ |a0b0|2 +∞∑j=1

(1

j

j∑m=0

∣∣∣∣∣m∑k=0

am−kbk

∣∣∣∣∣)2

= |a0b0|2 +∞∑j=1

(1

j

j∑m=0

|(a ∗ b) (m)|)2

.

Thus the proof is complete. �Now we show that M (Bw(�

2), Bw(�2)) ∩ T ⊆ M(�2) ∩ T .

Theorem 11. If M is a Toeplitz matrix with M = (mjk) ,mjk = cj−k

where (j, k = 0, 1, 2, 3, ...) and M ∈ M (Bw(�2), Bw(�

2)) there exists μa bounded, complex, Borel measure on T with μ(n) = cn for n =0,±1,±2, .... Moreover

‖μ‖ ≤ ‖M‖M(Bw(�2),Bw(�2)) .

Proof. We follow the standard method [2] for solving the ”problem

of moments”. Let p(t) =∞∑

n=−∞pne

int be an arbitrary trigonometric


polynomial and denote by P the Toeplitz matrix generated by p. Then∣∣∣∑ cnpn

∣∣∣ = limN→∞

∣∣∣∣∣∑n

cnpn

(1− |n|

N

)∣∣∣∣∣ = limN→∞

∣∣⟨(M ∗ P ) x(N), x(N)⟩∣∣

where

x(N)(j) =

{1√N+1

for 0 ≤ j ≤ N

0 for j > N

We consider now the linear functional

pΛ→∑n

cnpn

on the subspace of C (T) generated by polynomials

|Λ(p)| =∣∣∣∣∣∑

n

cnpn

∣∣∣∣∣ = limN→∞

∣∣⟨(M ∗ P ) x(N), x(N)⟩∣∣ ≤ ∥∥(M ∗ P ) x(N)

∥∥�2≤

≤ ‖(M ∗ P )‖Bw(�2) ≤ ‖M‖M(Bw(�2),Bw(�2))·‖P‖Bw(�2) = ( by Theorem 9 ) =

‖M‖M(Bw(�2),Bw(�2)) · ‖p‖∞which implies that

‖Λ‖ ≤ ‖M‖M(Bw(�2),Bw(�2)) .

It is clear now that this map is well-defined and continuous andthe existence of a measure satisfying all the requirements of the the-orem follows easily from the Hahn-Banach and Riesz representationtheorems. �Using an extension of the F. and M. Riesz theorem from [10] we

have:

Corollary 12. If M ∈ M (Bw(�2), Bw(�

2)) is a Toeplitz matrix M =(mjk), mjk = cj−k (k, j = 0, 1, 2, ...), and for n < 0, cn = 0 with n �∈ E,where E is a set of type Λ(1) ( see [10]) then cn → 0 as n → ∞.

Proof. This follows immediately from Riemann-Lebesgue lemma. �


References

[1] J. Arazy, Some remarks on interpolation theorems and the boundedness of thetriangular projection in unitary matrix spaces, Integral Equations OperatorTheory 1(1978), 453–495.

[2] S. Banach, Theorie des operations lineaires, second edition, Chelsea, NewYork, 1963.

[3] S. Barza, V.D. Lie and N. Popa, Approximation of infinite matrices by ma-triceal Haar polynomials, Ark. Mat. 43 (2005), no. 2, 251–269.

[4] S. Barza, L. E. Persson and N. Popa, A Matriceal Analogue of Fejer’s theory,Math. Nach 260(2003), 14–20.

[5] G. Bennett, Schur multipliers, Duke Math. J. 44(1977), 603–639.[6] G. Bennett, Factorizing the Classical Inequalities, Memoirs of the American

Mathematical Society, Number 576, 1996.[7] M. J. Carro and J. Soria, Weighted Lorentz spaces and the Hardy operator, J.

Funct. Anal. 112(1993), 480–494.[8] M. J. Carro, J. A. Raposo and J. Soria, Recent Developments in the Theory

of Lorentz Spaces and Weighted Inequalities, to appear in Memoirs of theAmerican Mathematical Society.

[9] S. Kaczmarz and G. Steinhaus, Theory of orthogonal series (Russian), Gosu-darstv. Izdat. Fiz.-Mat. Lit., Moscow 1958.

[10] W. Rudin, Trigonometric Series with Gaps, J. Math. and Mechanics, Vol. 92(1960), 203–227.

[11] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Stu-dia Math. 96(1990), 145–158.

[12] A. L. Shields, An analogue of a Hardy-Littlewood-Fejer inequality for uppertriangular trace class operators, Math. Z. 182(1983), 473–484.

[13] V. D. Stepanov, The weighted Hardy’s inequality for nonincreasing functions,Trans. Amer. Math. Soc. 338(1993), 173–186.

[14] K. Zhu, Operator theory on Banach function spaces, Marcel Dekker, New York,1990.

Department of Mathematics and Informatics, Technical University

of Civil Engineering Bucharest, 124 Lacul Tei Boulevard, Bucharest

020396

E-mail address: anca [email protected]

Department of Mathematics and Informatics,, Technical University

of Civil Engineering Bucharest, 124 Lacul Tei Boulevard, Bucharest

020396

E-mail address: liviu [email protected]

Paper 2

SCHUR MULTIPLIERS CHARACTERIZATION OF ACLASS OF INFINITE MATRICES

A.-N. MARCOCI, L.-G. MARCOCI, L.-E. PERSSON AND N. POPA

Abstract. Let Bw(�p) denote the space of infinite matrices A for

which A(x) ∈ �p for all x = {xk}∞k=1 ∈ �p with |xk| ↘ 0. In thispaper we characterize the upper triangular positive matrices fromBw(�

p), 1 < p < ∞, by using a special kind of Schur multipliersand the G. Bennett factorization technique. Also some relatedresults are stated and discussed.

1. Introduction

In this paper we deal with infinite matrices A, whose entries alk, fork ∈ Z and l ∈ Z+, are indexed with respect to the kth diagonal andwith the lth place on this diagonal. In what follows, sometimes weshall describe an infinite matrix by A = (alk)k∈Z,l∈Z+ more precisely

A =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

a10 a11 a12 a13 · · ·a1−1 a20 a21 a22

. . .

a1−2 a2−1 a30 a31. . .

a1−3 a2−2 a3−1 a40. . .

.... . . . . . . . . . . .

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠We started our study motivated by the paper [MM], where the first

two authors introduced the space Bw(�2) of those infinite matrices A

for which A(x) ∈ �2 for all x = {xk}∞k=1 ∈ �2 with |xk| ↘ 0.This space is of interest because the matrix version of the Wiener

algebra A(T), denoted by A(�2), which consists of all infinite matricesA = (alk)k∈Z,l∈Z+ such that supl∈Z+

∑k∈Z |alk| < ∞, is not contained in

the matrix version C(�2) of the space of all continuous functions C(T)(see [BPP] for the definition and the properties of C(�2)).

2000 Mathematics Subject Classification. 15A48, 15A60, 47B35, 26D15.Key words and phrases. Infinite matrices, Schur multipliers, discrete Sawyer du-

ality principle, Bennett factorization, Wiener algebra and Hardy type inequalities.1

2 A.-N. MARCOCI, L.-G. MARCOCI, L.-E. PERSSON AND N. POPA

Such an example is given by the matrix

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 0 0 0 · · ·0 0 1 0 0 0 0 · · ·0 0 1 0 0 0 0 · · ·0 0 0 0 0 1 0 · · ·0 0 0 0 0 1 0 · · ·0 0 0 0 0 1 0 · · ·0 0 0 0 0 0 0 · · ·...

......

......

......

. . .

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠where on the n(n+1)

2-column there are n entries equals to 1 placed on

the n(n−1)2

+ 1, . . . , n(n+1)2

- rows and 0 otherwise. Clearly we have

supl∈Z+

∑k∈Z

|alk| = 1, hence A ∈ A(�2) and∣∣∣Aen(n+1)

2

∣∣∣22= n for all en =

(0, . . . , 0, 1︸︷︷︸n

, 0, . . . ).

It yields that A(�2) ⊂ Bw(�2) (see Proposition 2 in [MM]), where

Bw(�2) is the Banach space with respect to the norm

‖A‖Bw(�2) = sup‖x‖2≤1, |xk|↘0

‖A(x)‖2.

We remark that �2dec = {x = (xk) ↘ 0, x ∈ �2} is a cone and the

solid hull of this cone, denoted by so(�2dec), coincides with the Banach

space d(2) = {x; ∑∞n=1 supk≥n |xk|2 < ∞}. The spaces d(p), p ≥ 1

are introduced in [B], where it is described how they are connected toHardy type inequalities (for historical information and results of thistype we refer to the books [KMP] and [KP]). Here so(�2dec) = {y =

(yk) ∈ �2 such that |yk| ≤ xk for all k ∈ N, where xk ↘ 0 in �2}.Let A be a positive matrix, that is such that all the elements of the

sequence A(x) are positive whenever x = (xj)j, is a sequence havingonly a finite number of nonzero positive elements. Clearly, if A ∈Bw(�

2), then A ∈ B(d(2), �2), that is A is a bounded linear operatorfrom d(2) into �2.

The next Lemma, which may be regarded as a discrete version ofa special case of the Sawyer duality principle [Sa](see also [KP]) wasobtained and applied in [MM].

SCHUR MULTIPLIERS 3

Lemma 1.1. It yields that

sup|xn|↘0

|∑∞n=1 anxn|

(∑∞

n=1 |xn|2)1/2= sup

|xn|↘0

∑∞n=1 |an||xn|

(∑∞

n=1 |xn|2)1/2

≈

⎛⎝ ∞∑n=1

(1

n

n∑k=1

|ak|)2⎞⎠1/2

,


For the investigations in this paper we need a coresponding (discreteSawyer type) result for every p > 1 and not only for p = 2 as in Lemma1.1 (see our Lemma 2.4).

In this paper we consider the space Bw(�p) consisting of infinite ma-

trices A for which A(x) ∈ �p for all x = {xk}∞k=1 ∈ �p with |xk| ↘ 0(1 < p < ∞). In Theorem 2.1 we characterize the upper triangular pos-itive matrices from Bw(�

p) by using a special kind of Schur multipliers.Also some related results are formulated in Section 2. The proofs canbe found in Section 3. We pronounce that our proofs are heavily de-pending on various important factorization results by G. Bennett [B]and Lemma 2.4.

2. Main results

First let us recall the definition of Schur multipliers.If A = (ajk) and B = (bjk) are matrices of the same size(finite or

infinite) their Schur product (or Hadamard product) is defined to bethe matrix of elementwise products

A ∗B = (ajkbjk).

There is, however, much justification for the term ”Schur product” andwe refer the reader to [B1] and [St] for an historical discussion. Thisconcept was first investigated by Schur in his paper [S] and has sincearisen in several different areas of analysis: [Po], [SS1], [SS2](complexfunction theory); [B], [KwP](Banach spaces); [SW], [P], [BP](operatortheory); [BPP], [BKP](matriceal harmonic analysis) and [St](multi-variate analysis).

If X and Y are two Banach spaces of matrices we define Schurmultipliers from X to Y as the space M(X, Y ) = {M : M ∗ A ∈Y for every A ∈ X}, equipped with the natural norm

‖M‖ = sup‖A‖X≤1

‖M ∗ A‖Y .


We use a matrix operation introduced in [BLP], which extends togeneral matrices, the usual product of a Toeplitz matrix A and a com-plex scalar c.

Namely, let c = (c1, c2, . . . ) be a sequence of complex numbers. Wedenote by [c] the matrix whose entries [c]lk are equal to cl, for l ≥ 1 andk ∈ Z.

We observe that, for a Toeplitz matrix A and for a constant sequencec = (c1, c1, . . . ), the matrix [c] ∗ A coincides with the usual productbetween the complex number c1 and the matrix A. Hence we denotedin [BLP] the product [c] ∗A by c�A and considered it as an externalproduct between a matrix and a sequence of complex numbers.

In what follows, using the results about multipliers from [B], wewill characterize the upper triangular positive matrices from Bw(�

p) bystudying the behaviour of the matrix [c]. Here Bw(�

p) denotes the spaceof those infinite matrices A for which A(x) ∈ �p for all x = {xk}∞k=1 ∈ �p

with |xk| ↘ 0. It is clear that for p > 1 this is a Banach space withrespect the norm

‖A‖Bw(�p) = sup‖x‖p≤1, |xk|↘0

‖A(x)‖p.

Here, as usual,

�p = {x = {xk}∞k=1 :

( ∞∑k=1

|xk|p) 1

p

< ∞}.

Moreover, let

d(p) = {x = {xk}∞k=1 :

( ∞∑k=1

supn≥k

|xn|p) 1

p

< ∞}.

Our first result reads:

Theorem 2.1. Let B be an upper triangular matrix. Then B ∈B(�p, ces(p)), 1 < p < ∞, if and only if B ∗ [c] ∈ B(�p, �1), for allc ∈ d(p).

Here

ces(p) = {x = {xk}∞k=1 with∞∑n=1

(1

n

n∑k=1

|xk|)p

< ∞}

denotes the Banach space equipped with the norm

‖x‖ces(p) =( ∞∑

n=1

(1

n

n∑k=1

|xk|)p)1/p

.

SCHUR MULTIPLIERS 5

Now we can state our main result concerning the characterization ofthe matrices belonging to Bw(�

p).

Theorem 2.2. A lower triangular positive matrix A belongs to Bw(�p),

1 < p < ∞, if and only if A∗ ∗ [c] ∈ B(�q, �1), where 1p+ 1

q= 1 for all

c ∈ d(p), where A∗ is the usual adjoint of the matrix A.

Besides the ces(p)-spaces who have already attracted a fair deal ofattention in the literature, an important role is played by �p, d(p) andalso g(p), defined by

g(p) =

⎧⎨⎩x = {xk}∞k=1 : supn≥1

(1

n

n∑k=1

|xk|p)1/p

< ∞

⎫⎬⎭ .

Therefore we also state the following result where ces(p) in Theorem2.2 is replaced by any of these spaces and �q ·d(p) is the sequence spaceof coordinatewise products (see [B] for further details).

Theorem 2.3. Let 1 < p < ∞, 1p+ 1

q= 1 and B be an upper triangular

matrix. Then

(1) B ∈ B(�p, d(p)) if and only if B ∗ [c] ∈ B(�p, �1) for all c ∈ces(q).

(2) B ∈ B(�p, �p) if and only if B ∗ [c] ∈ B(�p, �1) for all c ∈ �q.(3) B ∈ B(�p, g(p)) if and only if B ∗ [c] ∈ B(�p, �1) for all c ∈

�q · d(p).

Our proof of Theorem 2.1 (and thus of Theorem 2.2) is heavily de-pending on the following extension of Lemma 1.1 of independent inter-est.

Lemma 2.4. If p > 1, then

sup|xn|↘0

∣∣∣∣ ∞∑n=1

anxn

∣∣∣∣( ∞∑n=1

|xn|p) 1

p

= sup|xn|↘0

∞∑n=1

|an||xn|( ∞∑n=1

|xn|p) 1

p

≈( ∞∑

n=1

(1

n

n∑k=1

|ak|)q) 1

q

,

where (an)n and (xn)n are sequences of complex numbers and 1p+ 1

q= 1

.

Remark 2.5. If 1 < p < ∞ and 1p+ 1

q= 1, then

d(q)× = ces(p).


Here d(q)× is the associate space of d(q), that is

d(q)× = {a = (an)n; such that∞∑n=1

|anxn| < ∞ for all (xn)n ∈ d(q)}.

This result which gives us the Kothe dual of d(p) has been obtainedalso by G. Bennett in [B] using more technical methods, like factoriza-tion of some classical inequalities. This problem was first investigatedby Jagers in 1974 in the paper [J].

Finally we note that

so(�pdec

) = d(p)

(�pdec

denotes the subspace of �p consisting of non-increasing sequences)

and, hence, our results in particular implies Corollary 12.17 in paper[B] by G. Bennett.

3. Proofs

We first present a proof of the crucial Lemma 2.4, which is based onthe following result of E. Sawyer [Sa]. For p ≤ 1 a similar result hasbeen proved by M. J. Carro and J. Soria in their paper [CS].

Lemma 3.1. Let w = {w(n)}∞n=1, v = {v(n)}∞n=1 be weights on N∗, let

S = supf↘

∞∑n=0

f(n)v(n)( ∞∑n=0

f(n)pw(n)

) 1p

and v =∑∞

n=0 v(n)χ[n,n+1), w =∑∞

n=0 w(n)χ[n,n+1) and V (t) =∫ t

0v(s)ds,

W (t) =∫ t0w(s)ds.

If 1 < p < ∞, then

S ≈

⎛⎝∫ ∞

0

(V (t)

W (t)

)q−1

v(t)dt

⎞⎠ 1q

≈(∫ ∞

0

(V (t)

W (t)

)q

w(t)dt

) 1q

+V (∞)

W1p (∞)

where 1p+ 1

q= 1.

Here, as usual, the relation f ≈ g means that there are two positiveconstants C0 and C1 so that C0f(t) ≤ g(t) ≤ C1f(t), t ∈ [0,∞).

SCHUR MULTIPLIERS 7

Proof of Lemma 2.4. We denote

S = sup|xn|↘0

∞∑n=1

|an||xn|( ∞∑n=1

|xn|p) 1

p

.

According to Lemma 3.1 we have that

S ≈

⎛⎝ ∞∫0

(V (t)

W (t)

)q

w(t)dt

⎞⎠ 1q

+V (∞)

W1p (∞)

,

where v(n) = |an|, w(n) = 1, f(n) = |xn| for every nonnegative integern. In this case

v =∞∑n=0

v(n)χ[n,n+1) =∞∑n=0

|an|χ[n,n+1), where a0 = 0.

Therefore, for t ∈ (j, j + 1), it yields that

V (t) =

∫ t

0

v(s)ds =

∫ j

0

v(s)ds+

∫ t

j

v(s)ds

=

j−1∑m=0

∫ m+1

m

v(s)ds+

∫ t

j

v(s)ds

=

j−1∑m=0

|am|+ |aj|(t− j),

V (∞) =

∫ ∞

0

v(s)ds =

∫ ∞

0

∞∑n=0

|an|χ[n,n+1)(s)ds =∞∑n=1

|an|

and

W (∞) =

∫ ∞

0

w(s)ds = ∞,

since w(s) =∞∑n=0

χ[n,n+1)(s).

Letting vM =M∑n=0

|an|χ[n,n+1),


VM =∞∫0

vM(s)ds =∞∫0

M∑n=0

|an|χ[n,n+1)(s)ds = =M∑n=1

|an| < ∞, we get

that

S = sup|xn|↘0

∞∑n=1

|an||xn|( ∞∑n=1

|xn|p) 1

p

≈ supM

⎡⎣(∫ ∞

0

(VM(t)

t

)q

dt

) 1q

+VM(∞)

W1q (∞)

⎤⎦≈(∫ ∞

0

(V (t)

t

)q

dt

) 1q

.

But

∫ ∞

0

(V (t)

t

)q

dt =∞∑j=1

∫ j+1

j

⎛⎜⎜⎜⎝j−1∑m=0

|am|+ |aj|(t− j)

t

⎞⎟⎟⎟⎠q

≈∞∑j=1

(1

j

j∑m=1

|am|)q

,

which implies that

S ≈( ∞∑

n=1

(1

n

n∑k=1

|ak|)q) 1

q

.

The proof is complete. �

Proof of Theorem 2.1. For clearness we first prove the theorem for thespecial case p = q = 2. Note that A∗ is an upper triangular matrix.

Let Cdef= {c} be the upper triangular matrix obtained from [c] taking

the triangular projection PT , which acts as follows:

PT (A) =

{aij if i ≤ j0 otherwise.

(See [BLP].)Let B be an upper triangular matrix from B(�2, ces(2)). We have

B(x) =(∑∞

j=1 bijxj

)∞i=1

∈ ces(2), for all x = (xj)∞j=1 ∈ �2. But (B ∗

SCHUR MULTIPLIERS 9

C)(x) =((∑∞

j=1 bijxj)ci)∞i=1

is the product of two sequences, one from

ces(2), and the other one completely arbitrary. By Proposition 15.4 in[B] we have that

d(2) = I(2, 2)def= {m :

∑∞k=1(ik−ik−1)|mik |2 < ∞; for each sequence

i of integers with i0 = 0 < i1 < i2 < . . . }.Then, by using the table 29 on page 70 in [B], we get that (B∗C)(x) ∈

�1, where c ∈ d(2) = I(2, 2) and x ∈ �2. Hence B ∗ C ∈ B(�2, �1).Conversely, let B ∗ C ∈ B(�2, �1) for each c ∈ d(2). By Holder’s

inequality we have that �1 = �2 · �2, and, in view of Theorem 3.8 in [B],it follows that �2 = g(2) · d(2), where

g(2) =

{x; sup

n

∑n

k=1 |xk|2n

< ∞}.

Hence �1 = (�2 · g(2)) · d(2) and, according to Theorem 4.5 in [B],it yields that �1 = ces(2) · d(2). On the other hand, by Proposition14.5 in [B] ces(2) has d(2)-cancellation property, that is the inclusiony · d(2) ⊂ ces(2) · d(2) implies that y ∈ ces(2).

Now, by hypotheses, for each x ∈ �2, we have that

(B ∗ C)(x) =

((

∞∑j=1

bijxj)ci

)i

∈ �1 = ces(2) · d(2),

for all c ∈ d(2). By the cancellation property it follows that

B(x) =

( ∞∑j=1

bijxj

)i

∈ ces(2),

that is, by the closed graph theorem, B ∈ B(�2, ces(2)).Now we consider the case p �= 2.If B ∈ B(�p, ces(p)), c ∈ d(q), q > 1, and 1

p+ 1

q= 1, then as in the

proof of the case p = q = 2 we have that d(q) = I(q, q) and, in view ofthe table on page 70 in [B], it follows that

(B ∗ C)(x) ∈ �1, for all x ∈ �p,

that is B ∗ C ∈ B(�p, �1).Conversely, let B ∗ C ∈ B(�p, �1) for all c ∈ d(q). Then, similarly as

in the proof of the case p = q = 2 we find that

�1 = �p · �q = �p · g(q) · d(q) = (by Theorem 4.5 in [B]) = ces(p) · d(q).Since ces(p) has d(q)-cancellation property (see Proposition 14.5 in [B]) it follows that B ∈ B(�p, ces(p)).



Proof of Theorem 2.2. First let us note that by using Lemma 2.4 itfollows that A ∈ Bw(�

p) if and only if A∗ ∈ B(�q, ces(q)), for 1p+ 1

q= 1.

It remains to apply Theorem 2.1. �Proof of Theorem 2.3. (1). If B ∈ B(�p, d(p)), c ∈ ces(q) with 1

p+ 1

q= 1

and x ∈ �p we have that

(B ∗ [c])(x) =(( ∞∑

j=1

bijxj

)ci

)∞

i=1

=(yic

i)∞i=1

∈ d(p) · ces(q) =

(by Corollary 12.17 in [B]) = d(p) · d(p)∗ ⊂ �1.

Hence, by the closed graph theorem, it yields that

B ∗ [c] ∈ B(�p, �1).

Conversely, if B ∗ [c] ∈ B(�p, �1) for all c ∈ ces(q), then, denoting by

(yi)i =(∑∞

j=1 bijxj

)i∈N

, we have that (yici)i ∈ �1 for all c ∈ ces(q).

Thus

(yi)i ∈ ces(q)∗ = (by Corollary 12.17 in [B]) = d(p),

that is

B ∈ B(�p, d(p)).

(2). If B ∈ B(�p, �p), x ∈ �p, c ∈ �q with 1p+ 1

q= 1, then, by Holder’s

inequality, B ∗ [c] ∈ B(�p, �1).Conversely, let (yic

i)i ∈ �1 for all c ∈ �q. Then (yi)i ∈ �p and, conse-quently,

B ∈ B(�p, �p).

(3). If B ∈ B(�p, g(p)) and c ∈ �q · d(p), then, by using the previousnotations, we find that

(yici)i ∈ g(p) · �q · d(p) = ( by Theorem 3.8 in [B]) = �p · �q ⊂ �1.

Conversely, let (yi)i · �q · d(p) ∈ �1 = (by Theorem 3.8 in [B]) =g(p) · d(p) · �q. Consequently we have to show that

(yi)i ∈ g(p).

This fact follows clearly if g(p) has the d(p) · �q-cancellation property.We note that by using Proposition 14.5 in [B], we get that (yi)i·d(p) ∈

g(p) · d(p) · �p.Indeed, let (zi)i ∈ d(p) be fixed. Then (yizi)i · �q ∈ �p · �q. Since �p

has the �q-cancellation property (see Proposition 14.5 in [B]) it followsthat

(yizi)i ∈ �p = g(p) · d(p), for all (zi)i ∈ d(p);

SCHUR MULTIPLIERS 11

in other words (yi)i · d(p) ∈ g(p) · d(p). Using now the fact that g(p)has d(p)-cancellation property, it follows that (yi)i ∈ g(p). The proof iscomplete.

�

References

[B] G. Bennett, Factorizing the Classical Inequalities, Memoirs of the AmericanMathematical Society, Number 576, 1996. Zbl 0857.26009

[B1] G. Bennett, Schur multipliers, Duke Math.J. 44(1977), 603-639. Zbl0389.47015

[BKP] S. Barza, D. Kravvaritis and N. Popa, Matriceal Lebesgue spaces and Holderinequ ality, J. Funct. Spaces Appl. 3(2005), 239-249. Zbl 1093.42002

[BP] C. Badea and V. Paulsen, Schur multipliers and operator-valued Foguel-Hankel operators, Indiana Univ. Math. J. 50(2001), 1509-1522. Zbl1031.47019

[BPP] S. Barza, L.E. Persson and N. Popa, A Matriceal Analogue of Fejer’s theory,Math. Nach. 260(2003), 14-20. Zbl 1043.15020

[BLP] S. Barza, V.D. Lie and N. Popa, Approximation of infinite matrices bymatriceal Haar polynomials, Ark. Mat. 43 (2005), no. 2, 251–269. Zbl1097.15022

[CS] M.J. Carro and J. Soria, Weighted Lorentz spaces and the Hardy operator,J. Funct. Anal. 112(1993), 480-494. Zbl 0784.46022

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of Civil Engineering, 124 Lacul Tei Boulevard, Bucharest 020396, RO-

MANIA E-mail: anca [email protected] E-mail: liviu [email protected]

Department of Mathematics, Lulea University of Technology, SE-

97 187 Lulea, SWEDEN E-mail: [email protected]

Department of Mathematics, University of Bucharest and Insti-

tute of Mathematics of Romanian Academy, P.O. BOX 1-764 RO-014700

Bucharest, ROMANIA E-mail: [email protected]

Paper 3

A NEW CHARACTERIZATION OFBERGMAN-SCHATTEN SPACES AND A DUALITY

RESULT

L. G. MARCOCI, L. E. PERSSON, I. POPA, N. POPA

Abstract. Let B0(D, �2) denote the space of all upper triangularmatrices A such that limr→1−(1 − r2)‖(A ∗ C(r))′‖B(�2) = 0. We

also denote by B0,c(D, �2) the closed Banach subspace of B0(D, �2)consisting of all upper triangular matrices whose diagonals arecompact operators.

In this paper we give a duality result between B0,c(D, �2) andthe Bergman-Schatten spaces L1

a(D, �2). We also give a charac-terization of the more general Bergman-Schatten spaces Lp

a(D, �2),1 ≤ p < ∞, in terms of Taylor coefficients, which is similar to thatof M. Mateljevic and M. Pavlovic [12] for classical Bergman spaces.

1. Introduction

The Bloch and Bergman spaces have been studied for a long time incomplex analysis and in the last twenty years the interest concerningthese spaces has increased. A direction of research was that to studyvector valued analytic function, but considered from a Banach pointof view. In this way appeared a series of papers e.g. by J. A. Arregui,O. Blasco [2] and [3] and O. Blasco [6]–[9]. In what follows we con-sider the Bloch and Bergman spaces in the framework of matrices e.g.infinite matrix valued functions. We use the powerful device of Schurmultipliers and its characterizations in the case of Toeplitz matrices toprove the main theorems. The extension to the matriceal frameworkis based on the fact that there is a natural correspondence betweenToeplitz matrices and formal Fourier series associated to 2π-periodicfunctions (see e.g [1], [4], [11] and [14]).Let A = (ajk) and B = (bjk) be matrices of the same size (finite or

infinite). Then their Schur product (or Hadamard product) is definedto be the matrix of elementwise products:

A ∗B = (ajkbjk).

2000 Mathematics Subject Classification. 47B10, 15A60, 26D15.Key words and phrases. Infinite matrices, Toeplitz matrices, Schur multipliers,

Bergman-Schatten spaces, Bloch spaces, duality.1

2 L. G. MARCOCI, L. E. PERSSON, I. POPA, N. POPA

If X and Y are two Banach spaces of matrices we define Schur mul-tipliers from X to Y as the space

M(X, Y ) = {M : M ∗ A ∈ Y for every A ∈ X},equipped with the natural norm

‖M‖ = sup‖A‖X≤1

‖M ∗ A‖Y .

In the case X = Y = B(�2), where B(�2) is the space of all linear andbounded operators on �2, the space M(B(�2), B(�2)) will be denotedM(�2) and a matrix A ∈ M(�2) will be called Schur multiplier. Wemention here an important result due to G. Bennett [5], which will beoften used in this paper.

Theorem 1.1. The Toeplitz matrix M = (cj−k)j,k, where (cn)n∈Z is asequence of complex numbers, is a Schur multiplier if and only if thereexists a bounded and complex Borel measure μ on (the circle group) Twith

μ(n) = cn, for n = 0,±1,±2, ....

Moreover, we then have that

‖M‖ = ‖μ‖.We will denote by Ak, the kth − diagonal matrix associated to A

(see [4]). For an infinite matrix A = (aij) and an integer k we denoteby Ak the matrix whose entries a′ij are given by

a′ij =

{aij if j − i = k

0 otherwise.

Let A = (aij)i,j≥0, be an infinite matrix with complex entries and letn ≥ 0. The matrix A is said to be of n-band type (see [4]) if aij = 0 for|i− j| > n.In what follows we will recall some definitions from [13], which we

will use in this paper. We consider on the interval [0, 1) the Lebesguemeasurable infinite matrix-valued functions A(r). These functions maybe regarded as infinite matrix-valued functions defined on the unit discD using the correspondence A(r) → fA(r, t) =

∑∞k=−∞ Ak(r)e

ikt, where

Ak(r) is the kth-diagonal of the matrix A(r), the preceding sum is aformal one and t belongs to the torus T. This matrix A(r) is called an-alytic matrix if there exists an upper triangular infinite matrix A suchthat, for all r ∈ [0, 1), we have Ak(r) = Akr

k, for all k ∈ Z. In whatfollows we identify the analytic matrices A(r) with their correspondingupper triangular matrices A and we call them also analytic matrices.

BERGMAN-SCHATTEN SPACES AND A DUALITY RESULT 3

We will denote by Cp, 0 < p < ∞, the Schatten class operators (seee.g. [16]). We also recall the definition of Bergman-Schatten spaces(see e.g. [13]). Let 1 ≤ p < ∞. We denote

Lp(D, �2) ={r → A(r) which are strong measurable Cp−valued functionsdefined on [0, 1) such that

‖A‖Lp(D,�2) :=

(2

∫ 1

0

‖A(r)‖pCpr dr

) 1p

< ∞}.

Let Lpa(D, �2) = {r → A(r), where A(r) = A ∗ C(r) and A are up-

per triangular matrices with ‖A‖Lp(D,�2) < ∞}. Here C(r) denotes theToeplitz matrix associated to the Cauchy kernel 1

1−r, for 0 ≤ r <

1. By Lpa(D, �2) we mean the space of all upper triangular matri-

ces such that ‖A‖Lp(D,�2) < ∞. We identify Lpa(D, �2) and Lp

a(D, �2)and call Lp

a(D, �2) Bergman-Schatten classes. For p = ∞ we denoteL∞(D, �2) = {r → A(r) being a w∗ − measurable function on [0, 1) :‖A‖L∞(D,�2) := ess sup0≤r<1 ‖A(r)‖B(�2) < ∞} and L∞(D, �2) is thesubspace of L∞(D, �2) consisting of all strong measurable functions on[0, 1). We also consider

L∞a (D, �2) := {A analytic matrix : ‖A‖L∞

a (D,�2) :=

:= sup0≤r<1

‖C(r) ∗ A‖B(�2) = ‖A‖L∞(D,�2) < ∞}.

An important tool in this paper is the Bergman projection. It isknown (see e.g. [13]) that for all functions A(r) ∈ L2(D, �2) defined on[0, 1) and for all i, j ∈ N we have that

[P (A(·))](r)(i, j) ={2(j − i+ 1)rj−i

∫ 10aij(s) · sj−i+1ds, if i ≤ j,

0 otherwise.

Definition 1.2. The matriceal Bloch space B(D, �2) is the space of allanalytic matrices A with A(r) ∈ B(�2), 0 ≤ r < 1, such that

‖A‖B(D,�2) = sup0≤r<1

(1− r2)‖A′(r)‖B(�2) + ‖A0‖B(�2) < ∞

where B(�2) is the usual operator norm of the matrix A on the sequencespace �2 and A′(r) =

∑∞k=0 Akkr

k−1.

A matrix A ∈ B(D, �2) is called a Bloch matrix. It is clear that theToeplitz matrices which belong to the set of analytic matrices B(D, �2)appears as an extension of the classical Bloch space of functions.


A very useful theorem is the following (see e.g. [13]):

Theorem P Both P : L∞(D, �2) → B(D, �2) and P : L∞(D, �2) →B(D, �2) are bounded surjective operators.

The main results in this paper are presented, proved and discussedin Sections 2 and 3 below.

In Section 2 we give a characterization of matrices in the little Blochspace B0(D, �2) using the Bergman projection. One of the main resultsin this section is a new duality result (see Theorem 2.9). Also somerelated results are formulated and proved. In Section 3, we begin tostate and prove with three technical lemmas, which are necessary toprove the main result of this section namely a characterization of theBergman-Schatten spaces Lp

a(D, �2), 1 ≤ p ≤ ∞, in terms of Taylorcoefficients (see Theorem 3.4).

2. A matrix version of the little Bloch space

Now we introduce another space of matrices, the so-called little Blochspace of matrices.

Definition 2.1. The space B0(D, �2) is the space of all upper triangularinfinite matrices A such that limr→1−(1 − r2)‖(A ∗ C(r))′‖B(�2) = 0,where C(r) is the Toeplitz matrix associated with the Cauchy kernel.

Clearly B0(D, �2) is a closed subspace of B(D, �2) if the former isendowed with the norm of B(D, �2).

We denote by E the Toeplitz matrix having all its entries equal to1. First we state the following Lemma of independent interest:

Lemma 2.2. Let A ∈ B(D, �2) and Ar(s) = A(rs) = A(r)∗P (s) for all0 ≤ r < 1 and 0 ≤ s < 1, where P (s) is the Toeplitz matrix associatedto the Poisson kernel, that is

P (s) =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

1 s0 s2 s3 · · ·s 1 s r20

. . .

s2 s 1 s. . .

s3 s2 s 1. . .

.... . . . . . . . . . . .

⎞⎟⎟⎟⎟⎟⎟⎟⎠.

Then it follows that Ar is a matrix belonging to B0(D, �2) for all 0 ≤r < 1.

Proof. First we note that

lims→1

(1− s2)‖A′r(s)‖B(�2) = lim

s→1(1− s2)r‖A′(rs)‖B(�2)


and, by using well known facts about multipliers (see e.g. [5]) andelementary calculations, we find that

‖A′(rs)‖B(�2) = ‖∞∑k=0

kAk(rs)k−1‖B(�2) = ‖

∞∑k=0

kAkrk−1sk−1‖B(�2)

= ‖A′(r) ∗∞∑k=0

sk−1Ek‖B(�2) ≤ ‖A′(r)‖B(�2) · ‖∞∑k=0

sk−1Ek‖M(�2)

= ‖A′(r)‖B(�2) · ‖∞∑k=0

sk−1eikθ‖M(T) = ‖A′(r)‖B(�2) ·1

s‖ 1

1− seiθ‖L1(T)

= ‖A′(r)‖B(�2) ·1

s

∫ π

−π

1

|1− seiθ|dθ ≤

≤ ‖A‖B(D,�2) ·1

s(1− r2)

∫ π

−π

1

|1− seiθ|dθ.

Thus, by making some straightforward calculations, we find that

lims→1

(1− s2)‖A′r(s)‖B(�2) ≤

≤ ‖A‖B(D,�2) ·r

1− r2lims→1

(1− s2)

s·∫ π

−π

1

|1− seiθ|dθ =

= ‖A‖B(D,�2) ·r

1− r2lims→1

(1− s2) ln1

1− s= 0.

The proof is complete. �Our first result in this Section reads:

Theorem 2.3. Let A ∈ B(D, �2). Then A ∈ B0(D, �2) if and only iflimr→1− ‖Ar − A‖B(D,�2) = 0.

Proof. By Lemma 2.2 it follows that Ar ∈ B0(D, �2) and we use thefact that B0(D, �2) is a closed subspace of B(D, �2) in order to concludethat the condition is sufficient.

Conversely, let A ∈ B0(D, �2). Then, for every ε > 0 there exists0 < δ < 1 such that (1− s2)‖A′(s)‖B(�2) < ε, for every δ2 < s < 1. Weremark that

‖Ar − A‖B(D,�2) = sup0≤s<1

(1− s2)‖A′r(s)− A′(s)‖B(�2) ≤

≤ supδ<s<1

(1− s2)‖A′r(s)−A′(s)‖B(�2) + sup

0≤s≤δ

(1− s2)‖A′r(s)−A′(s)‖B(�2).

For δ < r < 1 the first term is smaller than

(1− r2s2)‖A′(rs)‖B(�2) + (1− s2)‖A′(s)‖B(�2) < 2ε.


The second term converges to 0 whenever r → 1−. Indeed, for0 ≤ s ≤ δ < δ′ < 1, letting u = s

δ′ and making some straightforwardcalculations, we get that

‖A′r(s)− A′(s)‖B(�2) = ‖rA′(rs)− A′(s)‖B(�2)

= ‖rA′(s) ∗∞∑k=0

rk−1Ek − A′(s)‖B(�2) = ‖A′(s) ∗∞∑k=0

(rk − 1)Ek‖B(�2)

= ‖A′(u) ∗∞∑k=0

(rk − 1)(δ′)k−1Ek‖B(�2) ≤

≤ ‖A′(u)‖B(�2)‖∞∑k=0

(rk − 1)(δ′)k−1Ek‖M(�2) ≤

≤ ‖A′(u)‖B(�2) · (1− r)

∫ π

−π

1

|1− rδ′eiθ| · |1− δ′eiθ|dθ

2π≤

≤ ‖A′(u)‖B(�2)

1− r

(1− rδ′)(1− δ′)= ‖A′(

s

δ′)‖B(�2) ·

(1− r)

(1− rδ′)(1− δ′).

Then

sups≤δ

(1− s2)‖A′r(s)− A′(s)‖B(�2) ≤

≤ sups≤δ≤δ′

[(1−( sδ′

)2)‖A′( sδ′

)‖B(�2) ·

1− s2

1− s2

δ′2

]· (1− r)

(1− δ′)(1− rδ′)≤

≤ ‖A‖B(D,�2) ·1− δ2

1− δ2

δ′2· (1− r)

(1− δ′)(1− rδ′).

Consequently

limr→1−

sups≤δ

(1− s2)‖A′r(s)− A′(s)‖B(�2) = 0,

i.e.

limr→1−

‖Ar − A‖B(D,�2) = 0.


Corollary 2.4. B0(D, �2) is the closure of all matrices of finite bandtype in the Bloch norm. In particular, this implies that B0(D, �2) is aseparable space.


Proof. Let A ∈ B0(D, �2) and An =∑n

k=0 Ak. Then, by Theorem 2.3,it yields that for every ε > 0 there is r0 < 1 such that ‖Ar0−A‖B(D,�2) <ε/2.We note that r → A(r) for r ∈ [0, 1) is a continuous B(�2)-valued

function on [0, s] for s < 1.Indeed, let 0 < sn ≤ s0 < 1 and sn → s0. Then,

‖A(sn)− A(s0)‖B(�2) =∥∥∥[C (sn

s′

)− C(s0s′

)]∗ A(s′)

∥∥∥B(�2)

≤

≤ ‖A(s′)‖B(�2)

∥∥∥C (sns′

)− C(s0s′

)∥∥∥M(�2)

,

where sn → s0 < s′ < 1. Hence, by putting δ = s0s< 1 and reasoning

as in the proof of the previous theorem, we get that

‖A(sn)− A(s0)‖B(�2) ≤

≤ ‖A(s′)‖B(�2) ·∥∥∥∥∥

∞∑k=0

δk

((sns0

)k

− 1

)eikθ

∥∥∥∥∥M(T)

→ 0.

Now, for a fixed r0 < 1 we have that

sup0≤s≤1

‖Ar0(s)‖B(�2) = M(r0) = ‖A(r0)‖B(�2) < ∞

for all analytic matrices.Thus, for r0 < r′ < 1 and by using the notation

Cn(r0r′

)=

n∑k=0

Ck

(r0r′

),

we find that

‖Ar0(·)− (Ar0)n(·)‖L∞(D,�2) = ess sup

s<1‖(A− An)(r0s)‖B(�2) =

= ‖(A− An)(r0)‖B(�2) = ‖[C(r0r′

)− Cn

(r0r′

)]∗ A(r′)‖B(�2) ≤

≤∥∥∥C (r0

r′

)− Cn

(r0r′

)∥∥∥M(�2)

· ‖A(r′)‖B(�2) =

=

∥∥∥∥∥∞∑

k=n+1

(r0r′

)keikθ

∥∥∥∥∥M(T)

· ‖A(r′)‖B(�2) → 0.

Consequently,

‖Ar0(·)− (Ar0)n(·)‖L∞(D,�2) → 0

and

‖A(·)− (Ar0)n(·)‖B(D,�2) ≤ ε,


whenever r0 < 1 is fixed as before and n is sufficiently large. Sinceε > 0 is arbitrary and (Ar0)

n =∑n

k=0(Ar0)k is a matrix of finite bandtype it follows that B0(D, �2) is the closure of all matrices of finite bandtype in the Bloch norm. The proof is complete. �

The next theorem express a natural relation between the Bergmanprojection and the Bloch spaces. More exactly, our first main result inthis Section is the following equivalence theorem:

Theorem 2.5. Let A ∈ B(D, �2). Then the following assertions areequivalent:

1) A ∈ B0(D, �2).2) There is a continuous B(�2)- valued function r → B(r) defined

on [0, 1] such that P (B(·))(r) = A(r).3) There is r → B(r) which is a continuous B(�2)- valued function

such that limr→1 B(r) = 0 and satisfying P (B(·))(r) = A(r).

Proof. To prove that 1) implies 3), let us take A ∈ B0(D, �2). Wedefine A1(r) :=

∑∞k=2 Akr

k, with r < 1. Thus A′1(r) =

∑∞k=2 kAkr

k−1

and A(r) = A0 + A1r + A1(r). We take now

B2(r) = (1− r2)T ∗ P (r) ∗ A′1(r),

where T = (tij)i,j is a Schur multiplier which will be defined later on.Thus, by the definition of the Bergman projection P , we get that

[PB2(·)](r)(i, j) =

=

{tijaij(j − i+ 1)(j − i)rj−i 1

[ 32(j−i+1)]

3(j−i)+12

for j − i ≥ 2

0 otherwise .

Consequently, by taking tij =

{ 3[3(j−i)+1]4(j−i)

for all j �= i, i, j ≥ 194

if j = i, i ≥ 1,

it follows that T is a Schur multiplier and P [B2(·)] = A1(·). Let nowB(r) = 2(1−r2)A0+3(1−r2)rA1+B2(r). It is clear that [PB(·)](r) =A(r). But, since A ∈ B0(D, �2), it follows that B2(r) and, consequently,B(r) is a continuous B(�2)- valued function such that limr→1 B(r) = 0.Thus 3) holds.

It is obvious that 3) implies 2).It remains to prove that 2) implies 1). Let 2) hold and choose B(r) ∈

B(�2) be such that

[PB(·)](r) = A(r) for r ∈ [0, 1].

Assume that r → B(r) is a continuous B(�2)- valued function on[0, 1] and let M = sup0≤r≤1 ‖B(r)‖B(�2) < ∞.


Let 0 ≤ r0 < 1 be fixed and let us consider Ar0(r) given by theformula

Ar0(r)(i, j) =

{(j − i+ 1)(rr0)

j−i(2∫ 1

0bij(s)s

j−i+1ds) if j − i ≥ 0

0 otherwise.

Consequently, according to Theorem P, we find that

Ar0(·) = P [P (r0) ∗B(·)] = P [Br0(·)] ∈ B(D, �2),

where P (r0) is the Toeplitz matrix associated with the Poisson kernel.Let C(D, �2) denote the space of all continuous B(D, �2)- valued

functions defined on [0, 1]. We will prove that the function s →P (P (r0) ∗ B(s)) belongs to C(D, �2) if B is a continuous B(�2)-valuedfunction. Moreover, we will show that

(1) limr→1

sups∈[0,1]

‖Ar(s)− A(s)‖B(D,�2) = 0.

This, in its turn, implies that limr→1 Ar = A in B(D, �2). Thus, byTheorem 2.3, it follows that A ∈ B0(D, �2).

Let s, s0 ∈ [0, 1]. Then

‖P (r0)∗[B(s)−B(s0)]‖B(�2) ≤ ‖∑k∈Z

r|k|0 eikθ‖M(T)·‖B(s)−B(s0)‖B(�2) → 0

for s → s0 and B(s) is a continuous function on [0, 1]. Here we haveused G. Bennett’s Theorem 1.1 and the fact that

‖∑k∈Z

r|k|0 eikθ‖M(T) = ‖ 1− r20

|1− r0eiθ|2‖M(T) ≤ 1.

Thus, the function s → P [P (r0) ∗B(s)] belongs to C(D, �2).Hence, it only remains to prove that (1) holds.

‖P (r) ∗B(s)− B(s)‖B(�2) ≤ ‖∑k∈Z

(r|k| − 1)eikθ‖M(T) · ‖B(s)‖B(�2)

≤ M · ||∑k∈Z

(r|k| − 1)eikθ||M(T) for all s ∈ [0, 1].

Denoting by μr(θ) the measure∑

k∈Z(r|k|−1)eikθ, then, for a trigono-

metric polynomial φ(θ) =∑m

n=−m aneinθ, we have that

μr(φ) =m∑

n=−m

(r|n| − 1)an and |μr(φ)| ≤ |φ(r)− φ(1)| ≤ 2‖φ‖,

where φ(r) is the value of the Poisson extension of φ in the point r.Consequently μr is a measure with a norm smaller than 2. But

limr→1 μr(φ) = 0 for all trigonometric polynomials φ. Thus w∗ −


limr→1 μr = 0 in M(T) and then is is clear that limr→1 ‖μr‖ = 0 andby Theorem P the relation (1) is proved. Thus also the implication 2)⇒ 1) is proved and the proof is complete. �By using similar ideas as in [13] we can obtain the following result:

Theorem 2.6. P2 is a continuous operator (precisely a continuousprojection ) from L1(D, �2) onto L1

a(D, �2), where(2)

[P2A(·)](r)(i, j) ={

2Γ(j−i+4)(j−i)!Γ(4)

rj−i∫ 1

0(1− s2)2aij(s)s

j−i(2sds) if j ≥ i,

0 otherwise .

Proof. The topological dual of L1(D, �2) is L∞(D, �2) with respect tothe duality pair:

< A(·), B(·) >=

∫ 1

0

tr(A(s)[B(s)]∗)2sds,

where A(·) ∈ L∞(D, �2), B(·) ∈ L1(D, �2) (see e.g. Theorem 8.18.2in [10]). Using a duality argument it is sufficient to prove that P ∗

2 :L∞(D, �2) → L∞(D, �2) is bounded.

Now we are looking for the adjoint P ∗2 of P2. We note that

< P ∗2A(·), B(·) >=

∫ 1

0

∞∑i=1

∞∑j=1

(P ∗2A(·))(r)(i, j)bij(r)(2rdr)

=∞∑i=1

∞∑j=1

∫ 1

0

(P ∗2A(·))(r)(i, j)bij(r)(2rdr).

On the other hand it yields that

< P ∗2A(·), B(·) >=< A(·), P2B(·) >=

∫ 1

0

trA(r)(P2B)∗(r)(2rdr)

=∞∑i=1

∞∑j=1

∫ 1

0

A(r)(i, j)(P2B)(r)(i, j)(2rdr)

=∞∑i=1

∞∑j=i

Γ(j − i+ 4)

(j − i)!Γ(3)

(∫ 1

0

[A(s)](i, j)sj−i(2sds)

)×

(∫ 1

0

bij(s)sj−i(1− s2)2(2sds)

).

Now let us consider {Ik}, a sequence of intervals such that

limk→∞

μ(Ik) = 0, dμ = 2sds and r ∈ Ik.


For every k we take B(s)(i, j) = χIk(s)/(μ(Ik)) and B(s)(l, k) = 0,(l, k) �= (i, j) for every (i, j) ∈ N× N.By Lebesgue’s differentiation theorem (see e.g. [15]) we have that

(P ∗2A(·))(r)(i, j) =

=

{Γ(j−i+4)(j−i)!Γ(3)

rj−i(1− r2)2∫ 10A(s)(i, j)sj−i(2sds) if j ≥ i,

0 if j < i,

a.e. for all r ∈ [0, 1).We will now prove that P ∗

2 : L∞(D, �2) → L∞(D, �2) is a boundedoperator. In order to prove that we first note that

‖A(r)‖2L∞(D,�2) = ess sup0≤r<1

‖A(r)‖2B(�2)

= ess sup0≤r<1

sup∑∞j=1 |hj |2≤1

∞∑i=1

|∞∑j=1

aij(r)hj|2.

Consequently,

‖P ∗2A(·)‖2L∞(D,�2) =

= ess sup0≤r<1

sup∑∞j=1 |hj |2≤1

∞∑i=1

|∞∑j=1

hjrj−iΓ(j − i+ 4)

(j − i)!Γ(3)(1− r2)2·

∫ 1

0

aij(s)sj−i(2sds)|2 = ess sup

0≤r<1sup

‖h‖�2≤1

(1− r2)4∞∑i=1

|∫ 1

0

( ∞∑j=i

aij(s)·

((rs)j−iΓ(j − i+ 4)

(j − i)!Γ(3))hj

)(2sds)|2 ≤ ess sup

0≤r<1(1− r2)4·

sup‖h‖�2≤1

[

∫ 1

0

(∞∑i=1

|∞∑j=i

aij(s)(rs)j−i (j − i+ 3)!

(j − i)!hj|2)

12 (sds)]2.

Since the Toeplitz matrix C(rs) = ((cij)(rs))∞i,j=1, where

cij(rs) = cj−i(rs) =

=

{(rs)j−i(j − i+ 3)(j − i+ 2)(j − i+ 1) if j ≥ i

0 otherwise ,

is a Schur multiplier with

‖C(rs)‖L1(T) = ‖ 6

(1− rseiθ)4‖L1(T) = 6

∞∑n=0

(n+ 1)2(rs)2n,


we get that

sup∑∞j=1 |hj |2≤1

( ∞∑i=1

|∞∑j=i

ais(s)(rs)j−i(j − i+ 3)(j − i+ 2)(j − i+ 1)hj|2

) 12

= ‖A(s) ∗ C(rs)‖B(�2) ≤ 6‖A(s)‖B(�2) ·∞∑n=0

(n+ 1)2(rs)2n.

Consequently,‖P ∗

2A(·)‖2L∞(D,�2) ≤

ess supr<1

(1− r2)4

[∫ 1

0

3‖A(s)‖B(�2) ·∞∑n=0

(n+ 1)2r2ns2n(2sds)

]2≤

≤ 9 · ess supr<1

(1− r2)4‖A(·)‖2L∞(D,�2)

( ∞∑n=0

(n+ 1)r2n

)2

∼

∼ ess supr<1

(1− r2)442

(1− r2)4‖A(·)‖2L∞(D,�2) ∼ ‖A(·)‖2L∞(D,�2),

which shows in turn that P ∗2 : L∞(D, �2) → L∞(D, �2) is bounded. The

proof is complete. �Now let us denote by C0(D, �2) the space of all continuous B(�2)-

valued functions B(r) on [0, 1] such that limr→1 B(r) = 0 in the normof B(�2). Next we state the following Lemma of independent interest:

Lemma 2.7. Let V = (P2)∗, i.e, (P2A(·))∗(r)(i, j) =

=

{(j−i+3)(j−i+2)(j−i+1)

2rj−i(1− r2)2

∫ 10aij(s)s

j−i(2sds) if j − i ≥ 0

0 otherwise.

Then V is an isomorphic embedding of B0(D, �2) in C0(D, �2).

Proof. According to Theorem 2.5, for B ∈ B0(D, �2) we can find someA(·) ∈ C0(D, �2) such that [PA(·)](r) = B(r). Clearly, we have that

P ∗2B = P ∗

2PA = T 1 ∗ (P ∗2A1),

where A1(r) = T ∗ A(r), for T = (tj−i)i,j with

tj−i =

{2(j−i+1)j−i+2

for j − i �= −2,

0 otherwise .

T is a Schur multiplier and the same is true for T 1 = (t1j−i)i,j, where

t1j−i =

{j−i+2

2(j−i+1)for j − i �= 0,

0 otherwise.


Thus, ‖A1(r)‖B(�2) ∼ ‖A(r)‖B(�2) for all r ∈ [0, 1].Hence, we obtain that

‖P ∗2A1(r)‖2B(�2) =

= sup‖h‖�2≤1

∞∑i=1

|∞∑j=i

hjrj−iΓ(j − i+ 4)

(j − i)!Γ(3)(1− r2)2

∫ 1

0

a1ij(s)sj−i(2sds)|2 ≤

≤ sup‖h‖�2≤1

(1−r2)4[

∫ 1

0

(∞∑j=1

|∞∑j=i

a1ij(s)(rs)j−iΓ(j − i+ 4)

2(j − i)!hj|2)

12 (2sds)]2 =

(1−r2)4 sup‖h‖�2≤1

[

∫ 1

0

(∞∑i=1

|∞∑j=i

aij(s)(rs)j−i(j−i+1)2(j−i+3)hj|2)

12 (2sds)]2.

The Toeplitz matrix C(r, s) = (cj−i(r, s))i,j, where

cj−i(r, s) =

{(rs)j−i(j − i+ 3)(j − i+ 1)2 if j ≥ i,

0 otherwise,

is a Schur multiplier, since∑∞

k=0(rs)k(k + 3)(k + 1)2eikθ ∼ 1

(1−rseiθ)4

and∫ π

−π1

(1−rseiθ)4dθ2π

∼∑∞n=0 n

2(rs)2n.

Therefore, we have that

‖P ∗2A1(r)‖B(�2) ≤ C(1− r2)2

∫ 1

0

‖A(s)‖B(�2) · (∞∑n=0

n2(rs)2n)(2sds).

Since lims→1 ‖A(s)‖B(�2) = 0, for all ε > 0, there exists δ > 0 suchthat ‖A(s)‖ < ε for all s ≥ δ and, consequently,

‖P ∗2A1(r)‖B(�2) ≤ C(ε+ ‖A(·)‖C(D,�2) ·

(1− r2)2δ2

(1− r2δ2)2).

It follows that limr→1 ‖P ∗2A1(r)‖B(�2) = 0 and since T 1 is a Schur mul-

tiplier it follows that P ∗2B ∈ C0(D, �2) and ‖P ∗

2B‖B(�2 ≤ C‖A(·)‖C(D,�2).Moreover, in view the proof of Theorem 2.5, we can find an A(·) ∈

C0(D, �2) such that

‖A(·)‖C0(D,�2) ≤ C(‖B0‖B(�2) + ‖B(·)‖B(D,�2)),

where C > 0 is an absolute constant. By now also using the argumentsin the proof of Theorem (2.6) it follows that P ∗

2 : B0(D, �2) → C0(D, �2)is bounded.

On the other hand, if A ∈ B0(D, �2), then since A(r) is an analyticmatrix it is obvious that A(r) = [PA(·)](r) = (P [P ∗

2A(·)])(r) for allr ∈ [0, 1). Thus, by using Theorem P, we conclude that there existsa constant C > 0 such that ‖A(·)‖B(D,�2) ≤ C‖P ∗

2A(·)‖C(D,�2), which


implies that P ∗2 : B0(D, �2) → C0(D, �2) is an isomorphic embedding.


Remark 2.8. From now on we identify B0(D, �2) with the space B0(D, �2)of all analytic matrices A ∗ C(r) for A ∈ B0(D, �2).

We denote now by B0,c(D, �2) the closed Banach subspace of B0(D, �2)consisting of all upper triangular matrices whose diagonals are com-pact operators. We are now ready to prove that the little Bloch spaceB0,c(D, �2) in fact is the predual of the Bergman-Schatten space. Moreexactly, our last main result in this Section is the following dualityresult:

Theorem 2.9. It yields that B0,c(D, �2)∗ = L1a(D, �2) with respect to

the usual duality, whenever B0(D, �2) is equipped with the norm inducedby B(D, �2).

Proof. Let A ∈ L1a(D, �2). Then B →

∫ 10tr[B(s)A∗(s)](2sds) defines a

linear and bounded functional on B0,c(D, �2) (see e.g. Theorem 22 in[13]). Conversely, let us assume that F is a bounded linear functionalon B0,c(D, �2). Then we shall prove that there is a matrix C fromL1a(D, �2) such that

(3) F (B) =

∫ 1

0

tr[B(r)C∗(r)](2rdr),

for B from a dense subset of B0(D, �2).By Lemma 2.7 it follows that P ∗

2 : B0(D, �2) → C0(D, �2) is an iso-morphic embedding. Thus X = P ∗

2 (B0,c(D, �2)) is a closed subspacein C0(D,C∞) and F ◦ (P ∗

2 )−1 : X → C is a bounded linear functional

on X, where C0(D,C∞) is the subset in C0(D, �2) whose elements areC∞-valued functions. By the Hahn-Banach theorem F ◦ (P ∗

2 )−1 can be

extended to a bounded lineal functional on C0(D,C∞).Let Φ : C0(D,C∞) → C denote this functional. It follows that

C0(D,C∞) = C0[0, 1]⊗εC∞ and, thus, Φ is a bilinear integral map,that is there is a bounded Borel measure μ on [0, 1]× UC1 , where UC1

is the unit ball of the space C1 with the topology σ(C1, C∞), such that

Φ(f ⊗ A) =

∫[0,1]×UC1

f(r)tr(AB∗)dμ(r, B)

for every f ∈ C0[0, 1] and A ∈ C∞.


Thus, for the matrix∑n

k=0 Ak ∈ B0,c(D, �2), identified with the ana-lytic matrix

∑n

k=0 Akrk, we have that

F (n∑

k=0

Ak) = F (n∑

k=0

rkAk) = [F ◦ (P ∗2 )

−1][P ∗2 (

n∑k=0

rkAk)]

= Φ(n∑

k=0

(k + 3)(k + 2)

2rk(1− r2)2Ak)

=

∫[0,1]×UC1

n∑k=0

tr[((k + 3)(k + 2)

2rk(1− r2)2Ak)B

∗]dμ(r, B)

def= < μ(r, B), tr(

n∑k=0

(k + 3)(k + 2)

2rkAk)B

∗(1− r2)2 > .

On the other hand, we wish to have that

F (n∑

k=0

Ak) =

∫ 1

0

tr(n∑

k=0

skAk)(C(s)∗)(2sds)

=

∫ 1

0

tr(n∑

k=0

s2kAkC∗k)(2sds) =

n∑k=0

trAk(C∗

k

k + 1).

Therefore, letting A = ei,i+k, denote the matrix having 1 as the singlenonzero entry on the ith-row and the (i + k)th-column, for i ≥ 1 andj ≥ 0, we have that

Ck =< μ(r, B),(k + 1)(k + 2)(k + 3)

2rk(1− r2)2Bk >, k = 0, 1, 2, . . . .

Then, it yields that ∫ 1

0

‖C(s)‖C1(2sds) =

=

∫ 1

0

‖∫[0,1]×UC1

n∑k=0

(k + 1)(k + 2)(k + 3)

2(sr)k(1− r2)2Bk·

dμ(r, B)‖C1(2sds) ≤∫[0,1]×UC1

[

∫ 1

0

‖n∑

k=0

(k + 1)(k + 2)(k + 3)

2(rs)k·

(1− r2)2Bk‖C1(2sds)]d|μ|(r, B) ≤∫[0,1]×UC1

[

∫ 1

0

‖n∑

k=0

(k + 1)(k + 2)

2·

(k + 3)(rs)k(1− r2)2eik(·)‖L1(T)‖B‖C1(2sds)]d|μ|(r, B) ≤

≤∫[0,1]×UC1

[

∫ 1

0

∫ 2π

0

(1− r2)2

|1− rseiθ|4dθ

2π(2sds)]d|μ|(r, B)


∼∫[0,1]×UC1

∫ 1

0

(1− r2)2∞∑k=0

(n+ 1)2(sr)2n(2sds)d|μ|(r, B)

=

∫[0,1]×UC1

(1− r2)2∞∑n=0

(n+ 1)r2nd|μ|(r, B) = ‖μ‖ < ∞.

Consequently, C ∈ L1a(D, �2) and we get the relation (3) by using the

fact that the set of all matrices∑n

k=0 Ak is dense in B0,c(D, �2). Theproof is complete. �

3. A new characterization of Bergman-Schatten spaces

In this Section we give a characterization of the space Lpa(D, �2) in

terms of Taylor coefficients which is similar to those obtained by M.Mateljevic and M. Pavlovic in [12]. For the proof of our main result(Theorem 3.4) we need the following three technical Lemmas.

Lemma 3.1. Let A =n∑

k=m

Ak, 0 ≤ m ≤ n. Then

‖A‖Cprn ≤ ‖A(r)‖Cp ≤ ‖A‖Cpr

m, (0 < r < 1)

Proof. Let us take B[r] =n∑

k=m

A−krn−k.

Then

‖A(r)‖Cp = ‖A(r)∗‖Cp = ‖n∑

k=m

A−krk‖Cp =

∥∥∥∥B [1r]∥∥∥∥

Cp

rn.

Since 1r> 1 and the function seit → ‖B(seit)‖Cp , 0 < s < 1, t ∈

[0, 2π] is a subharmonic function, we have that∥∥∥∥B [1r]∥∥∥∥

Cp

(∫ 2π

0

∥∥∥∥B [1r](eit)

∥∥∥∥pCp

dt

2π

) 1p

≥(∫ 2π

0

‖B[1]eit‖pCp

dt

2π

) 1p

= ‖B[1]‖Cp = ‖A∗‖Cp = ‖A‖Cp .

This proves the left hand side inequality. The proof of the right handside inequality is similar so we omit the details. �Lemma 3.2. Let A be an upper triangular matrix,

σk(A) =n∑

k=0

(1− k

n+ 1

)Ak,

the Cesaro mean of the order k and

‖σn(A)‖p = sup0<r<1

‖σn(A)(r)‖Cp , (n = 0, 1, 2, ...).


Then

‖σk(A)‖p rk ≤ ‖A(r)‖Cp ≤ (1− r)2∞∑n=0

‖σn(A)‖p(n+ 1)rn.

Proof. First we observe that

‖A(r)‖Cp ≥ ‖σn(A)(r)‖Cp , for every r ∈ (0, 1), n ∈ N.

Since ‖n∑

k=−n

(1− |k|

n+1

)eikt‖L1(T)‖Fn(t)‖L1(T) ≤ 1,

Fn ∈ M(�2) = M(C1) ⊂ M(Cp) ⊂ M(C2)

for 1 ≤ p ≤ 2 and M(Cp) = M(Cp′),1p+ 1

p′ = 1, for 2 ≤ p < ∞ it

follows that ‖σn(A)(r)‖Cp = ‖A(r) ∗ Fn‖Cp ≤ ‖A(r)‖Cp . By using thisinequality and Lemma 3.1 we find that

‖A(r)‖Cp ≥ ‖σn(A)(r)‖Cp ≥ rn‖σn(A)‖Cp ≥ rn‖σn(A)‖p,and the left hand side of the inequality is proved. The proof of theright hand of the inequality follows by using the formula

A(r) = (1− r)2∞∑n=0

σn(A)(n+ 1)rn

and Minkovski’s inequality. The proof is complete. �Lemma 3.3. Let A be an upper triangular matrix, 0 ≤ k < n andp ≥ 1. Then we have that

(4) (n− k + 1)‖σk(A)‖p ≤ (n+ 1)‖σn(A)‖p.Proof. First we note that ‖σn(A)‖p = sup0<r<1 ‖σn(A)(r)‖Cp . Since

‖σk(B)‖Cp ≤ ‖B‖Cp

it follows that

‖σn(A)‖p ≥ ‖σkσn(A)‖p = ‖σk(A)−r

n+ 1σ′k(A)‖p

≥ ‖σk(A)‖p −1

n+ 1‖σ′

k(A)‖p ≥

[ by Bernstein’s inequality ] ≥ ‖σk(A)‖p −k

n+ 1‖ σk(A) ‖p,

and the inequality (4) is proved. �

Our main result in this Section reads:


Theorem 3.4. Let A be an analytic matrix. Then A ∈ Lpa(D, �2) if

and only if∞∑n=0

1

(n+ 1)2‖σn(A)‖pp < ∞.

Proof. First we prove that

A ∈ Lpa(D, �2)

if and only if

(5)

(∫ 1

0

‖A(r)‖pCpdr

) 1p

< ∞.

If(∫ 1

0‖A(r)‖pCp

dr) 1

p< ∞, then it is clear that A ∈ Lp

a(D, �2). Con-

versely if A ∈ Lpa(D, �2), then ‖A(r)‖Lp(D,�2) < ∞.

Let Eθ be the Toeplitz matrix corresponding to the Dirac measureδθ, i.e.,

Eθ = (ekj)k,j≥1, ekj = ei(j−k)t.

Then Eθ ∈ M(�2) = M(C1) ⊂ M(Cp), 1 ≤ p < ∞.Since reiθ → ‖A(r) ∗ Eθ‖Cp is subharmonic on D it follows that∫ 2π

0‖A(r) ∗ Eθ‖pCp

dθ ≤∫ 2π

0‖A(√r) ∗ Eθ‖pCp

dθ. Therefore∫ 1

0

∫ 2π

0

‖A(r) ∗ Eθ‖pCpdθdr ≤

∫ 1

0

∫ 2π

0

‖A(√r ∗ Eθ)‖pCpdθdr

and ∫ 1

0

‖A(r)‖pCpdr ≤

∫ 1

0

‖A(√r)‖pCpdr = 2

∫ 1

0

‖A(s)‖pCpsds < ∞,

and (5) is proved.Now let A ∈ Lp

a(D, �2). Then, by the first inequality in Lemma 3.2,we have that

‖A(r)‖pCp= (1− r)

∞∑n=0

‖A(r)‖pCprn ≥ (1− r)

∞∑n=0

‖σn(A)‖pprn(p+1).

Now integration yields that

∞ >

∫ 1

0

‖A(r)‖pCpdr ≥

∫ 1

0

(1− r)∞∑n=0

‖σn(A)‖pprn(p+1)dr

≥ C−1

∞∑n=0

1

(n+ 1)2‖σn(A)‖pp.


Conversely, suppose that∞∑n=0

‖σn(A)‖pp(n+ 1)2

< ∞.

Let xn =∑∞

k=0(k+1)(n−k+1)‖σk(A)‖p. Then, by summing by partsas in (4.4) in [12], we find that

(6)∞∑n=0

‖σn(A)‖p(n+ 1)rn = (1− r)2∞∑n=0

xnrn.

On the other hand, by using Lemma 3.3, we see that

xn ≤ C(n+ 1)3‖σn(A)‖pand, therefore,

∞∑n=0

1

(n+ 1)3p+2xpn < ∞.

We use now Lemma 4.8 in [12] with q = p, φ(r) = r1p , r ∈ (0, 1] and

obtain that ∫ 1

0

[(1− r)4

∞∑n=0

xnrn

]pdr < ∞.

Moreover by using also (6) we arrive at∫ 1

0

(1− r)4p

(1

(1− r)2

∞∑n=0

‖σn(A)‖p(n+ 1)rn

)p

dr < ∞.

It follows that∫ 1

0

(1− r)2p

( ∞∑n=0


)p

dr < ∞.

From the right hand side inequality in Lemma 3.2, we finally obtainthat∫ 1

0

‖A(r)‖ppdr ≤∫ 1

0

(1− r)2p

( ∞∑n=0


)p

dr < ∞.

The proof is complete. �Acknowledgements: The first named author thanks Lulea Uni-

versity of Technology and Department of Mathematics for their finan-cial support, hospitality and for the warm and friendly atmosphere dur-ing his visit. The first and the last authors were partially supported bythe CNCSIS grant ID-PCE 1905/2008.


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[10] R. E. Edwards, Functional analysis; Theory and Applications, Holt, Rinehartand Winston, New York, 1965.

[11] A. N. Marcoci and L. G. Marcoci, A new class of linear operators on �2 andSchur multipliers for them, J. Funct. Spaces Appl. 5(2007), 151–164.

[12] M. Mateljevic and M. Pavlovic, Lp-behaviour of the integral means of analyticfunctions, Studia Math. 77(1984), 219–237.

[13] N. Popa, Matriceal Bloch and Bergman-Schatten spaces, Rev. Roumaine Math.Pures Appl. 52(2007), 459–478.


[15] W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987.[16] K. Zhu, Operator theory in Banach function spaces, Marcel Dekker, New York,

1990.




of Civil Engineering Bucharest, RO-020396 Bucharest, ROMANIA




of Civil Engineering Bucharest, RO-020396 Bucharest, ROMANIA E-mail: [email protected]


Department of Mathematics, University of Bucharest and Insti-

tute of Mathematics of Romanian Academy, P.O. BOX 1-764 RO-014700

Bucharest, ROMANIA E-mail: [email protected]

Paper 4

BESOV-SCHATTEN SPACES AND HANKELOPERATORS

A.-N. MARCOCI, L.-G. MARCOCI, L.-E. PERSSON AND N. POPA

Abstract. Let Bp(�2) denote the Besov-Schatten space of all up-

per triangular matrices A such that

‖A‖Bp(�2) =

[∫ 1

0


] 1p

< ∞.

In this paper we present and prove a duality result between Bp(�2)

and Bq(�2), 1 < p < ∞, 1

p +1q = 1 and also a similar result for the

limit cases p = 1 and q = 1. We also give a sufficient condition fora Hankel operator to be nuclear.

1. Introduction

Analytic Besov spaces first found a direct application in operatortheory in Peller’s paper [9]. A comprehensive account of the theory ofBesov spaces is given in Peetre’s book [7]. In what follows we considerthe Besov-Schatten spaces in the framework of matrices e.g. infinitematrix valued functions. The extension to the matriceal frameworkis based on the fact that there is a natural correspondence betweenToeplitz matrices and formal series associated to 2π-periodic functions(see e.g. [1], [3], [5], [14]). In particular the Besov-Schatten spaceB1(�

2) can be used to prove that the associated Hankel operator isnuclear. We use the powerfull device Schur multipliers and its characte-rizations in the case of Toeplitz matrices to prove some of the mainresults.

The Schur product (or Hadamard product) of matrices A = (ajk)j,k≥0

and B = (bjk)j,k≥0 is defined as the matrix A ∗B whose entries are theproducts of the entries of A and B:

A ∗B = (ajkbjk)j,k≥0.

2000 Mathematics Subject Classification. 47B10, 47L20, 47B35, 30H25.Key words and phrases. Besov-Schatten spaces, Schur multipliers, Hankel

operator.1


If X and Y are two Banach spaces of matrices we define Schur multi-pliers from X to Y as the space

M(X, Y ) = {M : M ∗ A ∈ Y for every A ∈ X},equipped with the natural norm

‖M‖ := sup‖A‖X≤1

‖M ∗ A‖Y .

In the case X = Y = B(�2), where B(�2) is the space of all linear andbounded operators on �2, the space M(B(�2), B(�2)) will be denotedM(�2) and a matrix A ∈ M(�2) will be called Schur multiplier. Wemention here the following important result due to G. Bennett [4],which will be often used in this paper:

Theorem 1.1. The Toeplitz matrix M = (cj−k)j,k, where (cn)n∈Z is asequence of complex numbers, is a Schur multiplier if and only if thereexists a bounded and complex Borel measure μ on (the circle group) Twith

μ(n) = cn, for n = 0,±1,±2, ....

Moreover, we then have that

‖M‖ = ‖μ‖.We will denote by Cp, 0 < p < ∞, the Schatten class of operators

(see e.g. [15]). Let us sumarize briefly some well-known properties ofthe classes M(Cp), which will be very often used in what follows. If1 < p < ∞, then M(Cp) = M(Cp′), where

1p+ 1

p′ = 1 and M(�2) =

M(C1). Next, interpolating between the classes Cp, we can easily seethat M(Cp1) ⊂ M(Cp2) if 0 < p1 ≤ p2 ≤ 2 (see e.g. [2]). We willdenote by Ak, the k

th-diagonal matrix associated to A (see [3]). For aninfinite matrix A = (aij) and an integer k we denote by Ak the matrixwhose entries a′ij are given by

a′ij =

{aij if j − i = k

0 otherwise.

In what follows we will recall some definitions from [11] (see also [6]),which we will use in this paper. We consider on the interval [0, 1)the Lebesgue measurable infinite matrix-valued functions A(r). Thesefunctions may be regarded as infinite matrix-valued functions definedon the unit disc D using the correspondence

A(r) → fA(r, t) =∞∑

k=−∞Ak(r)e

ikt,

BESOV-SCHATTEN SPACES AND HANKEL OPERATORS 3

where Ak(r) is the kth-diagonal of the matrix A(r), the preceding sumis a formal one and t belongs to the torus T. This matrix A(r) is calledanalytic matrix if there exists an upper triangular infinite matrix Asuch that, for all r ∈ [0, 1), we have Ak(r) = Akr

k, for all k ∈ Z. Inwhat follows we identify the analytic matrices A(r) with their corre-sponding upper triangular matrices A and we call them also analyticmatrices.We also recall the definition of the matriceal Bloch space and the

so-called little Bloch space of matrices (see [6]). The matriceal Blochspace B(D, �2) is the space of all analytic matrices A with A(r) ∈ B(�2),0 ≤ r < 1, such that

‖A‖B(D,�2) = sup0≤r<1

(1− r2)‖A′(r)‖B(�2) + ‖A0‖B(�2) < ∞,

where B(�2) is the usual operator norm of the matrix A on the sequencespace �2 and A′(r) =

∑∞k=0 Akkr

k−1.The space B0(D, �2) is the space of all upper triangular infinite matri-

ces A such that limr→1−(1−r2)‖(A∗C(r))′‖B(�2) = 0, where C(r) is theToeplitz matrix associated with the Cauchy kernel 1

1−r, for 0 ≤ r < 1.

An important tool in this paper is the Bergman projection. It isknown (see e.g. [11]) that for all strong measurable Cp-valued functions

r → A(r) defined on [0, 1) with∫ 1

0‖A(r)‖pCp

2rdr < ∞ and for alli, j ∈ N we have that

[P (A(·))](r)(i, j) ={2(j − i+ 1)rj−i

∫ 10aij(s) · sj−i+1ds, if i ≤ j,

0 otherwise.

We recall now a Lemma from [6] that we will use in the following.

Lemma 1.2. Let V = (P2)∗, i.e, (P2A(·))∗(r)(i, j) =

=

{(j−i+3)(j−i+2)(j−i+1)

2rj−i(1− r2)2

∫ 10aij(s)s


0 otherwise.

Then V is an isomorphic embedding of B0(D, �2) in C0(D, �2).

The paper is organized as follows: In Section 2 we give a characteriza-tion of matrices in the Besov-Schatten space Bp(�

2) using the Bergmanprojection. The main result in Section 3 is a new duality result (seeTheorem 3.2). Finally, in Section 4 we give a sufficient condition for aHankel operator to be nuclear.

2. Besov-Schatten spaces

Now we introduce a new space of matrices namely the so calledBesov-Schatten space.


Definition 2.1. Let 1 ≤ p < ∞ and a positive measure on [0, 1) givenby

dλ(r) :=2rdr

(1− r2)2.

The Besov-Schatten matrix space Bp(�2) is defined to be the space of

all upper triangular infinite matrices A such that

‖A‖Bp(�2) :=

[∫ 1

0


] 1p

< ∞.

On Bp(�2) we introduce the norm

‖A‖ := ‖A0‖Cp + ‖A‖Bp(�2).

We also introduce the notation Lp(D, dλ, �2) for the space of allstrongly measurable functions r → A(r) defined on the measurablespace ([0, 1), dλ) with Cp-values such that

‖A‖Lp(D,dλ,�2) :=

(∫ 1

0

‖A(r)‖pCpdλ(r)

) 1p

< ∞.

We need the following useful Lemma in what follows (see [15], p.53):

Lemma 2.2. Let z ∈ D, c is real, t > −1, and

Ic,t :=

∫D

(1− |w|2)t|1− zw|2+t+c

dA(w).

(1) If c < 0, then Ic,t(z) is bounded in z;(2) If c > 0, then

Ic,t(z) ∼1

(1− |z|2)c (|z| → 1−);

(3) If c = 0, then

I0,c(z) ∼ log1

1− |z|2 (|z| → 1−).

In the next theorem we express a natural relation between the Berg-man projection and the Besov-Schatten spaces. More precisely, ourmain result of this section is the following equivalence theorem:

Theorem 2.3. Let 1 ≤ p < ∞ and A be an upper triangular matrixsuch that the Cp- valued function r → A′′(r) is continuous on [0, r0)for some r0, 0 < r0 < 1. Then the following assertions are equivalent:

(1) A ∈ Bp(�2);

(2) (1− r2)2A′′(r) ∈ Lp(D, dλ, �2);(3) A ∈ PLp(D, dλ, �2), where P is the Bergman projection.


Proof. It is obvious that (1) is equivalent to (2). We observe that theBergman projection may be described as follows:

P (A(·)) =∞∑k=0

(k + 1)

∫ 1

0

[A(s)]ksk(2sds),

where A(·) ∈ Lp(D, �2). Then

P((1− r2)2Akr

k)(s) =

2skAk

(k + 2)(k + 3),

for all k ≥ 0, and all Ak ∈ Cp.It follows that each matriceal polynomial is in PLp(D, dλ, �2) for all

1 ≤ p < ∞.Suppose that A is an upper triangular matrix with Ak ∈ Cp for all

k ≥ 0. We write

A =4∑

k=0

Ak + A1,

where A1 :=∑∞

k=5 Ak.If (1− r2)2A′′(r) ∈ Lp(D, dλ, �2), then we have that

Φ(r) :=4∑

k=0

(k + 2)(k + 3)

2(1− r2)2Akr

k +(1− r2)2(A1)′′(r)

2!r2

is in Lp(D, dλ, �2) and, moreover, that A = PΦ.Indeed, for 0 < r < r0, r → (A1)′′(r) is a continuous function.

Therefore ∫ r0

0

‖(A1)′′(s)‖pCp

s2pds < ∞,

so that Φ ∈ Lp(D, dλ, �2).Moreover A = PΦ since

∞∑k=5

∫ 1

0

k(k − 1)Aksk−2(1− s2)2

s2(k + 1)sk+1ds =

=∞∑k=5

(k − 1)k(k + 1)Ak

∫ 1

0

s2k−3(1− s2)2ds =∞∑k=5

Ak.

Thus we have proved that (2) implies (3).It remains to prove that (3) implies (2). Suppose that (3) holds and

let A = PΦ for some Φ(·) ∈ Lp(D, dλ, �2). Then we have that

(1− r2)2A′′(r) = (1− r2)2∫ 1

0

[Φ(s) ∗ 6s2

(1− rs)4

]2sds.


Using Fubini’s Theorem and Lemma 2.2 we obtain that∫ 1

0

(1− r2)2‖A′′(r)‖C1

2rdr

(1− r2)2≤

≤∫ 1

0

[∫ 1

0

‖Φ(s)‖C1

∫ 2π

0

6s2dθ

(1− rseiθ)42sds

]2rdr =

=

∫ 1

0

6s2‖Φ(s)‖C1

[∫ 1

0

1

2π

∫ 2π

0

dθ

|1− rseiθ|4]2rdr ∼

∼∫ 1

0

‖Φ(s)‖C1

12s3

(1− s2)2ds ≤ 6

∫ 1

0

‖Φ(s)‖C1dλ(s) < ∞.

Consequently, A ∈ L1(D, dλ, �2) and this proves that (3) implies (2)in the case p = 1. The proof in the case 1 < p < ∞ is similar withthe classical case of functions (see e.g. Theorem 5.3.3. in [15]). LetT (rs) = ((tij)(rs))

∞i,j=1 be the Toeplitz matrix with

tij(rs) = tj−i(rs) =

=

{s2(rs)j−i(j − i+ 3)(j − i+ 2)(j − i+ 1) if j ≥ i

0 otherwise.

Since T (rs) is a Schur multiplier with ‖T (rs)‖M(�2) = ‖T (rs)‖L1(T) =

‖ 6s2

(1−rseiθ)4‖L1(T) and M(�2) = M(C1) ⊂ M(Cp), 1 ≤ p < ∞ we get that

(1− r2)2‖A′′(r)‖Cp = (1− r2)2‖∫ 1

0

[φ(s) ∗ 6s2

(1− rs)4](2sds)‖Cp ≤

≤ (1− r2)2∫ 1

0

‖φ(s)‖Cp‖6s2

(1− rseiθ)4‖L1(T)(2sds) =

= (1− r2)2∫ 1

0

‖φ(s)‖Cp(1− s2)2‖ 6s2

(1− rseiθ)4‖L1(T)dλ(s) := Sφ(r).

From Schur’s Theorem (see e.g. [15]) it follows that Sφ(r) is boundedon Lp([0, 1), dλ) which in its turn implies that

(1− r2)2A′′(r) ∈ Lp(D, dλ, �2)

for 1 < p < ∞. Thus also the implication (3)⇒(2) is proved and theproof is complete. �


3. The dual of Besov-Schatten spaces

Our aim in this Section is to characterize the Banach dual spaces ofBesov-Schatten spaces.

First we prove the following lemma of independent interest:

Lemma 3.1. Let V = (P2)∗, that is

[V (A(·))](r)(i, j) :={(j−i+3)(j−i+2)(j−i+1)

2rj−i(1− r2)2

∫ 1

0aij(s)s


0 otherwise.

Then V is an embeding from Bp(�2) into Lp(D, dλ, �2) for all p ≥ 1,

if Bp(�2) = PLp(D, dλ, �2) is equipped with the quotient norm.

Proof. Suppose that A ∈ Bp(�2) and B(·) ∈ Lp(D, dλ, �2) with A =

PB(·). Since

P (B(·))(r)(i, j) ={2(j − i+ 1)rj−i

∫ 10bij(s)s

j−i+1ds if j − i ≥ 0

0 otherwise,

it is easy to see that

PV = P and V P = V

on Lp(D, dλ, �2). Therefore V (A) = V (B(·)) for all A ∈ Bp(�2) and

B(·) ∈ Lp(D, dλ, �2).We will now prove that V is a bounded operator on Lp(D, dλ, �2).

We first prove this fact for p = 1. By Fubini’s theorem we have that

‖V (A(·))‖L1(D,dλ,�2) =

∫ 1

0

‖[V (A(·))]‖C1dλ(r) =

∫ 1

0

‖∞∑k=0

(k + 3)(k + 2)(k + 1)

2rk(1− r2)2

∫ 1

0

Ak(s)sk(2sds)‖C1dλ(r) ≤

∫ 1

0

∫ 1

0

‖∞∑k=0

(k + 3)(k + 2)(k + 1)

2rk(1−r2)2Ak(s)s

k+1‖C1(2ds)dλ(r) =

∫ 1

0

[

∫ 1

0

‖∞∑k=0

(k + 3)(k + 2)(k + 1)

2rkAk(s)s

k(1−s2)2‖C1(2rdr)]dλ(s) =

=

∫ 1

0

∫ 1

0

‖A(s) ∗ C(rs)‖C1(2rdr)dλ(s),


where C(rs) = (cij(rs))∞i,l=1 means the Toeplitz matrix given by

cij(rs) = cj−i(rs) =

{(rs)j−is2(1− s2)2 (j−i+3)(j−i+2)(j−i+1)

2if j ≥ i

0 otherwise .

Since the Toeplitz matrix C(rs) is a Schur multiplier with

‖C(rs)‖M(�2) = ‖6s2(1− s2)2

(1− rseiθ)4‖L1(T),

according to Lemma2.2 it follows that∫ 1

0

∫ 1

0

‖A(s) ∗ C(rs)‖C1(2rdr)dλ(s) ≤

≤∫ 1

0

‖A(s)‖C1

∫ 1

0

‖C(rs)‖M(�2)(2rdr)dλ(s) ∼

∼∫ 1

0

‖A(s)‖C1dλ(s).

Consequently, V is bounded on L1(D, dλ, �2). For 1 < p < ∞ we havethat

‖V A(·)(r)‖Cp ≤∫ 1

0

‖∞∑k=0

(k + 3)(k + 2)(k + 1)

2rksk(1− r2)2(1− s2)2Ak(s)‖Cpdλ(s) =

=

∫ 1

0

‖A(s) ∗ T (rs)‖Cpdλ(s),

where T (rs) = (tj−i(rs))i,j is a Toeplitz matrix and

tj−i(rs) =

{(rs)j−i(1− s2)2(1− r2)2 (j−i+3)(j−i+2)(j−i+1)

2if j ≥ i

0 otherwise .

T (rs) is a Schur multiplier and therefore∫ 1

0

‖A(s) ∗ T (rs)‖Cpdλ(s) ≤∫ 1

0

‖A(s)‖Cp(1− r2)2(1− s2)2‖ 6

(1− rseiθ)4‖L1(T) := SA(r).

From Schur’s Theorem(see e.g. [15]) we obtain that SA(r) is boundedon Lp([0, 1), dλ). Hence V is bounded on Lp(D, dλ, �2), 1 ≤ p < ∞,and there is a constant C > 0 such that

‖V (A(·))‖Lp(D,dλ,�2) ≤ C‖B(·)‖Lp(D,dλ,�2)


for all A = PB(·). Taking the infimum over B, we get that

‖V (A)‖Lp(D,dλ,�2) ≤ C‖A‖Bp(�2).

Thus V : Bp(�2) → Lp(D, dλ, �2) is bounded.

On the other hand, since PV = P and V P = V on Lp(D, dλ, �2) weget easily that A = PV (A) for all A ∈ Bp(�

2). Thus,

‖A‖Bp(�2) = inf{‖B(·)‖Lp(D,dλ,�2) : A = PB} ≤ ‖V A‖Lp(D,dλ,�2)

and, hence, V : Bp(�2) → Lp(D, dλ, �2) is an embedding. The proof is

complete. �

Now we can formulate and prove the duality of Besov-Schatten spaces.

Theorem 3.2. Under the pairing

< A,B >=

∫ 1

0

tr(V (A)[V (B)]∗)dλ(r)

we have the following dualities:(1) Bp(�

2)∗ ≈ Bq(�2) if 1 < p < ∞ and 1

p+ 1

q= 1;

(2) B0,c(D, �2)∗ ≈ B1(�2) and B1(�

2)∗ ≈ B(D, �2).

Proof. Since V is an embedding from Bp(�2) into Lp(D, dλ, �2) for all

1 ≤ p < ∞, Holder’s inequality shows that Bq(�2) ⊂ Bp(�

2)∗ for 1 ≤p < ∞ and B1(�

2) ⊂ B∗0,c(D, �2).

Suppose that F is a bounded linear functional on the Besov-Schattenspace Bp(�

2) with 1 ≤ p < ∞. Then F ◦ V −1 : V Bp(�2) → C can be

extended to a bounded linear functional on Lp(D, dλ, �2). Thus thereexists C(·) ∈ Lp(D, dλ, �2) such that ‖C(·)‖Lp(D,dλ,�2) = ‖F ◦ V −1‖ and

(F ◦ V −1)(B) =

∫ 1

0

tr(B(r))[C(r)]∗dλ(r), B(·) ∈ Lp(D, dλ, �2).

In particular, if B(·) = V (A) with A ∈ Bp(�2), then

F (A) =

∫ 1

0

tr((V A)(r))[C(r)]∗dλ(r).

Let B = P (C); then B ∈ Bq(�2) and it is easy to check that

F (A) =

∫ 1

0

tr((V A)(r)[(V B)(r)]∗)dλ(r), A ∈ Bp(�2),

with ‖B‖Bq(�2) ≤ ‖C(·)‖Lp(D,dλ,�2) = ‖F ◦ V −1‖ ≤ ‖V −1‖‖F‖. Thisproves the duality Bp(�

2)∗ ≈ Bq(�2) for 1 ≤ p < ∞.

It remains to prove the duality B∗0,c(D, �2) ≈ B1(�

2).


Let us assume that F is a bounded linear functional on B0,c(D, �2).Then we shall prove that there is a matrix C from B1(�

2) such that

(1) F (B) =

∫ 1

0

tr[V B(r)(V C)∗(r)]dλ(r),

for B from a dense subset of B0(D, �2). By Lemma 1.2 it followsthat V : B0(D, �2) → C0(D, �2) is an isomorphic embedding. ThusX = V (B0,c(D, �2)) is a closed subspace in C0(D,C∞) and F ◦ (V )−1 :X → C is a bounded linear functional on X, where C0(D,C∞) is thesubset in C0(D, �2) whose elements are C∞-valued functions. By theHahn-Banach theorem F ◦ (V )−1 can be extended to a bounded linearfunctional on C0(D,C∞).

Let Φ : C0(D,C∞) → C denote this functional. It follows thatC0(D,C∞) = C0[0, 1]⊗εC∞ and, thus, Φ is a bilinear integral map,that is there is a bounded Borel measure μ on [0, 1]× UC1 , where UC1

is the unit ball of the space C1 with the topology σ(C1, C∞), such that

Φ(f ⊗ A) =

∫[0,1]×UC1

f(r)tr(AB∗)dμ(r, B)

for every f ∈ C0[0, 1] and A ∈ C∞.Thus, for the matrix

∑n

k=0 Ak ∈ B0,c(D, �2), identified with the ana-lytic matrix

∑n

k=0 Akrk, we have that

F (n∑

k=0

Ak) = F (n∑

k=0

rkAk) = [F ◦ (V )−1][V (n∑

k=0

rkAk)] =

= Φ(n∑

k=0

(k + 3)(k + 2)

2rk(1− r2)2Ak) =

=

∫[0,1]×UC1

n∑k=0

tr[((k + 3)(k + 2)

2rk(1− r2)2Ak)B

∗]dμ(r, B)

def= < μ(r, B), tr(

n∑k=0

(k + 3)(k + 2)

2rkAk)B

∗(1− r2)2 > .


On the other hand, we wish to have that

F (A) =

∫ 1

0

trV (A)(V (C))∗dλ(s) =

=

∫ 1

0

tr(n∑

k=0

(k + 3)(k + 2)

2skAk)(V (C))∗(2sds) =

=

∫ 1

0

tr(n∑

k=0

s2k(k + 3)2(k + 2)2

4(1− s2)2AkC

∗k)(2sds) =

=n∑

k=0

trAk

((k + 3)(k + 2)

2(k + 1)C∗

k

).

Therefore, letting A = ei,i+k, denote the matrix having 1 as the singlenonzero entry on the ith-row and the (i + k)th-column, for i ≥ 1 andj ≥ 0, we have that

Ck =< μ(r, B), (k + 1)rk(1− r2)2Bk >, k = 0, 1, 2, . . .

Then, it yields that ∫ 1

0

‖C ′′(s)‖C12sds =

=

∫ 1

0

‖∫[0,1]×UC1

n∑k=2

(k + 1)!

(k − 2)!sk−2rk(1− r2)2Bkdμ(r, B)‖C1(2sds) ≤

≤∫[0,1]×UC1

[

∫ 1

0

‖n∑

k=2

(k + 1)!

(k − 2)!(rs)k−2r2(1− r2)2Bk‖C1(2sds)]d|μ|(r, B)

≤∫[0,1]×UC1

[

∫ 1

0

‖n∑

k=2

(k + 1)!

(k − 2)!(rs)k−2r2(1− r2)2eik(.)‖L1(T)‖B‖C1(2sds)

≤∫[0,1]×UC1

(∫ 1

0

∫ 2π

0

r2(1− r2)2

|1− rseiθ|4dθ

2π(2sds)

)d|μ|(r, B) ∼

∼∫[0,1]×UC1

r2(1− r2)21

(1− r2)2d|μ|(r, B) ≤ ‖μ‖ < ∞.

Consequently, C ∈ B1(�2) and we get the relation (1) by using the fact

that the set of all matrices∑n

k=0 Ak is dense in B0,c(D, �2). The proofis complete.

�


4. Nuclear Hankel operators and the space M1

Let us denote with T2 the subspace of C2 consisting of all uppertriangular matrices. We denote by P+ the triangular projection, P+A =∑

k∈Z Ak and P− = I − P+, where I is the identity on T2. Let Φ bean infinite matrix such that for all matrices A =

∑n

k=0 Ak, n ∈ N, wehave P−(ΦA) ∈ C2.

Examples of such matrices are either all matrices representing linearbounded operators on �2, or matrices Φ such that P−Φ = 0.

We define the matrix version of Hankel operator to be

HΦ : T2 → (T2)− := C2 � T2,

on the dense subspace in T2 of all matrices A =∑n

k=0 Ak, n ∈ N suchthat ΦA ∈ C2 by

HΦ(A) = P−(ΦA).The matrix Φ is called the symbol operator of Hankel operator HΦ.

We briefly indicate the connection between the matrix version ofHankel operator and the vector Hankel operator in the sense describedby V. Peller in [10].

Let f be a function on the torus T such that for any function g ∈H2(T), fg belongs to L2(T). Examples of such functions are analyticfunctions from the classical Besov space B1 (see e.g. [15]).If Φ is a matrix as above, then the Hankel operator in the sense

described by V. Peller,

Hf⊗Φ : H2(T2) → (H2)−(T2 −)

coincides with the tensor product Hf⊗HΦ between the classical Hankeloperator Hf and the matrix version of Hankel operator HΦ.

We mention here that the matrix version of Hankel operator was firstconsidered by S. C. Power in 1986 [13] and recently studied in moredetail in [12].

We give now a sufficient condition in order that the Hankel operatorHA to be nuclear.

We use the space M1 considered by A. Pelczynski and F. Sukochevin [8], that is

M1 = {A such that ‖A‖M1 :=∑k∈Z+

‖Ak‖C1 < ∞}.

This space was investigated in [8] from the viewpoint of Schur mul-tipliers with range consisting of matrices with absolutely summableentries.

We intend to show that for an upper triangular matrix A such thatA′ =

∑∞k=0 kAk ∈ M1, the operator HA is nuclear. It is easy to see that


an upper triangular matrix A as above belongs in fact to the Besov-Schatten matrix space B1(�

2). Thus, our next Theorem gives a partialanswer to the following conjecture.

Conjecture 4.1. If A ∈ B1(�2) then HA∗ is a nuclear operator.

Let 1 ≤ p < ∞. Then let us denote by Sp the Schatten class of alloperators from T2 into T2−.

Then we have the following result:

Theorem 4.2. Let A be an upper triangular infinite matrix such thatA′ ∈ M1. Then it follows that the Hankel operator HA∗ : T2 → T2− isnuclear. Moreover, there is a constant C > 0, such that

‖HA∗‖S1 ≤ ‖A′‖M1 .

Proof. The proof uses induction with respect to n. Of course if A = A0,

thenHA∗ = 0, and A′ = 0. Now let A = A1 =

⎛⎜⎜⎜⎜⎝0 a11 0 0 · · ·0 0 a21 0

. . .

0 0 0 a31. . .

.... . . . . . . . . . . .

⎞⎟⎟⎟⎟⎠ .

Then HA∗(B) = (I−P )(A∗B), where B ∈ T2, P is the upper triangularprojection and I is the identity matrix. Then

(I − P )

⎛⎜⎜⎜⎜⎜⎜⎜⎝

⎛⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 · · ·a11 0 0

. . .

0 a21 0. . .

0 0 a31. . .

.... . . . . . . . .

⎞⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎝

b10 b11 b12 · · ·0 b20 b21

. . .

0 0 b30. . .

0 0 0. . .

.... . . . . . . . .

⎞⎟⎟⎟⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎟⎠=

=

⎛⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 · · ·a11b

10 0 0 0

. . .

0 a21b20 0 0

. . .

0 0 a31b30 0

. . ....

. . . . . . . . . . . .

⎞⎟⎟⎟⎟⎟⎟⎟⎠.


Consequently, HA∗(B) = A∗1 ∗ τ−1B0, for all B ∈ T2, where

τ−1

⎛⎜⎜⎜⎜⎜⎜⎜⎝

⎛⎜⎜⎜⎜⎜⎜⎜⎝

b10 b11 b12 · · ·0 b20 b21

. . .

0 0 b30. . .

0 0 0. . .

.... . . . . . . . .

⎞⎟⎟⎟⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎟⎠=

⎛⎜⎜⎜⎜⎜⎜⎜⎝

b11 b12 b13 · · ·b10 b21 b22

. . .

0 b20 b31. . .

0 0 b30. . .

.... . . . . . . . .

⎞⎟⎟⎟⎟⎟⎟⎟⎠.

Denoting by Ekj , where j ∈ Z, and k ≥ 1, the infinite matrix having

as entries 1 on the kth place on the jth-diagonal, and 0 otherwise, weget easily that

HA∗(B) =∞∑k=1

ak1 < Ek0 , B > Ek

−1 =∞∑k=1

ak1 < Ek0 ⊗ Ek

−1, B >,

where∑∞

k=1 |ak1| = ‖A‖M1 < ∞.It follows that HA∗ is a nuclear operator for A = A1 ∈ M1 and

‖HA∗‖S1 ≤ ‖A‖M1 .We consider now the case n = 2, that is A = A1 +A2. We have that

HA∗(B) = (I − P )

⎛⎜⎜⎜⎜⎜⎜⎜⎝

⎛⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 · · ·a11 0 0

. . .

a12 a21 0. . .

0 a22 a31. . .

.... . . . . . . . .

⎞⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎝

b10 b11 b12 · · ·0 b20 b21

. . .

0 0 b30. . .

0 0 0. . .

.... . . . . . . . .

⎞⎟⎟⎟⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎟⎠=

=

⎛⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 · · ·a11b

10 0 0 0

. . .

a12b10 a12b

11 + a21b

20 0 0

. . .

0 a22b20 a22b

21 + a31b

30 0

. . ....

. . . . . . . . . . . .

⎞⎟⎟⎟⎟⎟⎟⎟⎠.

Then we obtain that

HA∗ =∞∑k=1

ak2Ek0 ⊗ Ek

−2 +∞∑k=1

ak1Ek0 ⊗ Ek

−1 +∞∑k=1

ak2Ek1 ⊗ Ek+1

−1 ,

which implies that HA∗ is a nuclear operator and

‖HA∗‖S1 ≤ ‖A1‖C1 + 2‖A2‖C1 = ‖A′‖M1 .

Let us consider now the case n = 3 which contains all the difficultiesof the general case.


If A = A1 + A2 + A3, then

HA∗(B) =

= (I − P )

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 · · ·a11 0 0 0

. . .

a12 a21 0 0. . .

a13 a22 a31 0. . .

0 a23 a32 a41. . .

.... . . . . . . . . . . .

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎝

b10 b11 b12 b13 · · ·0 b20 b21 b22

. . .

0 0 b30 b31. . .

0 0 0 b40. . .

.... . . . . . . . . . . .

⎞⎟⎟⎟⎟⎟⎟⎟⎠

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0 0 · · ·a11b

10 0 0 0 0

. . .

a12b10 a12b

11 + a21b

20 0 0 0

. . .

a13b10 a13b

11 + a22b

20 a13b

12 + a22b

21 + a31b

30 0 0

. . .

0 a23b20 a23b

21 + a32b

30 a23b

32 + a32b

31 + a41b

40 0

. . ....

. . . . . . . . . . . . . . .

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.

Consequently

HA∗ =∞∑k=1

ak1Ek0 ⊗ Ek

−1 +∞∑k=1

ak2(Ek0 ⊗ Ek

−2 + Ek1 ⊗ Ek+1

−1 )+

+∞∑k=1

ak3(Ek0 ⊗ Ek

−3 + Ek1 ⊗ Ek+1

−2 + Ek2 ⊗ Ek+2

−1 ).

Hence

‖HA∗‖S1 ≤ ‖A1‖C1 + 2‖A2‖C1 + 3‖A3‖C1 = ‖A′‖M1

and the theorem can be proved in general by using an induction pro-cedure. �

References

[1] J. Arazy, Some remarks on interpolation theorems and the boundedness of thetriangular projection in unitary matrix spaces, Integral Equations OperatorTheory 1(1978), no. 4, 453–495.

[2] A. B. Alexandrov and V. V. Peller, Hankel and Toeplitz-Schur multipliers,Math. Ann. 324(2002), 277–327.


[4] G. Bennett, Schur multipliers, Duke Math. J. 44(1977), 603–639.[5] A. N. Marcoci and L. G. Marcoci, A new class of linear operators on �2 and

Schur multipliers for them, J. Funct. Spaces Appl. 5(2007), 151–164.


[6] L. G. Marcoci, L. E. Persson, I. Popa and N. Popa, A new characterization for aBergman-Schatten spaces and a duality result, J. Math. Anal. Appl. 360(2009),No.1, 67–80.

[7] J. Peetre, New thoughts on Besov spaces, Duke Univ. Press., Durham, N.C.,1976.

[8] A. Pelczynski and F. Sukochev, Some remarks on Toeplitz multipliers andHankel matrices, Studia Math. 175(2006), 175–204.

[9] V. Peller, Hankel operators on class Cp and their applications (rational approx-imation, Gaussian processes, the problem of majorization aperators), Math.USSR-Sbornik 41(1982), 443–479.

[10] V. Peller, Hankel Operators and Their Applications , Springer- Verlag, NewYork, 2003.

[11] N. Popa, Matriceal Bloch and Bergman-Schatten spaces, Rev. Roumaine Math.Pures Appl. 52(2007), 459–478.

[12] N. Popa, Schur multipliers between Banach spaces of matrices, Proc. ofthe Sixth Congress of Romanian Mathematicians, Bucharest, 2007, Edit.Academiei Romane, Bucharest, 2009, 373–380.

[13] S. C. Power, Factorization in analytic operator algebras, J. Funct. Anal.67(1986), 413–432.


[15] K. H. Zhu, Operator Theory in Function Spaces, Marcel Dekker, 1990.


of Civil Engineering Bucharest, RO-020396 Bucharest, ROMANIA E-mail: [email protected], E-mail: [email protected]



Institute of Mathematics of Romanian Academy, P.O. BOX 1-764

RO-014700 Bucharest, ROMANIA E-mail: [email protected]

Paper 5

A CLASS OF LINEAR OPERATORS ONQUASI-MONOTONE SEQUENCES IN �2 AND ITS

SCHUR MULTIPLIERS

LIVIU GABRIEL MARCOCI

Abstract. In this paper we introduce and investigate a new classof linear operators. In particular, we give concrete characteriza-tions for some cases and prove some new results concerning Schurmultipliers.

1. Introduction

We started our study motivated by the papers [3], [7], [16] and [17].In the paper [16], the authors introduced the space Bw(�

2) of thoseinfinite matrices A for which A(x) ∈ �2 for all x = (xn)n ∈ �2 with|xn| ↘ 0. This Banach space of infinite matrices can be regarded asa weaker version of the space B(�2), the space of all bounded opera-tors from �2 into �2. It is a weaker version because it consists of thoseoperators which maps the sequences which are decreasing in modulusfrom �2 into �2. This space actually appeared in the study of matricealanalogues of classical function spaces like C(T) (the continuous func-tions on the torus), the Wiener algebra A(T) and the Lebesgue spaceL1(T).

2000 Mathematics Subject Classification. 15A60, 47B35, 26D15.Key words and phrases. Infinite matrices, Schur multipliers, discrete Sawyer

duality principle.1

2 LIVIU GABRIEL MARCOCI

It is easy to see that B(�2) ⊂ Bw(�2) and that the inclusion is proper.

It is interesting that these two spaces coincides on the subspace con-sisting of Toeplitz matrices. Let

Bαw(�

2) = {A infinite matrix ;Ax ∈ �2 for every

x = (xn)n ∈ �2 with|xn|nα

↘ 0, α ≥ 0}.

Thus B0w(�

2) = Bw(�2).

The Schur product of two matrices is defined by

A ∗B = (aij · bij)i,j≥1 ,

where A = (aij)i,j≥1 , B = (bij)i,j≥1 . We denote by

M(�2)={M : M ∗ A ∈ B

(�2)for every A ∈ B

(�2)}

the space of all Schur multipliers equipped with the following norm

‖M‖ = sup‖A‖B(�2)≤1

‖M ∗ A‖B(�2) .

For an infinite matrix A = (aij) and an integer k we denote byAk = (a′ij), where

a′ij =

{aij if j − i = k,

0 otherwise,

i.e. we have that

Ak =

⎛⎜⎜⎜⎜⎝0 0 . . . a1k 0 . . . 0 . . .0 0 . . . 0 a2 k+1 . . . 0 . . .. . . . . . . . . . . . . . . . . . . . . . . .0 0 . . . 0 0 . . . ak 2k−1 . . .. . . . . . . . . . . . . . . . . . . . . . . .

⎞⎟⎟⎟⎟⎠ .

A CLASS OF LINEAR OPERATORS ON QUASI-MONOTONE SEQUENCES 3

Ak is called the Fourier coefficient of k:th order of the matrix A (seee.g. [3]). In particular, if a = (ak)k≥1 is a sequence we write that

A0 =

⎛⎜⎜⎜⎜⎝a1 0 0 0 · · ·0 a2 0 0 · · ·0 0 a3 0 · · ·0 0 0 a4 · · ·· · · · · · · · · · · · · · ·

⎞⎟⎟⎟⎟⎠and we sometimes say that the diagonal matrix A0 is given by thesequence a = (ak)k≥1. The definition of the matrix Ak given by thesequence a = (ak)k≥1 is similar.The paper is organized as follows: In Section 2 we give some prelim-

inaries. One of the main tools in this paper is to use Sawyer’s dualityTheorem in the discrete case. We include also some additional defini-tions and results that we need in the sequel. In Section 3 we presentthe main results of this paper. In Section 4 we present the proofs ofthe main results. Finally, for the readers convenience, we include anappendix with proofs of some Theorems from Section 2.

2. Preliminaries

The following standard notations will be used very often: Letterssuch asX or Y are always σ-finite measure spaces andM(X) (M+(X))denotes the space of measurable (resp. measurable and nonnegative)functions onX. We denote by R+ = (0,∞). The letters w, w, w0, ... areused for weight functions in R+ (nonnegative locally integrable func-tions in R+). For a given weight w we write W (r) =

∫ r

0w(t)dt < ∞,

0 ≤ r < ∞. If f is a positive nonincreasing function we will writef ↘. The following Lemma (see e.g. [9]) will be very useful.

Lemma 2.1. Let 0 < p < ∞ and v ≥ 0 be a measurable function inR+. Let V (r) =

∫ r0v(s)ds, 0 ≤ r < ∞. Then, for every decreasing

function f we have that∫ ∞

0

f p(s)v(s)ds =

∫ ∞

0

ptp−1V (μf (t))dt.


Next we recall an important result due to E. Sawyer which will beused in what follows. The original proof can be seen in [21] and otherproofs in [8] and [25]. For completeness we present in our appendix aproof, which is a combination of the original proof and the one from[8].

Theorem 2.2 (Sawyer’s Theorem). Suppose that 1 < p < ∞ andthat v(x) and g(x) are nonnegative measurable functions on R+, withv locally integrable. Then

sup

∫∞0

f(x)g(x)dx

(∫∞0

f(x)pv(x)dx)1p

≈

(1)

(∫ ∞

0

(∫ x

0

g(t)dt

)p′

v(x)

(∫ x

0

v(t)dt

)−p′

dx

) 1p′

+

∫∞0

g(t)dt

(∫∞0

v(t)dt)1p

,

where the supremum is taken over all nonnegative and nonincreasingfunctions f . Moreover, the right hand side of (1) can be replaced withthe integral(∫ ∞

0

(∫ x

0

g(t)dt

)p′−1(∫ x

0

v(t)dt

)1−p′

g(x)dx

) 1p′

.

The expression f ≈ g will indicate the existence of two positiveconstants a and b such that af ≤ g ≤ bf .

For p ≤ 1, we also have (see [8]) the following result.

Theorem 2.3. Suppose p ≤ 1 and that v(x) and g(x) are nonnegativemeasurable functions on R+ with v locally integrable. Then

sup

∫∞0

f(x)g(x)dx

(∫∞0

f(x)pv(x)dx)1p

≈ supr>0

(∫ r

0

g(x)dx

(∫ r

0

v(x)dx

)− 1p

),

where the supremum is taken over all nonnegative and nonincreasingfunctions f .


The discrete version of Sawyer’s formula is given by the next Theo-rem (see [10]). We first need some preliminaries. In the paper [10] theauthors considered the problem of characterizing the boundedness of

T : (Lp0(X) ∩ L, ‖ · ‖p0) → Lp1(Y ),

that is

(2) ‖Tf‖Lp1 (Y ) ≤ C‖f‖Lp0 (X), f ∈ L,

where L is a subclass of M(X) and T is an operator (usually linearor sublinear). In some cases inequality (2) holds for every f ∈ L ifand only if it holds for the characteristic functions f = χA ∈ L. Thishappens (see [8] and [24]) in the case L = Lp0(w0), X = (R+, w0(t)dt)and Y = (R+, w1(t)dt) in the range of indices 0 < p0 ≤ 1, p0 ≤ p1 < ∞and operators of the type

(3) Tf(r) =

∫ ∞

0

k(r, t)f(t)dt, r > 0,

for positive kernels k. In [10], they obtained a more general principlethat can be applied to a more general class of operators than thoseof integral type and for a bigger class of functions that includes themonotone functions. We recall now some definitions from [10].

Definition 2.4. We say that ∅ �= L ⊂ M(X) is a regular class in Xif, for every f ∈ L,

(i) |αf | ∈ L, for every α ∈ R,(ii) χ{|f |>t} ∈ L, for every t > 0, and(iii) there exists a sequence of simple functions (fn)n ⊂ L such that

0 ≤ fn(x) ≤ fn+1(x) → |f(x)| a.e. x ∈ X.

Example 2.5. (i) If X is an arbitrary measure space, every functionallattice in X (i.e., a vector space L ⊂ M(X) with g ∈ L if |g| ≤ |f |,f ∈ L) with the Fatou property (see [5]), is a regular class. In particularthe Lebesgue space Lp(X), 0 < p ≤ ∞ is a regular class.

(ii) In R+ the set of positive decreasing functions is a regular classand the same holds for the increasing functions.


Definition 2.6. Let Y be a measure space, L a regular class and

T : L → M(Y )

an operator.(i) T is sublinear if

|T (α1f1 + · · ·+ αnfn)(y)| ≤ |α1Tf1(y)|+ · · ·+ |αnTfn(y)|a.e. y ∈ Y , for every α1, . . . , αn ∈ R and every f1, . . . , fn ∈ L suchthat α1f1 + · · ·+ αnfn ∈ L.(ii) T is monotone if

|Tf(y)| ≤ |Tg(y)|a.e. y ∈ Y , if |f(x)| ≤ |g(x)| a.e. x ∈ X, f, g ∈ L.(iii) We say that T is order continuous if it is monotone and if, for

every sequence (fn)n ⊂ L with 0 ≤ fn(x) ≤ fn+1(x) → f(x) ∈ L a.e.x ∈ X, we have that

limn

|Tfn(y)| = |Tf(y)|a.e. y ∈ Y .

Remark 2.7. The identity operator f �→ f is obviously order contin-uous.

We recall an important result which will be used in what follows andwhere the boundedness of certain operators is characterized by theirrestriction to the set of characteristic functions.

Theorem 2.8 (see e.g. [10]). Let L ⊂ M(X) be a regular class andT : L → M(Y ) be an order continuous sublinear operator. If 0 < p0 ≤1, p0 ≤ p1 < ∞ then we have that

supf∈L

‖Tf‖Lp1 (Y )

‖f‖Lp0 (X)

= supχB∈L

‖TχB‖Lp1 (Y )

‖χB‖Lp0 (X)

.

This result is actually a Corollary of a more general Theorem (seeTheorem 1.2.11 in [10]).We present now the discrete version of Sawyer’s formula as it was

obtained in [10].


Theorem 2.9. Let w = (w(n))n, v = (v(n))n be weights in N∗ and let

S = supf↘0

∑∞n=0 f(n)v(n)

(∑∞

n=0 f(n)pw(n))

1p

.

(i) If 0 < p ≤ 1, then

S = supn≥0

V (n)

W1p (n)

,

with W defined by W (n) =∑n

k=0 w(k), n = 0, 1, 2, ... and V analo-gously.(ii) If 1 < p < ∞, then

S ≈

⎛⎝∫ ∞

0

(V (t)

W (t)

)p′−1

v(t)dt

⎞⎠ 1p′

≈

⎛⎝∫ ∞

0

(V (t)

W (t)

)p′

w(t)dt

⎞⎠ 1p′

+V (∞)

W1p (∞)

,

where v is the weight in R+ defined by

v =∞∑n=0

v(n)χ[n,n+1)

and V (t) =∫ t

0v(s)ds and analogously for w and W .

Moreover, the implicit constants in the symbol ≈ only depend on p.For the readers convenience we include a proof also of this theorem inour appendix (see [10]).

3. Main results

We begin with a characterization of main diagonal matrices.


Theorem 3.1. Let α ≥ 0, A = A0 be given by the sequence a = (an)nThen A ∈ Bα

w(�2) if and only if

supn≥1

(∑n

k=1 |ak|2k2α∑n

k=1 k2α

) 12

< ∞.

Moreover, the norm


n≥1

(∑n

k=1 |ak|2k2α∑n

k=1 k2α

) 12

.

Example 3.2. It is clear that B(�2) ⊂ Bαw(�

2) and using the previoustheorem for the matrix A = A0 given by the sequence a = (ak)

nk=1,

where

ak =

{2

p−12

−α, if k = 2p

0, otherwise,

we can easily show that the inclusion is proper. An easy computationshow us that ‖A‖Bα

w(�2) < ∞ but the sequence is not bounded. This factimplies that A /∈ B(�2).

Remark 3.3. Using the previous example we can conclude that:

B(�2) ⊂ Bαw(�

2)

and the inclusion is proper. It is known that B(�2) is closed underSchur multiplication (see e.g. [7]). Thus it is a natural question ifBα

w(�2) is closed under Schur multiplication. The answer is negative:

we can take for example the sequence defined in the Example 3.2 andwe can see that A ∗ A /∈ Bα

w(�2).

If α = 0 it is known that the space of all bounded operators on �2

is included in the space of all Schur multipliers from B0w(�

2) to B0w(�

2)(see e.g. [16]). The following Theorem shows that the result in fact istrue for α ≥ 0. Our next result reads:

Theorem 3.4. The space B(�2) is included in the space of all Schurmultipliers from Bα

w(�2) to Bα

w(�2) where α ≥ 0.


Our next result of this Section gives us the coincidence of the spacesB(�2) and Bα

w(�2) in the case of Toeplitz matrices.

Theorem 3.5. Let us denote T the set of all Toeplitz matrices. Thenwe have that

Bαw(�

2) ∩ T = B(�2) ∩ T .

For the next result we need the following Lemma from [16]:

Lemma 3.6.

sup|xn|↘0

∣∣∣∣ ∞∑n=1

anxn

∣∣∣∣( ∞∑n=1

|xn|2) 1

2

= sup|xn|↘0

∞∑n=1

|an| |xn|( ∞∑n=1

|xn|2) 1

2

≈

⎛⎝ ∞∑n=1

(1

n

n∑k=1

|ak|)2⎞⎠ 1

2

,


Theorem 3.7. Let A =

⎛⎜⎜⎜⎜⎝a0 a1 a2 a3 ...0 a0 a1 a2 ...0 0 a0 a1 ...0 0 0 a0 ...... ... ... ... ...

⎞⎟⎟⎟⎟⎠ be an upper triangular

positive Toeplitz matrix. Then A ∈ B (�2) if and only if the sublinearoperator TA is bounded from �2 into �2, where

TA (b) (j) =1

j

j∑m=0

|(a ∗ b) (m)| , TA (b) (0) = |a0b0|

and (a ∗ b)(m) =∑

k+l=m

akbl, a = (ak)k≥0, b = (bk)k≥0 ∈ �2.

Our next Theorem is a characterization of diagonal matrices, whichare Schur multipliers from Bα

w(�2) to Bα

w(�2).


Theorem 3.8. For M = M0 given by the sequence m = (mk)k≥1 wehave that

M ∈ M(Bαw(�

2), Bαw(�

2))

if and only if m = (mk)k≥1 ∈ �∞.

Moreover, in the case of Toeplitz matrices we have the followingembeddings:

Theorem 3.9. It yields that

M(Bαw(�

2), Bαw(�

2)) ∩ T ⊆ M(�2) ∩ T .

4. Proofs of the main results

Proof of Theorem 3.1. We will compute sup |xn|nα ↘

‖Ax‖�2‖x‖�2

. Since

sup|xn|nα ↘

‖Ax‖�2‖x‖�2

= sup|xn|nα ↘

(∑∞

n=1 |an|2|xn|2)12

(∑∞

n=1 |xn|2)12

we have that for yn = xn

nα ,

sup|xn|nα ↘

‖Ax‖�2‖x‖�2

= sup|yn|↘

(∑∞

n=1 |anynnα|2)12

(∑∞

n=1 |ynnα|2)12

.

Moreover, we consider

S := sup|yn|↘

(∑∞

n=1 |anynnα|2)12

(∑∞

n=1 |ynnα|2)12

.

Applying now Theorem 2.9 (i) for p = 1 and f(n) = |yn|2, v(n) =n2α|an|2, w(n) = n2α we obtain that, for every nonnegative integer n,

V (n) =n∑

k=1

v(k) =n∑

k=1

k2α|ak|2

and

W (n) =n∑

k=1

w(k) =n∑

k=1

k2α.


It follows that

S = supn≥1

(∑n

k=1 k2α|ak|2∑n

k=1 k2α

) 12

and


n≥1

(∑n

k=1 k2α|ak|2∑n

k=1 k2α

) 12

.

The proof is complete. �Proof of Theorem 3.4. Let us take A ∈ B(�2) and B ∈ Bα

w(�2). Then

we have that∑j

|∑k

ajkbjkxk|2 ≤∑j

(∑k

|ajk||bjk||xk|)2

(4)

≤∑j

(∑k

|ajk|2)∑

k

(|bjk|2|xk|2

)≤ sup

j

(∑k

|ajk|2)∑

j

(∑k

|bjk|2|xk|2).

To estimate the last term in (4) we consider the Rademacher func-tions rk(t) = sgn sin(2nπt) on [0, 1], for k ≥ 1 and we use the equality(see [12] p. 126) ∑

k

|yk|2 =∫ 1

0

|∑k

ykrk(t)|2dt.

It follows that∑j

(∑k

|bjk|2|xk|2)

=∑j

∫ 1

0

|∑k

bjkxkrk(t)|2dt

≤ esssupt∈[0,1]∑j

|∑k

bjkxkrk(t)|2

≤ ‖B‖2Bαw(�2)‖x‖22.

Thus, we obtain that


‖A ∗B‖Bαw(�2) ≤ ‖A‖2,∞ · ‖B‖Bα

w(�2).

Since ‖A‖2,∞ ≤ ‖A‖B(�2) the proof is complete.�

Proof of Theorem 3.5. It follows from the definition that

B(�2) ⊂ Bαw(�

2) ⊂ Bw(�2).

From Theorem 9 in [16] we have that the Toeplitz matrices from B(�2)coincides with Toeplitz matrices from Bw(�

2). Thus, the Toeplitz ma-trices from B(�2) coincides with those from Bα

w(�2). The proof is com-

plete.�

Proof. From Theorem 3.5 we have that A ∈ B(�2) if and only if A ∈

Bαw (�2) i.e. that

(∞∑j=0

ajxj+k

)k≥0

∈ �2 for any x ∈ �2 with|xj |nα ↘ 0.

This, in its turn, is equivalent to that

sup||b||�2≤1

∣∣∣∣∣∞∑k=0

( ∞∑j=0

ajxj+k

)bk

∣∣∣∣∣ < ∞

for all x ∈ �2,|xj |nα ↘ 0. We define the convolution cl =

l∑k=0

al−kbk and

it yields that

∞∑k=0

( ∞∑j=0

ajxj+k

)bk =

∞∑l=0

xlcl.


From Lemma 3.6 it follows that

sup|xl|↘0

∣∣∣∣∞∑l=0

xlcl

∣∣∣∣(∑l

|xl|2) 1

2

= sup|xl|↘0

∞∑l=0

|xl| |cl|(∑l

|xl|2) 1

2

≈ |a0b0|2 +∞∑j=1

(1

j

j∑m=0

∣∣∣∣∣m∑k=0

am−kbk

∣∣∣∣∣)2

= |a0b0|2 +∞∑j=1

(1

j

j∑m=0

|(a ∗ b) (m)|)2

.

Thus the proof is complete. �

Proof of Theorem 3.8. Suppose that M ∈ M(Bαw(�

2), Bαw(�

2)). Then

M ∗ A ∈ Bαw(�

2),

for every A ∈ Bαw(�

2). In particular,

M ∗ A0 ∈ Bαw(�

2),

for A0 = diag(a), with a = (ak)k≥1. We have that A0x ∈ �2 for every

x ∈ �2 with |xk|kα

↘ 0.Since (M ∗ A0)x ∈ �2 it implies that m = (mk)k≥1 ∈ �∞.Conversely, assume that m = (mk)k≥1 ∈ �∞. Observe first that

if A = (ajk)jk is an infinite matrix and A0 is the matrix defined asA0 = diag(akk), then

‖A0‖Bαw(�2) ≤ ‖A‖Bα

w(�2).


Indeed, using the same arguments as in proof of Theorem 3.4, whererk, k ≥ 1, are the Rademacher functions we have that

‖A0x‖22 =∑k

|akkxk|2

≤∑j

∑k

|ajkxk|2

=∑j

∫ 1

0

∣∣∣∣∣∑k

ajkxkrk(t)

∣∣∣∣∣2

dt

≤ esssupt∈[0,1]∑j

∣∣∣∣∣∑k

ajkxkrk(t)

∣∣∣∣∣2

≤ ‖A‖2Bαw)(�2)‖x‖22,

for every x = (xk)k≥1 ∈ �2 with |xk|kα

↘ 0.Let A be an infinite matrix such that A ∈ Bα

w(�2). Then

‖(M ∗ A)x‖22 = ‖(M ∗ A0)x‖22= ‖MA0x‖22≤ ‖m‖2∞ · ‖A0x‖22≤ ‖m‖2∞ · ‖A0‖2Bα

w(�2) · ‖x‖22≤ ‖m‖2∞ · ‖A‖2Bα

w(�2) · ‖x‖22,

for every x = (xk)k≥1 ∈ �2 such that |xk|kα

↘ 0. The proof is complete.�

Proof of Theorem 3.9. Let M = (mjk)jk be a Toeplitz matrix of theform

(5) mjk = cj−k, j, k = 0, 1, 2, · · ·such that M ∈ M(Bα

w(�2), Bα

w(�2)). Using the same techniques as

in [16], applying Theorem 3.5 we have that there exists a bounded,


complex, Borel measure μ on the circle group T with

μ(n) = cn for n = 0,±1,±2, · · · .Moreover,

‖μ‖ ≤ ‖M‖M(Bαw(�2),Bα

w(�2)).

Applying now Theorem 8.1 in [7] we have that the Toeplitz matrix (5)is in M(�2). The proof is complete. �

References

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[2] C. Badea and V. Paulsen, Schur multipliers and operator-valued Foguel-Hankel operators, Indiana Univ. Math. J. 50(2001), 1509–1522.


[4] S. Barza, V. D. Lie and N. Popa, Approximation of infinite matrices bymatriceal Haar polynomials, Ark. Mat. 43(2005), no. 2, 251–269.

[5] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press,Boston, 1988.

[6] G. Bennett, Factorizing the Classical Inequalities, Memoirs of the AmericanMathematical Society, Number 576, 1996.

[7] G. Bennett, Schur multipliers, Duke Math. J. 44(1977), 603–639.[8] M. J. Carro and J. Soria, Boundedness of some integral operators, Canad.

J. Math. 45(1993), 1155–1166.[9] M. J. Carro and J. Soria, Weighted Lorentz spaces and the Hardy operator,

J. Funct. Anal. 112(1993), 480–494.[10] M. J. Carro, J. A. Raposo and J. Soria, Recent Developments in the The-

ory of Lorentz Spaces and Weighted Inequalities, Memoirs of the AmericanMathematical Society, Number 877, 2007.

[11] A. A. Jagers, A note on Cesaro sequence spaces, Nieuw Arch. voor Wiskunde3(1974), 113–124.

[12] A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type, WorldScientific Publishing Co., Singapore/ New Jersey/ London/ Hong Kong,2003.

[13] A. Kufner, L. Maligranda and L. E. Persson, The Hardy Inequality. About itsHistory and Some Related Results, Vydavatelsky Servis Publishing House,Pilsen, 2007.


[14] S. Kwapien and A. Pelczynski, The main triangle projection in matrix spacesand its applications, Studia Math. 34(1970), 43–68.

[15] S. Kaczmarz and G. Steinhaus, Theory of orthogonal series (Russian), Go-sudarstv. Izdat. Fiz.-Mat. Lit., Moscow 1958.

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[18] V. Paulsen, Completely bounded maps and operator algebras, Cambridgestudies in advanced mathematics 78, Cambridge University Press, 2002.

[19] Chr. Pommerenke, Univalent Functions, Hubert, Gottingen 1975.[20] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces,

Studia Math. 96(1990), 145–158.[21] J. Schur, Bemerkungen zur theorie der beschrankaten bilinearformen mit

unendlich vielen verandlichen, J. Reine Angew. Math. 140(1911), 1–28.[22] H. S. Shapiro and A. L. Shields, On some interpolation problems for analytic

functions, Amer. J. Math. 83(1961), 513–532.[23] H. S. Shapiro and A. L. Shields, On the zeros of functions with finite Dirich-

let integral and some related function spaces, Math. Zeit. 80(1962), 217–229.[24] V. Stepanov, Integral operators on the cone of monotone functions, Insti-

tute for Applied Mathematics F-E.B. of the USSR Academy of Sciences,Khabarovsk, 1991.

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[26] G. P. H. Styan, Hadamard products and multivariate statistical analysis,Linear Algebra 6(1973), 217–240.

[27] A. L. Shields and J. L. Wallen, The commutants of certain Hilbert spaceoperators, Indiana Univ. Math. J. 20(1971), 777–799.

[28] K. Zhu, Operator theory on Banach function spaces, Marcel Dekker, NewYork, 1990.


Appendix

Proof of Theorem 2.2. By virtute of the monotone convergence theo-rem, we may assume that g is supported in a compact subset of (0,∞)with

∫∞0

g(x)dx = 1 and that 0 <∫∞0

v(t)dt < ∞, for all x > 0. Set

f(x) =

(∫ ∞

x

g(t)∫ t

0v(s)ds

dt

)p′−1

, 0 < x < ∞.

Then f is bounded and nonincreasing on (0,∞), and an integrationby parts shows that

∫ ∞

0

f(x)pv(x)dx = f(x)p∫ x

0

v(t)dt

∣∣∣∣∞0

−

−∫ ∞

0

pf(x)p−1f ′(x)(∫ x

0

v(t)dt

)dx

= p′∫ ∞

0

f(x)g(x)dx.

Thus the supremum on the left side of (1) is at least

(6) C

(∫ ∞

0

f(x)g(x)dx

) 1p′.


Now let xj satisfy∫ xj

0g(x)dx = 2j, −∞ < j ≤ 0. Then

∫ ∞

0

f(x)g(x)dx =0∑

j=−∞

∫ xj

xj−1

(∫ ∞

x

g(t)∫ t0v(s)ds

dt

)p′−1

g(x)dx ≥

≥−1∑

j=−∞

(∫ xj+1

xj

g(t)dt

)p′−1(∫ xj+1

0

v(t)dt

)1−p′(∫ xj

xj−1

g(t)dt

)≥

≥ C−2∑

j=−∞

(∫ xj+2

0

g(t)dt

)p′−1(∫ xj+1

0

v(t)dt

)1−p′(∫ xj+2

xj+1

g(t)dt

)≥

≥ C

∫ ∞

0

(∫ x

0

g(t)dt

)p′−1(∫ x

0

v(t)dt

)1−p′

g(x)dx

since∫ xj

0g(t)dt = 2j. Using this inequality in (6), we conclude that

(7) supf≥0

f↘

∫∞0

f(x)g(x)dx

(∫∞0

f(x)pv(x)dx)1p

≥

≥ C

[∫ ∞

0

(∫ x

0

g(t)dt

)p′−1(∫ x

0

v(t)dt

)1−p′

g(x)dx

] 1p′

.

However, integration by parts yields

(8)

∫ ∞

0

(∫ x

0

g(t)dt

)p′−1(∫ x

0

v(t)dt

)1−p′

g(x)dx =

=1

p′

(∫ x

0

g(t)dt

)p′−1(∫ x

0

v(t)dt

)1−p′∣∣∣∣∣∞

0

+

+1

p

∫ ∞

0

(∫ x

0

g(t)dt

)p′ (∫ x

0

v(t)dt

)−p′

v(x)dx =

=1

p

∫ ∞

0

(∫ x

0

g(t)dt

)p′v(x)

(∫ x

0v(t)dt)p′

dx+


+1

p′

(∫ ∞

0

g(t)dt

)p′ (∫ ∞

0

v(t)dt

)1−p′

,

where the second term above is zero if∫∞0

v(t)dt = ∞ (recall that∫ x

0g(t)dt = 1 for large x). Now, the inequality ”≥” from (1) can be

obtained from (7) and (8). For the reverse inequality we give a prooffrom [8]. Set

h(t) =

(∫ ∞

t

(∫ x

0

g(s)ds

)p′−1(∫ x

0

v(s)ds

)−p′

v(x)dx+

+

(∫∞0

g(s)ds∫∞0

v(s)ds

)p′−1) 1

p′

.

Then ∫ ∞

0

f(x)g(x)dx =

∫ ∞

0

f(x)g(x)h(x)h(x)−1dx ≤

≤(∫ ∞

0

f p(x)h−p(x)g(x)dx

) 1p(∫ ∞

0

hp′(x)g(x)dx

) 1p′.

Applying the Fubini theorem to the last factor in the previous inequal-ity we obtain the right hand side of (1). For the first factor, we observethat

h(x)−p ≤(∫ x

0

g(t)dt

)−1(∫ x

0

v(t)dt

)and, thus, by Lemma 2.1 we have that∫ ∞

0

f p(x)h−p(x)g(x)dx = p

∫ ∞

0

yp−1

∫ μf (y)

0

h−p(x)g(x)dxdy.

Integrating by parts the inner integral and erasing the negative termsone has that the previous expression can be bounded, up to multiplica-tive constants, by∫ ∞

0

yp−1

(∫ μf (y)

0

g(x)dx

)h−p(μf (y))dy ≤


≤∫ ∞

0

yp−1

(∫ μf (y)

0

v(x)dx

)dy ≈

∫ ∞

0

f p(x)v(x)dx.


Proof of Theorem 2.9. (i) is obtained applying Theorem 2.8 with p1 =1, p0 = p, X = Y = N∗, T = Id to the regular class L of decreasingsequences in N∗.(ii) can be deduced from Theorem 2.2 by observing that

(9) S = supf↘

∫∞0

f(t)v(t)dt

(∫∞0

f p(t)w(t)dt)1p

.

To see this, note that if f = (f(n))n is a decreasing sequence in N∗

and we define

f =∞∑n=0

f(n)χ[n,n+1) ∈ Mdec(R+).

It is obvious that∑∞n=0 f(n)v(n)

(∑∞

n=0 f(n)pw(n))

1p

=

∫∞0

f(t)v(t)dt

(∫∞0

f p(t)w(t)dt)1p

.

Therefore S is less than or equal to the second member in (9). Onthe other hand, if g ≥ 0 is a decreasing function in R+ and we define

the decreasing sequence f(n) = (∫ n+1

ngp(s)ds)

1p , n = 0, 1, ... then we

obtain that ∫ ∞

0

gp(t)w(t)dt =∑n

f(n)pw(n),


while, by Holder’s inequality, we have that∫ ∞

0

g(t)v(t)dt =∑n

v(n)

∫ n+1

n

g(t)dt

≤∑n

v(n)

(∫ n+1

n

gp(t)dt

) 1p

=∑n

v(n)f(n).

Hence, ∫∞0

g(t)v(t)dt(∫∞0

gp(t)w(t)dt) 1

p

≤∑∞

n=0 f(n)v(n)

(∑∞

n=0 f(n)pw(n))

1p

≤ S.

Thus (9) is proved and (ii) follows by applying Theorem 2.2. The proofis complete. �



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