Matrix Spaces and Schur Multipliers: Matriceal Harmonic Analysis

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MATRIX SPACES AND

SCHUR MULTIPLIERS Matriceal Harmonic Analysis

May 2, 2013

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MATRIX SPACES AND

SCHUR MULTIPLIERS Matriceal Harmonic Analysis

Lars-Erik Persson Luleå University of Technology, Sweden & Narvik University College, Norway

Nicolae Popa “Simion Stoilov” Institute of Mathematics, Romanian Academy, Romania & Technical University “Petrol si Gaze”, Romania

World Scientific NEW JERSEY

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LONDON

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TA I P E I

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Persson, Lars-Erik, 1944– author. Matrix spaces and Schur multipliers : matriceal harmonic analysis / by Lars-Erik Persson (Luleå University of Technology, Sweden & Narvik University College, Norway) & Nicolae Popa (“Simion Stoilov” Institute of Mathematics, Romanian Academy, Romania & Technical University “Petrol si Gaze”, Romania). pages cm Includes bibliographical references and index. ISBN 978-9814546775 (alk. paper) 1. Matrices. 2. Algebraic spaces. 3. Schur multiplier. I. Popa, Nicolae, author. II. Title. QA188.P43 2014 512.9'434--dc23 2013037182

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

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Printed in Singapore

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Preface

In the last two centuries the Fourier analysis, known also as harmonic analysis, experienced a strong development. Roughly speaking it consists mainly in the study of properties of periodical functions connected to their Fourier coeﬃcients. As examples of such properties we may consider the beautiful Fejer’s theory [31] about the convergence of the Fourier series to a function with respect to its Cesaro means, or the well-known Fej´er-HardyLittlewood inequality [96]. On the other hand many mathematicians have observed that there is a linear bijective correspondence between the periodical functions f on the torus T and its corresponding Toeplitz matrix, that is the inﬁnite matrix having the nth Fourier coeﬃcient of f on the nth diagonal submatrix parallel to the main diagonal, numbered by 0. (See for instance [94], [11].) For instance we began to think about this topic after reading the paper of J. Arazy [1], where a short remark about the analogy between Fourier coeﬃcients and diagonal submatrices was made. In fact, it appears that periodical functions on the torus are particular cases of inﬁnite matrices and it is tempting to develop a matrix version of the classical harmonic analysis, where instead of nth Fourier coeﬃcient of a periodical function you have to consider the nth diagonal of a given inﬁnite matrix. The present volume is dedicated to this goal, namely to formulate and prove some statements in this new matrix version of classical harmonic analysis in terms of “diagonal” submatrices of an inﬁnite matrix, which are analogous of the well-known statements in harmonic analysis. The current knowledge is presented in a uniﬁed way and also some new results are included to complete the picture to a fairly nice theory we hereby call matriceal harmonic analysis. Now we brieﬂy describe some of the most motivating results of this vii

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book. After an introduction and a presentation of principal notions, in Chapter 2 we present some results from the master dissertation of Victor Lie (see [50]). The main idea of this work is to consider an inﬁnite matrix as a sequence of functions (Lk )k≥1 and to exploit this interpretation in order to obtain a useful formula for the operator norm of an inﬁnite matrix, namely: ||A||B(2 ) = sup ||VLB ,h ||∞ , ||h||2 ≤1

where VLB ,h (x) =

∞

1/2 |(Lk ∗ h) (x)|

2

,

∀x ∈ [0, 1] and ∀h ∈ H02 ([0, 1]).

k=1

By using this formula we give new proofs of some classical results of inﬁnite matrix theory. For instance, it is possible to prove, with the same method, some scattered facts like: • For a Toeplitz matrix A, A ∈ B(2 ) if and only if fA ∈ L∞ ([0, 1]) (see Theorem 3.2), • (Bennett’s Theorem) For a Toeplitz matrix A, A ∈ M (2 ) if and only fA is a bounded Borel measure on [0, 1] (see Theorem 3.3), • (Nehari’s Theorem) For a Hankel matrix A, A ∈ B(2 ) if and only if gA ∈ BM O (see Theorem 3.5), • (Theorem of Kwapien-Pelczynski) If Pn is the main triangle projection of order n, then sup ||A||B(2 ) ≤1

||Pn (A)||B(2 ) = O(log n)

n→∞

(see Theorem 3.8). In Chapter 3 we give a matrix version of Fejer’s theory, introducing a subclass of inﬁnite matrices representing bounded linear operators on 2 , namely the class of continuous matrices C(2 ). This class consists of those inﬁnite matrices, which are approximable in the operator norm by matrices of ﬁnitely band-type. In particular, we present the results in [17], which were extended later in [48] and [46]. Just as one possible application of the matrix version of Fourier analysis we mention the approximation of an inﬁnite matrix in various ways. For

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instance the following natural question arise: Which inﬁnite matrices can be approximated in the operator norm by its ﬁnite-type band submatrices? In particular, an answer to this question, which extends the well-known Jordan’s theorem, is given in this chapter. Another type of approximation of inﬁnite matrices by a special type of matriceal polynomials is also given in Chapter 3 and it extends a well-known theorem of A. Haar from 1910 (see Theorem C on page 49): • Let A = (alk )l≥1, k∈Z be a matrix belonging to C(2 ) such that all def

sequences ak = (alk )l≥1 , k ∈ Z, located on the kth diagonal, belong to the class ms. Then, for any > 0 there is an n ∈ N∗ and sequences αk ∈ ms, k ∈ {0, . . . , n − 1}, such that ||A −

n−1

αk Hk ||B(2 ) < .

k=0

Here ms is a special linear subspace of the space of all bounded sequences ∞ , and means a product between two matrices, which extends the usual product of a scalar and a function. Next we mention that in 1983 A. Shields [84] stated and proved the following beautiful theorem: • Let M ∈ C1 have upper triangular form with respect to the orthonormal basis {en } (n = 1, 2, . . . ) of 2 . Then ∞ k |M (j, k)| ≤ π||M ||T1 , 1 +k−j j=1

k=1

with equality only when M = 0. Here T1 means the subspace of all trace class inﬁnite upper triangular matrices. The above result is analogous to the well-known Hardy-Littlewood-Fejer inequality. We present and discuss this theorem and some other related results in Chapter 4.

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For instance we present a new characterization of the elements of the space T1 with respect to the sequence of a special linear combination of their diagonals, namely: • Let A ∈ B(2 ) be an upper triangular matrix. Then the following assertions are equivalent: a) A ∈ T1 ;

b) sup n

n 1 1 ||sj A|| < ∞; an j=0 j + 1

c) sup ||Pn A|| < ∞. n

Here Pn A =

n n 1 1 1 sj A, where an = (n = 0, 1, 2, . . . ) an j=0 j + 1 j + 1 j=0

and sj A =

j k=0

Ak .

It is important to mention that in Chapter 4 and also in the sequel, often the obtained matriceal results are only analogous to but do not extend the known results from harmonic analysis. This happens because most of the results from these chapters refer to matrices connected to Schatten classes of matrices and consequently their proofs cannot be applied to Toeplitz matrices. Another topic which was touched some years ago is the study of matrix version of Hankel operators. Some especially interesting results about these Hankel operators were obtained by S. Power in [77]. For instance he showed there the following matrix version of Nehari’s theorem: • Let Φ be an inﬁnite matrix such that ΦA ∈ C2 for all ﬁnite bandtype matrices A. Then the following statements are equivalent: (a) HΦ is a bounded linear operator on T2 . (b) There is Ψ ∈ B(2 ) such that Ψk = Φk for all k < 0. (c) P− Φ ∈ BM OF (2 ).

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See Chapter 5 for all unexplained notions and notations in this theorem. In Chapter 5 we give a different and in our opinion more natural proof of this result and investigate this topic further. For example, we derive a sufficient condition in order that a matrix version of a Hankel operator to be nuclear. For a particular class of its symbol this condition is even necessary. A class of Banach spaces of analytic functions, which has received great attention in the last two decades is the class of Bergman spaces. See e.g. [94], [28] and the references given there. We investigate some properties of the class of Bergman-Schatten spaces in Chapter 6. For instance we present and discuss some inequalities valid in Bergman-Schatten spaces (see [76]) e.g. the following: • (Hausdorff-Young Theorem) For 1 ≤ p ≤ ∞, let q be the conjugate index, i.e. p1 + 1q = 1 (for p = 1 we have q = ∞). P 1/q ∞ q (i) If 1 ≤ p ≤ 2, then A ∈ Tp implies that ≤ n=0 ||An ||Tp

||A||Tp . (ii) If 2 ≤ p ≤ ∞, then {||An ||Tp } ∈ `q implies that ||A||Tp ≤ P 1/q ∞ q ||A || . n n=0 Tp

Another important result, which appears in Chapter 6 is a matriceal analogue of a result obtained by Mateljevic and Pavlovic [61] in 1984, (see [62]): • Let A be an upper triangular matrix. Then A ∈ L1a (D, `2 ) if and only if ∞ X ||σn (A)||C1 < ∞. (n + 1)2 n=1

Moreover, a matrix version of the usual Bloch space is introduced in Chapter 7. See [72]. The interest of this space consists mainly in the fact that it satisfies the equality B(D, `2 ) = H 1 (`2 ), BM OA(`2 ) ,

where B(D, `2 ) means the matriceal Bloch space, H 1 (`2 ), BM OA(`2 ) stand for the matrix version of the Hardy space H 1 , respectively, for the matrix version of the space BM OA, and (X, Y ) is the space of all Schur multipliers between spaces of infinite matrices X and Y.

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This equation is the matrix analogue of a result valid for Fourier multipliers of periodical functions proved in 1990 by Mateljevic and Pavlovic [60]. Moreover, in 1995 O. Blasco [12] proved that a vector valued version of the equality of Mateljevic and Pavlovic is valid, generally speaking, only for functions with values in Hilbert spaces. Consequently, this version of the above equality deserves some special attention. The equality is proved in Chapter 8. This chapter is dedicated to a very important tool in the theory, namely Schur multipliers, which represents the matrix version of classical Fourier multipliers. We also present, prove and discuss some other results concerning Schur multipliers between Banach spaces of infinite upper triangular matrices. We mention just the following matrix version of a well-known result of Paley: • If A =

P

An ∈ H 1 (`2 ), then P (A) :=

P∞

k=1

A2k ∈ H 2 (`2 ).

Here H 1 (`2 ) is a matrix version of Hardy space introduced in Chapter 4 and H 2 (`2 ) is the space of all upper triangular Hilbert-Schmidt matrices. Aknowledgement The second author was partially supported by the CNCSIS grant ID-PCE 1905/2008. Moreover, we are both grateful to Lule˚ a University of Technology for financial support for research visits to be able to finalize this book. We are also very grateful to Dr. Niklas Grip for helping and supporting us in both professional and practical ways. Special thanks are due to our colleagues Dr. A. N. Marcoci and Dr. L. G. Marcoci who read carefully a preliminary version of this book and made valuable suggestions and remarks. Finally, we emphasize that writing this volume had not been possible without the existence of an atmosphere favorable to scientific activity, the atmosphere existing in the Institute of Mathematics ”Simion Stoilow” of the Romanian Academy and at the Department of Mathematics at Lule˚ a University of Technology. The cover images of the midnight sun were taken by Elin Persson from the north harbour of Lule˚ a, Sweden, at midnight of 16th June 2013. These images reflect the empowering senses the authors felt when this special book was finalized under the skies of this famous light. Lule˚ a, June 2013, under the influence of the magic midnight sun atmosphere close to the Polar Circle. Lars-Erik Persson and Nicolae Popa

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Contents

Preface 1.

Introduction 1.1

2.

1

Preliminary notions and notations 1.1.1 Infinite matrices . . . . . . 1.1.2 Analytic functions on disk 1.1.3 Miscellaneous . . . . . . . 1.1.4 The Bergman metric . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

Integral operators in infinite matrix theory 2.1 2.2 2.3

3.

vii

9

Periodical integral operators . . . . . . . . . . . . . . . . . Nonperiodical integral operators . . . . . . . . . . . . . . Some applications of integral operators in the classical theory of infinite matrices . . . . . . . . . . . . . . . . . . . . 2.3.1 The characterization of Toeplitz matrices . . . . . 2.3.2 The characterization of Hankel matrices . . . . . . 2.3.3 The main triangle projection . . . . . . . . . . . . 2.3.4 B(`2 ) is a Banach algebra under the Schur product

Matrix versions of spaces of periodical functions 3.1 3.2 3.3 3.4

Preliminaries . . . . . . . . . . . . . . . . . . . . . . Some properties of the space C(`2 ) . . . . . . . . . . Another characterization of the space C(`2 ) and related results . . . . . . . . . . . . . . . . . . . . . . A matrix version for functions of bounded variation xiii

1 1 4 5 7

9 17 18 18 24 27 30 33

. . . . . .

34 34

. . . . . .

36 41

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xiv

3.6

4.5 4.6

5.3

7.

109

First properties of BM OA(`2 ) space . . . . . . . . . . . . 109 Another matrix version of BM O and matriceal Hankel operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Nuclear Hankel operators and the space M1,2 . . . . . . . 119 Schatten class version of Bergman spaces . . . . . Some inequalities in Bergman-Schatten classes . . A characterization of the Bergman-Schatten space Usual multipliers in Bergman-Schatten spaces . . .

A matrix version of Bloch spaces 7.1 7.2

8.

First properties of matriceal Hardy space . . . . . . . . . 65 Hardy-Schatten spaces . . . . . . . . . . . . . . . . . . . . 69 An analogue of the Hardy inequality in T1 . . . . . . . . . 75 The Hardy inequality for matrix-valued analytic functions 79 p 4.4.1 Vector-valued Hardy spaces HX . . . . . . . . . . 79 p 4.4.2 (H − `q )-multipliers and induced operators for vector-valued functions . . . . . . . . . . . . . . . 80 A characterization of the space T1 . . . . . . . . . . . . . 97 An extension of Shields’s inequality . . . . . . . . . . . . . 101

Matrix version of Bergman spaces 6.1 6.2 6.3 6.4

44 45 50 56 61 65

The matrix version of BM OA 5.1 5.2

6.

Approximation of infinite matrices by matriceal Haar polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . 3.5.2 About the space ms . . . . . . . . . . . . . . . . . 3.5.3 Extension of Haar’s theorem . . . . . . . . . . . . Lipschitz spaces of matrices; a characterization . . . . . .

Matrix versions of Hardy spaces 4.1 4.2 4.3 4.4

5.

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3.5

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121 . . . .

. . . .

. . . .

. . . .

121 132 136 141 149

Elementary properties of Bloch matrices . . . . . . . . . . 149 Matrix version of little Bloch space . . . . . . . . . . . . . 161

Schur multipliers on analytic matrix spaces

175

Bibliography

185

Index

191

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Chapter 1

Introduction

1.1

Preliminary notions and notations

In this section we collect some notions and facts of the theory of inﬁnite matrices, the theory of analytic functions on the disk and the circle, of vector-valued integration theory and of geometry of the disk etc. 1.1.1

Inﬁnite matrices

For an inﬁnite matrix A = (aij ), and an integer k we denote by Ak the matrix whose entries ai,j are given by

ai,j if j − i = k, . ai,j = 0 otherwise Then Ak will be called the kth-diagonal matrix associated to A. Sometimes we use also the notation a(i, j) for the entries of the matrix A. An important notion in the theory of matrices is the Schur product. Let A = (aij )i,j and B = (bij )i,j be two inﬁnite matrices. Then the Schur product C = (cij )i,j of A and B, denoted by A ∗ B, has the entries cij = aij bij for all i, j ∈ N. An inﬁnite matrix A such that A ∗ B ∈ Y for all B ∈ X, where X, Y are Banach spaces of inﬁnite matrices, is called a Schur multiplier from X into Y , and the space of all Schur multipliers from X into Y, endowed with the natural norm ||A||(X,Y ) =

sup ||B||X ≤1

is denoted by (X, Y ). 1

||A ∗ B||Y

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In the case X = Y = B(2 ), where B(2 ) is the space of all linear and bounded operators on 2 , the space (X, Y ) is denoted by M (2 ) (an explanation of this notation is given later in this section) and a matrix A ∈ M (2 ) is simply called a Schur multiplier. We consider on the interval [0, 1) the Lebesgue measurable inﬁnite matrix valued functions A(r). These functions may be regarded as inﬁnite matrix-valued functions deﬁned on the unit disk D using the correspon∞ ikt , where Ak (r) is the kthdence A(r) → fA (r, t) = k=−∞ Ak (r)e diagonal of the matrix A(r), the preceding sum is a formal one and t belongs to the torus T. We may consider fA (r, t), or fA (z), with z = reit , as a matrix valued function, or distribution, or just a formal series. Such a matrix A(r) is called an analytic matrix if there exists an upper triangular inﬁnite matrix A such that, for all r ∈ [0, 1), we have Ak (r) = Ak r k , for all k ∈ Z. In what follows we identify the analytic matrices A(r) with their corresponding upper triangular matrices A and call the latter also as analytic matrices. A special class of inﬁnite matrices is considered often in this book, namely the class of Toeplitz matrices. Let A = (aij )i,j≥1 be an inﬁnite matrix. If there is a sequence of complex numbers (ak )+∞ k=−∞ , such that aij = aj−i for all i, j ∈ N, then A is called a Toeplitz matrix. For simplicity we can identify a Toeplitz matrix with A = (ak )+∞ k=−∞ , and the class of all Toeplitz matrices is denoted by T . G. Bennett proved in 1977 the following interesting result (see Theorem 8.1-[11]) about Schur multipliers: Bennett’s Theorem The Toeplitz matrix M = (cj−k )j,k , where (cn )n∈Z is a sequence of complex numbers, is a Schur multiplier if, and only if, there exists a (bounded, complex, Borel) measure μ on (the circle group) T with μ (n) = cn

for n = 0, ±1, ±2, . . . .

Moreover, we then have ||M ||M (2 ) = ||μ||. Bennett’s theorem justiﬁes the notation M (2 ) since for a Toeplitz matrix the notions of Schur multiplier and Borel measure on the torus coincide.

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In the sequel we give some results about compact operators on the Hilbert space 2 . (See for instance [94].) For example the following decomposition formula is known. Schmidt Theorem If T is a self-adjoint compact operator on a Hilbert space H, then there exists a sequence of real numbers {λn } tending to 0 and there also exists an orthonormal set {en } in H such that ∞ Tx = λn (x, en )en n=1

for all x ∈ H. If the operator T is compact, but not necessarily self-adjoint, then we ﬁrst consider the polar decomposition T = V |T |, where |T | = (T ∗ T )1/2 is positive (and hence self-adjoint) and compact. By the above theorem, there is an orthonormal set {en } in H such that |T |x = λn (x, en )en , x ∈ H, n

where {λn } is a nonincreasing sequence of nonnegative numbers tending to 0. Let σn = V en for each n; then {σn } is still an orthonormal set and we have that λn (x, en )σn , x ∈ H. Tx = n

This is called the canonical decomposition of a compact operator T. The non-increasingly arranged sequence {λn } is called the sequence of singular values of T. The number λn is called the nth singular value of T. Now we introduce the Schatten class operators. Given 0 < p < ∞ and a separable Hilbert space H, we deﬁne the Schatten p-class of H, denoted by Cp (H) or simply Cp , to be the space of all compact operators T on H with its singular value sequence {λn } belonging to p (the p-summable sequence space). We are mainly concerned with the range 1 ≤ p < ∞. In this case, Cp is a Banach space with the norm

1/p p |λn | . ||T ||p = n

We recall that we can identify the operators with its corresponding matrices; so we may consider Cp as being spaces of matrices. C1 is also called the trace class of H, and C2 is usually called the HilbertSchmidt class.

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1.1.2

Analytic functions on disk

In this subsection we introduce the deﬁnitions of some important spaces of analytic functions on the disk. First of all we consider the classical Hardy space of functions on the disk. Let 0 < p ≤ ∞. We say that an analytic function f : D → C belongs to the Hardy space H p , for 1 ≤ p < ∞ if and only if ||f ||H p := 1/p 2π 1 iθ p |f (re )| dθ < ∞ and f ∈ H ∞ if and only if sup0≤r<1 2π 0 ||f ||H ∞ := sup0≤θ≤2π; 0≤r<1 |f (reiθ )| < ∞. The Hardy space was intensely studied in the last forty years. Duren’s book [23] and Garnett’s book [33] are frequently cited as references to H 1 . We mention only that Hardy and Littlewood proved in [39] the following inequality for functions f ∈ H 1 : (HL) 0

1

(1 − r)

0

1

|f (re2πiθ )|dθ

2

dr ≤ C||f ||2H 1 ,

where C > 0 is a constant independent of the choice of f. Another analytic function space studied in connection with Hardy space is the space of all analytic functions of bounded mean oscillation denoted by BM OA. One of the equivalent deﬁnitions of this space is as follows (see [33]): BM OA coincides with the space of all analytic functions f on the disk such that the following norm is ﬁnite:

||f ||∗ := sup

λ∈D

|f (z)|2 (1 − |z|2 )(1 − |λ|2 )|1 − λz|−2 dxdy

1/2 .

D

Now we recall the deﬁnition of the classical Bloch space for functions on the disk. The Bloch space B is the space of all analytic functions f : D → C such that ||A||B := sup (1 − |z|2 )|f (z)| + |f (0)| < ∞. z∈D

Another interesting space of analytic functions on the disk is the socalled little Bloch space. The little Bloch space of D, denoted by B0 , is the closed subspace of B consisting of functions f with (1 − |z|2 )f (z) → 0 (|z| → 1− ).

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Introduction

1.1.3

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Miscellaneous

We also need some notions of vector-valued integration theory. We say that a function f : D → B(2 ), is w∗ -measurable if A ◦ f is a Lebesgue measurable function on D for every A ∈ C1 , where C1 is the Schatten class of all operators with trace, and A is considered as a functional on B(2 ). The function f : D → B(2 ) is strongly measurable if it is a norm limit of a sequence of simple functions. (See [30] for more details about vectorvalued measurability.) Then we have the following particular case of Proposition 8.15.3-[30]: Proposition 1.1. Let f : D → B(2 ) be a w∗ -mesurable function. Then the function ||f || : z → ||f (z)||B(2 ) is measurable. Moreover, we state without proof the following theorem, which is a particular case of Theorem 8.18.2 [30] for E = C1 : Theorem 1.2. The topological dual of L1 (D, 2 ) may be identiﬁed with L∞ (D, 2 ) by the duality bilinear map: 1 < A(·), B(·) > := tr (A(s)[B(s)]∗ )2sds, 0

where A(·) ∈ L∞ (D, 2 ), B(·) ∈ L1 (D, 2 ). Here L1 (D, 2 ) and L∞ (D, 2 ) are deﬁned in Section 6.1 as particular cases of Banach spaces of vector-valued functions. We also need the following well-known lemma (see for instance [94]): Lemma 1.3. Let z ∈ D, c ∈ R, t > −1, and (1 − |w|2 )t dA(w). Ic,t (z) = 2+t+c D |1 − zw| Then we have that (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− );

1 1 − |z|2

(|z| → 1− ).

(3) if c = 0, then I0,t (z) ∼ log

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Matrix spaces and Schur multipliers: Matriceal harmonic analysis

Proof. Since t > −1, the integral Ic,t is deﬁned for all z ∈ D. Let λ = 1 (2 + t + c) . If λ is zero or a negative integer, then clearly c < 0 and Ic,t (z) 2 is bounded. If λ is not zero or a negative integer, then ∞ 1 Γ(n + λ) n n z w = (1 − zw)λ n!Γ(λ) n=0

and the rotation invariance of (1 − |w|2 )t dA(w) shows that ∞ (1 − |w|2 )t Γ(n + λ)2 2n dA(w) = |z| (1 − |w|2 )t |w|2n dA(w) 2λ (n!)2 Γ(λ)2 D |1 − zw| D n=0 ∞ Γ(n + λ)2 2n 1 = |z| (1 − r)t r n dr 2 Γ(λ)2 (n!) 0 n=0 =

∞ Γ(n + λ)2 Γ(t + 1)Γ(n + 1) 2n |z| (n!)2 Γ(λ)2 Γ(n + t + 2) n=0

=

∞ Γ(t + 1) Γ(n + λ)2 |z|2n . Γ(λ)2 n=0 n!Γ(n + t + 2)

By Stirling’s formula, Γ(n + λ)2 ∼ nc−1 n!Γ(n + t + 2) Clearly

∞ n=1

(n → ∞).

nc−1 |z|2n is bounded in z for c < 0. If c = 0, then ∞ |z|2n 1 = log . n 1 − |z|2 n=1

If c > 0, then nc−1 |z|2n ∼

1 (1 − |z|2 )c

since ∞ Γ(n + c) 2n 1 |z| = 2 c (1 − |z| ) n!Γ(c) n=0

and Γ(n + c)/n! ∼ nc−1 by Stirling’s formula. This completes the proof of the lemma.

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Introduction

1.1.4

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The Bergman metric

In this section we recall some geometric facts about the disk. More specifically we consider the Bergman metric on D, which will be useful in the study of matriceal Bloch space. The pseudo-hyperbolic metric. Recall that for any z ∈ D, ϕz is the Moebius transformation of D, which interchanges the origin and z, namely z−w φz (w) = , w ∈ D. 1 − zw The pseudo-hyperbolic distance on D is deﬁned by z−w , ρ(z, w) = |ϕz (w)| = 1 − zw

z, w ∈ D.

An important property of the pseudo-hyperbolic distance is that it is Moebius invariant, that is, ρ(ϕ(z), ϕ(w)) = ρ(z, w) for all ϕ ∈ Aut(D), the Moebius group of D, and all z, w ∈ D. The Bergman metric. The Bergman metric on D is given by β(z, w) =

1 + ρ(z, w) 1 log , 2 1 − ρ(z, w)

z, w ∈ D.

The Bergman metric is also Moebius invariant: β(ϕ(z), ϕ(w)) = β(z, w) for all ϕ ∈ Aut(D) and all z, w ∈ D.

Notes Most of the information in this chapter is classical but not so easy to ﬁnd collected in this form elsewhere. The Schur (or Hadamard) product is well known to specialists but there are few monographs which treat this matter. The authors know only the book of G. Pisier [81]. Bennett’s Theorem is also considered as part of folklore (see for example [8]). For an accessible proof see [11] and [8].

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In the next chapter we present another proof of this important result (see Theorem 2.14). More about Schmidt Theorem and Schatten class operators can be found in many excellent books, for instance [34]. Analytic function on the disk are treated also in many books as [68], [33] and [23]. We pay special attention to different spaces of analytic functions on the disk as Hardy space H 1 , which is intensively studied in [23] and [33]. More about the space of analytic functions of bounded mean oscillation BM OA may be found in [33]. A very interesting space is also the Bloch space of analytic functions B. A classical reference is [3]. Proposition 1.1, Theorem 1.2 and related facts about vector-valued integration theory are mainly taken from [30]. Lemma 1.3 may be found in [94]. Moreover, the subsection dedicated to the Bergmann metric is taken from [94].

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Chapter 2

Integral operators in inﬁnite matrix theory

The content of the present chapter is taken from the master dissertation of V. Lie at University of Bucharest under the supervision of the second author (see [50]). We start deﬁning and discussing some important devices we need in what follows in the study of inﬁnite matrices. The main idea is to consider an inﬁnite matrix as a sequence of functions (resp. distributions). This new point of view has the advantage to use the more reﬁned results from function theory in the theory of inﬁnite matrices. For instance in the ﬁrst section we deﬁne the important notions of square function and matriceal operator associated to a matrix. Using these notions we prove the ﬁrst main result, namely Theorem 2.8. In the second section the central result is the non-periodical analogue of Theorem 2.8 (see Proposition 2.2).

2.1

Periodical integral operators

Let the matrix ⎛

b11 ⎜ b21 ⎜ ⎜ .. B=⎜ ⎜. ⎜b ⎝ n1 .. .

b12 b22 .. . bn2 .. .

⎞ b13 . . . b23 . . . ⎟ ⎟ ⎟ .. . ...⎟ ⎟ ∈ B(2 ). bn3 . . . ⎟ ⎠ .. . ...

Since B ∈ B(2 ) it follows that, for all k ∈ N, the sequence (bkj )j≥1 ∈ 9

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2 (N). Therefore we can deﬁne the functions: Lk (B)(t) =

∞

bkj e2πijt ∈ H02 ([0, 1]),

j=1

and L∗k (B)(t) =

∞

bkj e2πijt · e−2πikt = Lk (B)(t)e−2πikt .

j=1

Consequently, to each row k in the matrix B it corresponds a unique function from the Hardy space, H02 ([0, 1]), of all analytic functions h(t) = ∞ 2πikt , ∀t ∈ [0, 1]. This function is denoted by Lk (B), and k=1 xk e Lk (B)(j) = bkj , for all k, j ≥ 1. For brevity we denote in what follows Lk (B) simple by Lk , and L∗k (B) by L∗k . Thus the matrix B can be written as follows: ⎛ 1 ⎞ 1 L1 (t)e−2πit dt 0 L1 (t)e−4πit dt . . . 0 1 ⎜ 1 ⎟ ⎜ 0 L2 (t)e−2πit dt 0 L2 (t)e−4πit dt . . . ⎟ ⎜. ⎟ .. ⎜ ⎟ B = ⎜ .. . ...⎟, ⎜ 1 ⎟ ⎜ Ln (t)e−2πit dt 1 Ln (t)e−4πit dt . . . ⎟ 0 ⎝ 0 ⎠ .. .. . . ... or B= ⎞ ⎛1 1 ∗ 1 ∗ ∗ −2πit −4πit L (t)dt L (t)e dt L (t)e dt . . . 1 1 1 0 01 ∗ 01 ∗ ⎟ ⎜ 1 ∗ L2 (t)dt L2 (t)e−2πit dt ...⎟ ⎜ 0 L2 (t)e2πit dt 0 0 ⎟ ⎜. .. ⎟ ⎜. ⎜. . ... ...⎟. ⎟ ⎜ 1 ∗ ⎜ Ln (t)e2πi(n−1)t dt 1 L∗n (t)e2πi(n−2)t dt 1 L∗n (t)e2πi(n−3)t dt . . . ⎟ 0 0 ⎠ ⎝ 0 .. .. .. . . . ... We use the both expressions of B in what follows. If x = (xj )j≥1 ∈ 2 , then ||Bx||22 =

1 0

2 L1 (t) x1 e−2πit + x2 e−4πit + . . . dt + . . .

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+

1 0

Ln (t) x1 e

Denote by h(t) the sum ||Bx||22 =

1 0

−2πit

∞ j=1

−4πit

+ x2 e

∞

k=1

2 + . . . dt + . . .

xj e2πijt ∈ H02 ([0, 1]). Then

2 L1 (t)h(−t)dt + · · · +

=

11

1 0

1 0

2 Ln (t)h(−t)dt + . . .

2 Lk (t)h(−t)dt ,

and, since Lk ∈ H02 ([0, 1]) for all k ≥ 1, we have that 1 1 Lk (t)h(−t)dt = Lk (t)g(−t)dt, 0

0

− g)(n) = 0 for all n ≥ 1. for all g ∈ L ([0, 1]) such that (h Thus, ∞ 2 1/2 1 ||B||B(2 ) = sup ||Bx||2 = sup Lk (t)h(−t)dt 2

||x||2 ≤1

||h||H 2 ≤1

=

sup ||g||L2 ≤1

∞ k=1

1 0

k=1

0

2 1/2 Lk (t)g(−t)dt .

Consequently, the space B(2 ) may be considered as a subspace of the ∞ space n=1 Hn , where Hn = H02 ([0, 1]) for all n ≥ 1. Moreover, if we denote by L := (L1 , L2 , . . . ) ∈

∞

Hn ,

n=1

and

||L||H 2 (∞) =

sup ||h||H 2 ≤1

∞ k=1

1 0

2 1/2 Lk (t)h(−t)dt ,

then it follows H02 (∞) := {L ∈

∞ n=1

Hn | ||L||H02 (∞) < ∞},

and H02 (∞), || · ||H02 (∞) is a Banach space.

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Moreover, the linear operator deﬁned by L : B(2 ), || ||B(2 ) → H02 (∞), || ||H02 (∞) L(B) = LB , where LB = (L1 (B), L2 (B), . . . ) is an isometry between B(2 ) and H02 (∞). In the sequel we introduce the periodical square function associated to a matrix. 2 Let L = (L1 , L2 , . . . ) ∈ ∞ n=1 Hn and h ∈ H0 ([0, 1]). We deﬁne VL,h : [0, 1] → R+ by: 1/2 ∞ 2 VL,h (x) = |(Lk ∗ h)(x)| . k=1

Proposition 2.1. Let L = (L1 , L2 , . . . ) be a ﬁxed element in We have that sup VL,h (0) = sup ||VL,h (·)||∞ . ||h||H 2 ≤1

∞

n=1 Hn .

||h||H 2 ≤1

First, it is clear that VL,h (0) ≤ ||VL,h ||∞ . For the converse, observe that VL,h0 (x) ≤ sup VL,h (x),

Proof.

||h||2 ≤1

where h0 ∈ H02 ([0, 1]) is ﬁxed with ||h0 ||2 ≤ 1. We deﬁne the isometric operator Sx (h) = hx , where hx (t) = h(x + t). Then sup VL,h (x) = sup VL,hx (0) = sup VL,hx (0) = sup VL,h (0), ||h||2 ≤1

||h||2 ≤1

||hx ||2 ≤1

||h||2 ≤1

and, hence, VL,h0 (x) ≤ sup VL,h (0), ||h||2 ≤1

for all h0 such that ||h0 ||2 ≤ 1, and for all x ∈ [0, 1]. In other words sup ||VL,h ||∞ = sup ||VL,h0 ||∞ ≤ sup VL,h (0). ||h||2 ≤1

||h0 ||2 ≤1

||h||2 ≤1

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Remark 2.2. a) By the previous discussion it follows that ||L||H02 (∞) =

sup ||h||H 2 ≤1

||VL,h ||∞ ,

and, consequently, H02 (∞) = {L ∈

∞ Y n=1

Hn |

sup ||h||H 2 ≤1

||VL,h ||∞ < ∞}.

b) It is clear from the definition that the value of VL,h in each point x ∈ [0, 1] has a definite meaning. Q Proposition 2.3. Let L = (L1 , L2 , . . . ) ∈ ∞ n=1 Hn , and B the matrix canonically associated to this element in the sense defined previously. Then the following assertions are equivalent: i) B ∈ B(`2 ). ii) LB = L ∈ H02 (∞). iii) For all h ∈ H02 ([0, 1]) we have VL,h ∈ Cs ([0, 1]), where Cs ([0, 1]) := {f : [0, 1] → C | ∃(fn )n≥1 such that fn : [0, 1] → C are continuous functions, (fn )n≥1 is a bounded sequence in the sup-norm and fn (x) → f (x) as n → ∞, for all x ∈ [0, 1]}. Proof. The implication i) ⇒ ii) follows from the definition of the operator L. i) ⇒ iii) We show that VL,h (x) ≤ M ||h||2 ,

for h ∈ H02 ([0, 1]), x ∈ [0, 1], and M := ||B||B(`2 ) . Of course, it is enough to prove that VL,h (x) ≤ M, for ||h||2 ≤ 1. If not, then there exist x0 ∈ [0, 1], and h0 ∈ H02 ([0, 1]), with ||h0 ||2 ≤ 1 such that

But, for h0 (t) =

P∞

k=1

VL,h0 (x0 ) > M. ak e2πikt , we have that

VL,h0 (x0 ) = (VL,h0 )x0 (0) = ||By0 ||2 ,

where y0 = (y01 , y02 , . . . ), and y0k = ak e2πikx0 . Since ||y0 ||2 = ||h0 ||H 2 it follows that there exists y0 , with ||y0 ||2 ≤ 1 such that ||By0 ||2 > ||B||B(`2 ) = M, which is a contradiction.

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Therefore, for each h ∈ H02 ([0, 1]), and for each x ∈ [0, 1], we have VL,h (x) ≤ M ||h||2 . Fixing h it follows that there exists C > 0 such that, for all x ∈ [0, 1], we have that VL,h (x) < C.

∞

2

Hence, the sum k=1 |(Lk ∗ h) (x)| converges absolutely and because Lk ∗ h ∈ C([0, 1]), for all k ≥ 1, VL,h ∈ Cs ([0, 1]). iii) ⇒ i) From the hypothesis we have that VL,h (0) < ∞ for all h ∈ H02 ([0, 1]). This fact means, by using the correspondences B ↔ (Lk )k≥1 = L, and x ↔ h, that, for each x ∈ N∗ , we have ||Bx||2 < ∞. In view of Banach Steinhaus Theorem it follows that B ∈ B(2 ).

We extend in what follows some of the notions introduced previously. Deﬁnition 2.4. Let B ∈ PM(2 ) := {A = (a ij )i,j≥1 such that ∃C > 0 with |aij | ≤ C ∀i, j ≥ 1}, and let LB := LB k k≥1 be its sequence of distributions. We call the matriceal distribution associated to the matrix B the expression LB ∈ D ([0, 1] × [0, 1]) given by the formula LB (t, x) =

∞

2πikx LB . k (t)e

k=1

∞ 2 2 Remark 2.5. i) If LB = LB k k≥1 ∈ n=1 H0 ([0, 1]), and h ∈ H0 ([0, 1]), then we observe that 2 1/2 1 1 LB (t, x)h(u − t)dt dx . VLB ,h (u) = 0

0

Hence, B ∈ B(2 ) ⇔

1

sup ||h||H 2 ≤1

0

1 0

2 1/2 LB (t, x)h(t)dt dx < ∞.

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Integral operators in inﬁnite matrix theory

ii) The notion of square function associated to a matrix may be extended from the class B(2 ) to the class PM(2 ) as follows: Let P([0, 1]) be the linear space of all analytic polynomials (i.e. trigonometrical polynomials having Fourier coeﬃcients of nonpositives indices equal to zero). Then we deﬁne, ∀ h ∈ P([0, 1]), 1 1 . VLB ,h := sup L (t, x)h(u − t)r(x)dtdx B r∈P([0,1]); ||r||H 2 ≤1

0

0

Deﬁnition 2.6. Let B ∈ PM(2 ) and let LB denote the matriceal distribution associated to the matrix B. Then, the operator TB : P([0, 1]) × P([0, 1]) → C([0, 1] × [0, 1]), deﬁned by

TB (r ⊗ h)(u, v) =

1

0

1

0

LB (t, x)r(u − x)h(v − t)dtdx,

is called the matriceal operator associated to B. Proposition 2.7. Let B ∈ PM(2 ). The following assertions are equivalent: i) B ∈ B(2 ). ii) There exists a continuous operator TB : H02 ([0, 1])⊗H02 ([0, 1]) → Cs ([0, 1] × [0, 1]), such that TB P([0,1])⊗P([0,1]) = TB . Proof. i) ⇒ ii) Let B ∈ B(2 ). Then it follows that, for all h ∈ H02 ([0, 1]), and for all x ∈ [0, 1], we have that (a)

VLB ,h (x) ≤ M ||h||2 ,

with M = ||B||B(2 ) . Let r, h ∈ P([0, 1]) be ﬁxed polynomials. Then: 1 1 ∞ 2πikx r(u − x)h(v − t)dxdt TB (r ⊗ h)(u, v) = LB k (t)e 0

=

∞ k=1

=

1 0

∞ k=1

0

k=1

LB k (t)h(v − t)dt

LB k ∗ h (v)

1 0

1 0

r(u − x)e2πikx dx

r(x)e−2πikx dx e2πiku .

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Hence, |TB (r ⊗ h)(u, v)| ≤ VLB ,h (v)||r||2 ≤ (by (a)) ≤ M ||h||2 ||r||2 , and ||TB (r ⊗ h)||∞ ≤ ||B||B(2 ) ||r ⊗ h||. Since P([0, 1]) is dense in the norm || ||2 , it follows that there exists a unique continuous extension TB like in the statement of the proposition. ii) ⇒ i) Let TB be as in the hypothesis. Then there exists M > 0 such that, for all r, h ∈ P([0, 1]), we have that ||TB (r ⊗ h)||∞ ≤ M ||r ⊗ h|| = M ||r||2 ||h||2 . We ﬁx r, h ∈ P([0, 1]). Then it follows that ∞ 1 LB r(x)e−2πikx dx , ||TB (r ⊗ h)||∞ ≥ |TB (r ⊗ h)(0, 0)| = k ∗ h (0) 0 k=1

and, therefore, ∞ 1 LB r(x)e−2πikx dx ≤ M ||r||2 ||h||2 . k ∗ h (0) 0 k=1

Next we take the supremum over all r ∈ P([0, 1]) with ||r||2 ≤ 1 and we get that VLB ,h (0) ≤ M ||h||2

∀h ∈ P([0, 1]).

Hence, ||B||B(2 ) = sup VLB ,h (0) ≤ M < ∞, ||h||2 ≤1

i.e. B ∈ B(2 ).

Consequently, we have: Theorem 2.8. Let B ∈ PM(2 ). The following assertions are equivalent: i) B ∈ B(2 ). ii) LB ∈ H02 (∞). iii) For all h ∈ H02 ([0, 1]), we have that VLB ,h ∈ Cs ([0, 1]). iv) sup||h||2 ≤1 ||VLB ,h ||∞ < ∞, and ||B||B(2 ) = sup||h||2 ≤1 ||VLB ,h ||∞ . v) There exists a continuous operator 2 2 T B : H0 ([0, 1])⊗H0 ([0, 1]) → Cs ([0, 1] × [0, 1]), such that

T B P([0,1]×[0,1]) = TB .

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Integral operators in inﬁnite matrix theory

Remark 2.9. a) The equivalence i)-iv) above holds also in the more general case B ∈ PM(2 ). b) By the discussion above we have that: ∞ 2 1/2 1 B Lk (t)h(−t)dt . ||B||B(2 ) = sup ||h||2 ≤1

k=1

0

Finally, if B ∈ M (2 ), A ∈ B(2 ), x = (xn )n≥1 ∈ 2 (N), and h ∈ ˆ H02 ([0, 1]), with h(n) = xn , for all n ≥ 1, then we have that 2 ∞ 1 B A . L (t)h(−t)dt ∗ L ||(B ∗ A)x||22 = k k 0

k=1

Hence, we ﬁnd that ||B||M (2 ) =

sup

sup

||LA ||H 2 (∞) ≤1 ||h||2 ≤1 0

2.2

∞

k=1

1

0

LB k

∗

LA k

2 1/2 (t)h(−t)dt .

Nonperiodical integral operators

In what follows we present another method to use the functions in the framework of matrix theory. Let B = (bij )i,j≥1 ∈ B(2 ) and x ∈ 2 (N∗ ). Then ||Bx||22 =

∞

2

|bk1 x1 + bk2 x2 + . . . | .

k=1

We deﬁne P(0, ∞) := {f : (0, ∞) → C a measurable function f (k,k+1] = ct a.e. ∀ k ∈ N}, and L2 (0, ∞) := P(0, ∞) ∩ L2 (0, ∞). Then we have a one-to-one correspondence between the class of all sequences from 2 (N∗ ) and the space L2 (0, ∞), given by x = (xk )k≥1 ∈ 2 (N∗ ) ↔ h ∈ L2 (0, ∞), with

h (k,k+1] = xk+1

∀k ∈ N.

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Deﬁnition 2.10. Let B ∈ PM(2 ). To B it corresponds a unique function C B (·, ·) given by i) C B : (0, ∞) × (0, ∞) → C, ii) C B (·, t) ∈ P(0, ∞), and C B (y, ·) ∈ P(0, ∞) ∀t, y > 0, iii) C B (y, t) = bkj if y ∈ (k − 1, k] and t ∈ (j − 1, j], for all k, j ≥ 1, where B = (bkj )k≥1; j≥1 . Then, we have that 2 ∞ ∞ 2 B ||Bx||2 = C (k, t)h(t)dt = k=1

0

∞ 0

0

∞

2 C (y, t)h(t)dt dy. B

Proposition 2.11. Let B ∈ PM(2 ) and C B (·, ·) be its associated function. The following assertions are equivalent: i) B ∈ B(2 ). ii) The operator TB : L2 (0, ∞) → L2 (0, ∞), given by ∞

TB (h)(y) = is a continuous operator.

2.3

C B (y, t)h(t)dt

0

Some applications of integral operators in the classical theory of inﬁnite matrices

In this section we use the previous results to give diﬀerent proofs for some classical theorems of inﬁnite matrix theory.

2.3.1

The characterization of Toeplitz matrices

We recall the deﬁnition of Toeplitz matrices. Deﬁnition 2.12. Let A ∈ PM(2 ). The matrix A is a Toeplitz matrix if there exists a sequence of complex numbers (an )n∈Z such that ⎞ ⎛ a2 . . . an . . . a0 a1 ⎜ a−1 a0 a1 . . . an−1 . . . ⎟ ⎟ ⎜ ⎟ ⎜ .. .. .. .. .. ⎜ (2.1) A = ⎜. . . . . ...⎟ ⎟. ⎟ ⎜a a a . . . a . . . −n −n+1 −n+2 0 ⎠ ⎝ .. .. .. .. .. . . . . . ...

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19

To such a matrix it is possible to associate a unique pseudomeasure fA given by +∞

fA (t) =

ak e2πikt ,

k=−∞

equality being taken in the distribution’s sense, and t ∈ [0, 1]. In what follows we ﬁnd the necessary and suﬃcient conditions in order that a Toeplitz matrix A belong to B(2 ), resp. to M (2 ). More speciﬁcally, we have: Theorem 2.13. Let A be a Toeplitz matrix like in (2.1). Then A ∈ B(2 ) ⇔ fA ∈ L∞ ([0, 1]). Proof. By Proposition 2.7 we have that, if TA is the matriceal operator associated to A, given by TA : P([0, 1]) ⊗ P([0, 1]) → C([0, 1] ⊗ [0, 1]), TA (r ⊗ h)(u, v) =

1

0

0

1

LA (t, x)r(u − x)h(v − t)dxdt,

then the following assertions are equivalent: 1) A ∈ B(2 ). 2) ||TA (r ⊗ h)||∞ ≤ C||r||2 ||h||2 , where C > 0 is an absolute constant and r, h ∈ H02 ([0, 1]) are arbitrary functions. Consequently, it is enough to prove that 2) ⇔ fA ∈ L∞ ([0, 1]). Therefore, let us consider the pseudomeasures LA k (t) =

∞

a−k+j e2πijt = L∗k A (t)e2πikt

∀ k ≥ 1.

j=1

The matriceal distribution associated to A is ∞ 2πikx LA . LA (t, x) = k (t) · e k=1

For r, h ∈ P([0, 1]) we have that ∞ 1 1 TA (r ⊗ h)(u, v) = L∗k A (t)e2πik(t+x) r(u − x)h(v − t)dxdt k=1

0

0

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= = 0

0 k=1 ∞ 1

2πikx

r(u − x)e −2πikx

r(x)e

Hence, 1

0

fA (t)h(v − t)

1

dx 0 1

dx

L∗k A (t)e2πikt h(v − t)dt =

0

k=1 1

1

0

1

0

L∗k A (t)e2πikt h(v

L∗k A (t)e2πikt h(v

r(x)e

k=1 1

− t)dt

− t)dt e2πiku .

fA (t)e2πikt h(v − t)dt

TA (r ⊗ h)(u, v) = ∞ 1 −2πikx

for all k ≥ 1.

2πik(u+t)

dx e

dt

0

fA (t)h(v − t)r(u + t)dt.

= 0

In this way we have to prove the equivalence of the following two conditions: i) fA ∈ L∞ ([0, 1]). ii) There exists a constant C > 0 such that, for all h, r ∈ P([0, 1]), we have that ||

1

0

fA (t)h(· − t)r(· + t)dt||∞ ≤ C||h||2 ||r||2 .

i) ⇒ ii) By Schwarz inequality and a trivial estimate we have that 1 1 ≤ ||f f (t)h(v − t)r(u + t)dt || |h(v − t)r(u + t)|dt ≤ A A ∞ 0

0

||fA ||∞ ||h||2 ||r||2 , for ﬁxed u, v ∈ [0, 1].

ii) ⇒ i) Since 1 fA (t)h(−t)r(u + t)dt = sup ||h||2 ≤1

0

0

by using ii) we have that: 1 |fA (t)r(t)|2 dt ≤ C Since

1

0

0

sup ||r||2 ≤1

0

1

1

|r(t)|2 dt

2

|fA (t)r(u + t)| dt

1/2 ,

∀r ∈ P([0, 1]).

|fA (t)r(t)|2 dt = ||fA ||2∞ ,

it follows that ||fA ||2∞ ≤ C, i.e. fA ∈ L∞ ([0, 1]).

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We give a diﬀerent proof of Bennett’s Theorem (see [11]). Theorem 2.14. Let A be a Toeplitz matrix. Then A ∈ M (2 ) ⇔ fA ∈ M ([0, 1]), where this last space is the space of all bounded complex measures on [0, 1]. Proof.

Let B ∈ B(2 ) be ﬁxed. It yields that 1 1 TA∗B (r ⊗ h)(u, v) = LA∗B (t, x)r(u − x)h(v − t)dxdt. 0

LA∗B (t, x) =

∞

0

∞ A ∗ B 2πikx LA (t)e Lk ∗ LB = (t)e2πik(x+t) , k ∗ Lk k

k=1

k=1

B for A ↔ (LA k )k≥1 , and B ↔ (Lk )k≥1 . Since ∗ A Lk ∗ LB (t) = L∗k A ∗ L∗k B (t), k

we have that TA∗B (r ⊗ h)(u, v) =

∞ k=1

=

1 0

1

0

∞ k=1

1 0

1 0

L∗k A ∗ L∗k B (t)e2πik(x+t) r(u − x)h(v − t)dxdt

fA ∗ L∗k B (t)e2πik(x+t) r(u − x)h(v − t)dxdt

⎛

1

= 0

⎞

⎜ 1 1 ⎟ ∞ ⎜ ⎟ B 2πik(x+s) ⎜ fA (s) ⎜ Lk (t − s)e r(u − x)h(v − t)dxdt⎟ ⎟ ds ⎝ 0 0 k=1 ⎠ ! LB (t−s,x+s)

1

= 0

TB (r ⊗ h)(u + s, v − s)ds.

Thus, we conclude that TA∗B (r ⊗ h)(u, v) =

1 0

TB (r ⊗ h)(u + s, v − s)ds,

for a Toeplitz matrix A and arbitrary B ∈ B(2 ).

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Moreover, by using Theorem 2.13, we ﬁnd that 1 fB h(v − t)r(u + t)dt, TB (r ⊗ h)(u, v) = 0

for a Toeplitz matrix B ∈ B(2 ). ⇐ Suppose that fA ∈ M ([0, 1]). By using Proposition 2.7 it is enough to show that: ∀B ∈ B(2 ), ∃CB > 0 such that ∀r, h ∈ P([0, 1]), it follows that ||TA∗B (r ⊗ h)||∞ ≤ CB ||r||2 ||h||2 .

(2.2)

But, for B ∈ B(2 ), TB : P([0, 1]) ⊗ P([0, 1]) → C([0, 1] × [0, 1]) can be extended continuously to an operator TB . Hence, we have that 1 fA (s)TB (r ⊗ h)(u + s, v − s)ds, TA∗B (r ⊗ h)(u, v) = 0

and, since fA ∈ M ([0, 1]), it follows that |TA∗B (r ⊗ h)(u, v)| ≤ ||fA ||M ([0,1]) ||TB (r ⊗ h)||∞ . But 1 ||r||2 ||h||2 . ||TB (r ⊗ h)||∞ ≤ CB

Consequently, ||TA∗B (r ⊗ h)||∞ ≤ CB ||r||2 ||h||2 , 1 ||fA ||M . where CB = CB ⇒ Let A, B be Toeplitz matrices with A ∈ M (2 ), and B ∈ B(2 ), ||B||B(2 ) ≤ 1. Then, we have that 1 fB (t)r(u + t)h(v − t)dt, TB (r ⊗ h)(u, v) = 0

and, by using Theorem 2.13 and the relation (2.2), it follows that there exists C > 0 such that, for each fB ∈ L∞ ([0, 1]), with ||fB ||∞ ≤ 1, we have, ∀r, h ∈ P([0, 1]), that 1 1 ≤ C||h||2 ||r||2 . f (s) f (t − s)h(−t)r(t)dtds (2.3) A B 0

0

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Next we recall two notions from classical Fourier analysis: the Cesaro kernel " #2 n |j| sin(n + 1)πt 1 1− e2πijt = ≥ 0, Kn (t) = n+1 n+1 sin πt j=−n and the Dirichlet kernel Dn (t) =

n

e2πijt =

j=−n

sin π(2n + 1)t . sin πt

We introduce in relation (2.3) 1 h(t) = √ Dn (t)e2πi(n+1)t = r(t). 2n + 1 Then r, h ∈ P([0, 1]), with ||h||2 = ||r||2 = 1, and (2.3) becomes 1 fA (s)σ2n+1 (fB )(−s)ds ≤ C ∀n ≥ 1, 0

where

σn (fB )(s) = (Kn ∗ fB ) (s) are the Cesaro sums of the order n associated to fB . Then, for all n ≥ 1, and for all fB ∈ L∞ ([0, 1]), with ||fB ||∞ ≤ 1, we have that 1 ≤ C, f (−s)σ (f )(s)ds (2.4) B 2n+1 A 0

and, hence, for all n ≥ 1, ||σ2n+1 (fA )||1 =

sup ||fB ||∞ ≤1

1 0

fB (−s)σ2n+1 (fA )(s)ds ≤ C.

Moreover, we deﬁne the sequence of functionals Sn : C([0, 1]) → C, by 1 σ2n+1 (fA )(s)g(s)ds. Sn (g) = 0

Then Sn are linear operators with ||Sn || ≤ C for all n ≥ 1, and, by Alaoglu’s Theorem, there exists a linear bounded operator S : C([0, 1]) → C such that Sn → S weakly . Applying now Riesz Theorem, there exists μ ∈ M ([0, 1]) such that 1 gdμ. S(g) = It is clear that

0

μ = fA ∈ M ([0, 1]).

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2.3.2

The characterization of Hankel matrices

Deﬁnition 2.15. Let A ∈ PM(2 ). Then A is called a Hankel matrix if there exists the complex sequence (an )n∈N∗ such that ⎞ ⎛ a1 a2 a3 an ⎜a a a ⎟ n ⎟ ⎜ 2 3 ⎟ ⎜ ⎟ ⎜ a3 an A=⎜ ⎟. ⎟ ⎜ an ⎟ ⎜ ⎠ ⎝ an To such a matrix we associate a unique pseudomeasure gA given by gA (t) =

∞

ak e2πikt where t ∈ [0, 1].

k=1

Like in the case of Toeplitz matrices we study whenever the matrix A belongs to the spaces B(2 ), or M (2 ). More speciﬁcally, we have that Theorem 2.16. Let A be a Hankel matrix. Then A ∈ B(2 ) ⇔ gA ∈ BM O. Let TA be the matriceal operator associated to A 1 1 TA (r ⊗ h)(u, v) = LA (t, x)r(u − x)h(v − t)dxdt

Proof.

0

∞

=

k=1

Since

0

1

1 0

0

LA k (t)h(v

− t)dt

LA k (t)h(v − t)dt =

1 0

1 0

2πikx

r(u − x)e

dx .

gA (t)e−2πi(k−1)t h(v − t)dt,

we have that TA (r ⊗ h)(u, v) = 0

1

gA (t)e

2πit

h(v − t)

∞ k=1

1

−2πikx

r(x)e 0

2πik(u−t)

dx e

dt.

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Thus,

TA (r ⊗ h)(u, v) =

1 0

gA (t)h(v − t)e2πit r(u − t)dt,

(2.5)

or, denoting by hv (t) = h(v + t), and by ru (t) = r(u + t), 1 TA (r ⊗ h)(u, v) = gA (t)e2πit (hv ru )(−t)dt. 0

Next we prove the implication ⇒ . If A ∈ B(2 ), we have that, r, h ∈ P([0, 1]), 1 2πit |TA (r ⊗ h)(0, 0)| = gA (t)e (hr)(−t)dt ≤ C||r||2 ||h||2 . 0

In view of the theory of Hardy spaces, this last fact is equivalent to the following inequality: 1 2πit ≤ C, g (t)e f (−t)dt sup A 1 0

f ∈H0 ,||f ||1 ≤1

i.e., to that ||gA ||BM O ≤ C. For the converse implication let gA ∈ BM O. We have that 1 2πit gA (t)e (hv ru )(−t)dt ≤ ||gA ||BM O ||hv ru ||H 1 |TA (r ⊗ h)(u, v)| = 0

≤ ||gA ||BM O ||h||2 ||r||2 , i.e., ||TA || ≤ ||gA ||BM O .

Theorem 2.17. Let A be a Hankel matrix. We have: i) A ∈ M (2 ) implies that gA ∈ M (H01 , H01 ), where this last space is the space of all Fourier multipliers of H01 . ii) gA ∈ M ([0, 1]) implies that A ∈ M (2 ). Let B ∈ B(2 ). Then ∞ 1 1 A 2πikx TA∗B (r ⊗ h)(u, v) = Lk ∗ LB dtdx k (t)h(v − t)r(u − x)e

Proof.

k=1

= 0

1

gA (s)e2πis

1 0

0

1 0

0

2πik(x−s) LB (t − s)e h(v − t)r(u − x)dtdx ds. k

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Thus,

TA∗B (r ⊗ h)(u, v) =

1 0

gA (s)e2πis TB (r ⊗ h)(u − s, v − s)ds.

(2.6)

i) We assume that A ∈ M (2 ). Then there exists C > 0 such that, for each Hankel matrix B, B ∈ B(2 ), with ||B||B(2 ) ≤ 1, we have that, for r, h ∈ P([0, 1]), (2.7) ||TA∗B (r ⊗ h)||∞ ≤ C||r||2 ||h||2 . Therefore, by using the relations (2.5), (2.6), and (2.7), we have that 1 1 2πis 2πit g (s)e g (t)e (r h )(−t)dt ds ≤ C||r||2 ||h||2 . A B u−s v−s 0 0 We take u = v = 0, and s + t = y in the relation above, obtaining that 1 1 2πiy gA (s) gB (y − s)e (rh)(−y)dy ds ≤ C||r||2 ||h||2 , 0 0 or, equivalently, 1 1 sup sup g (s) g (y − s)f (−y)dy ds B A 1 gB ∈BM O; ||gB ||BM O ≤1 f ∈H0 ; ||f ||1 ≤1

0

≤ C.

0

Thus, the operator SA : H01 → H01 , given by

SA (f )(s) =

0

1

gA (s − y)f (y)dy,

is a bounded operator if and only if gA ∈ M (H01 , H01 ). ii) According to relation (2.7), for every B ∈ B(2 ), and for gA ∈ M ([0, 1]), we have that ||TA∗B (r ⊗ h)||∞ ≤ ||gA ||M ||TB (r ⊗ h)||∞ . Since B ∈ B(2 ), it follows that ||TB (r ⊗ h)||∞ ≤ CB ||r||2 ||h||2 , therefore, ||TA∗B (r ⊗ h)||∞ ≤ ||gA ||M CB ||r||2 ||h||2 . Hence, A ∗ B ∈ B(2 ) ∀B ∈ B(2 ), that is A ∈ M (2 ). Remark 2.18. An equivalent condition to the statement that a Hankel matrix A ∈ M (2 ) was given by G. Pisier in [81]: A ∈ M (2 ) ⇔ gA ∈ M (H 1 (S 1 ), H 1 (S 1 )), 1 1 where H (S ) is the operator trace class-valued Hardy space.

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2.3.3

The main triangle projection

In what follows we present a new proof of a result in [49], namely: What is the growth rate with respect to n of the expression sup ||A||B(2 ) ≤1

||Pn (A)||B(2 ) ,

where

Pn (A) = (bij ) i, j ≥ 1, for bij =

aij if i + j ≤ n + 1 0 otherwise,

with A = (aij )i,j≥1 . Theorem 2.19. Let Pn the triangle projection of the order n. Then we have that sup ||A||B(2 ) ≤1

||Pn (A)||B(2 ) = O(log n)

for n → ∞.

Moreover, there exists C > 0 such that sup ||A||B(2 ) ≤1

||Pn (A)||B(2 ) ≥ C log n

∀n ∈ N∗ .

Proof. Let A ∈ B(2 ) and x ∈ 2 (N). Then, by using the correspondences ∞ 2πijt , as in Section 2.1, A ↔ (LA k )k≥1 , and x = (xj )j≥1 ↔ h(t) = j=1 xj e we have the following equality: 2 n 1 A , ||Pn (A)x||22 = L (t)h(−t)dt ∗ D n+1−k k 0

k=1

where Dn (t) =

n

e2πikt

k=−n

is the Dirichlet’s kernel. Therefore, ||Pn (A)x||22 =

n k=1

≤

n

Dn+1−k (s)

0

||Dn+1−k ||1

k=1

≤ sup ||Dn+1−k ||1 1≤k≤n

1

0

1

0

1

1 0

2 LA (t − s)h(−t)dt ds k

2 |Dn+1−k (s)| LA k ∗ h (−s) ds

sup |Dn+1−k (s)|

1≤k≤n

n

A L ∗ h (−s)2 k

k=1

ds

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$ $ $ $ $ VL ,h 2 . ≤ sup ||Dn+1−k ||1 $ |D (s)| sup n+1−k A ∞ $ $ 1≤k≤n

1≤k≤n

1

Since ||Dn ||1 = O(log n), for n → ∞, it remains to prove that 1 sup |Dk (s)| ds = O(log n) ⇔ 0 1≤k≤n

1

| sin ks| ds = O(log n). s 1≤k≤n sup

0

In order to prove this we take gs : [1, u] → R+ , given by | sin xs| , s

gs (x) = where s ∈ (0, 1]. There are two distinct cases: 1 , which implies that 1) s > 2n gs (x) < and 2) s ≤

1 2n

1 s

∀x ∈ [1, n],

which implies 0 < xs ≤ gs (x) ≤

Therefore,

1

sin ns s

| sin ks| ds ≤ s 1≤k≤n

∀x ∈ [1, n].

sup

0

1 2

= 0

= 12 , so that

n 2n

0

1 2n

sin ns ds + s

1 1 2n

1 ds s

sin s ds + log 2n = O(log n), s

for n suﬃciently large, which, in turn, implies that ||Pn (A)||B(2 ) ≤ C log n sup ||VLA ,h ||∞ . ||h||2 ≤1

According to Remark 2.2 we have that ||Pn (A)||B(2 ) ≤ C log n||A||B(2 ) . In order to prove that there exists B > 0, such that sup ||A||B(2 ) ≤1

||Pn (A)||B(2 ) ≥ B log n,

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we observe that sup ||A||B(2 ) ≤1

||Pn (A)||B(2 ) =

sup ||A||B(2 ) ≤1

||Tn (A)||B(2 ) ,

where A = (aij )i,j≥1 , and ⎛ a11 ⎜ ⎜ a21 ⎜ ⎜ .. ⎜ . Tn (A) = ⎜ n ⎜ an1 ⎜ n+1 ⎜ 0 ⎝ .. .

n

0 a22 .. . an2 0 .. .

0 ... 0 0 ... 0 .. .. .. . . . an3 . . . ann 0 0 0 .. .. .. . . .

⎞ 0 ... ⎟ 0 ...⎟ ⎟ .. ⎟ . ...⎟ ⎟. 0 ...⎟ ⎟ 0 ...⎟ ⎠ .. . ...

n+1

We consider next Hilbert’s matrix given by ⎛ ⎞ 0 1 12 31 . . . ⎜ −1 0 1 1 . . . ⎟ 2 ⎜ ⎟ H = ⎜ − 1 −1 0 1 . . . ⎟ , ⎝ 2 ⎠ .. . . . . . . . . . . . . . 1 2

n

and xn = (1, 1, . . . , 1, 0, . . . ). Then ||H||B(2 ) < ∞, and ||Tn (H)xn ||22

2

1 =1 + 1+ 2 2

2

2

2

1 1 + ··· + 1 + + ··· + 2 n−1

≥ C log 2 + log 3 + · · · + log n = C

n 2

2

log2 xdx ∼ Cn log2 n.

Therefore ||Tn (H)||2B(2 ) ≥

||Tn (H)xn ||22 n log2 n , ≥ C ||xn ||22 n

that is sup ||A||B(2 ) ≤1

||Tn (A)||B(2 ) ≥ ||H||−1 B(2 ) ||Tn (H)||B(2 ) ≥ C log n.

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B(2 ) is a Banach algebra under the Schur product

2.3.4

Next we give a diﬀerent proof of an old result of I. Schur (see also [11]). Theorem 2.20. B(2 , ∗) is a Banach algebra. B Proof. Let A, B ∈ B(2 ), and (LA k )k≥1, (Lk )k≥1, , be the sequences of functions associated to A, and to B, respectively. We have that 2 ∞ 1 A 2 B Lk ∗ Lk (t)h(−t)dt ||(B ∗ A)x||2 = 0

k=1 ∞ = k=1

≤

∞ k=1

1 0

1 0

LA k (s)

1 0

A 2 L (s) ds

2 − s)h(−t)dt ds

LB k (t

k

0

1

B L ∗ h (−s)2 ds , k

where x = (xj )j≥1 ∈ 2 (N), and h(t) =

∞

xj e2πijt ∈ H02 ([0, 1]).

j=1

Therefore, || (B ∗ A) ||2 ≤ sup k≥1

1 0

A 2 Lk (s) ds

1/2

1 0

VL2B ,h (s)ds

1/2

≤ ||A||B(2 ) ||VLB ,h ||2 . Finally, ||B ∗ A||B(2 ) ≤ ||A||B(2 ) sup ||VLB ,h ||2 ≤ ||A||B(2 ) ||B||B(2 ) . ||h||2 ≤1

Notes The main idea of this chapter is the interpretation of an inﬁnite matrix as a sequence of functions (or, more generally, distributions). We take in this way the advantage of a rich class of notions and techniques, which are usual in function theory. For instance, in Section 2.1 we deﬁne the notions of a

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square function, respectively of a matriceal operator associated to a given matrix A. The main result of Section 2.1 is of course Theorem 2.8. In Section 2.2 the central result is Proposition 2.11, which is the nonperiodical analogue of Theorem 2.8. Section 2.3 is dedicated to applications of results from Section 2.1. In this way we give diﬀerent proofs of some classical results from inﬁnite matrix theory: Theorem 2.13 is of course well-known (see [94]). Theorem 2.14 is known as Bennett’s Theorem (see [11]). Theorem 2.16, known as Nehari’s Theorem (see [65]), is here presented with a new proof. Finally, we mention Theorem 2.19, ﬁrst proved in [49]. Theorem 2.20 was discovered apparently by I. Schur (see [11] for a more general result).

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Chapter 3

Matrix versions of spaces of periodical functions

An interesting problem concerning inﬁnite matrices is the following: Let A ∈ B(2 ). When is the matrix A approximable by matrices of ﬁnitely band type in the operator norm || · ||B(2 ) ? In what follows we deal with this problem in some spaces of inﬁnite matrices which can be regarded as extensions of classical Banach spaces of functions C(T) and L1 (T). These Banach spaces together with the spaces B(2 ) and M (2 ) are of interest in order to develop some results extending known theorems of classical harmonic analysis in the framework of matrices. One main aim of the present chapter is to extend in the framework of matrices Fejer’s theory for Fourier series. (See for instance [96].) As it was stated in the Introduction we have a similarity between the expansion in the Fourier series f = k ak eikx of a periodical function f on the torus T and the decomposition A = k∈Z Ak , where Ak is the kth diagonal of A for k ∈ Z. Moreover, there is a similarity between the convolution product f ∗ g of two periodical functions and the Schur product of two matrices A and B, C = A ∗ B. First we mention the following results obtained by Fejer, which have been guiding for our investigation: ikθ (A) A function f (θ) = is continuous on T (that is f ∈ k∈Z ak e C(T)) if and only if the Cesaro sums k |k| eikθ σn (f ) = ak 1 − n+1 k=−n

converge uniformly on T to f. (B) A function f (θ) = k∈Z mk eikθ ∈ L1 (T) if and only if ||σn (f ) − f ||L1 (T) → 0 as n → ∞.

33

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3.1

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Preliminaries

In view of Fejer’s result (A) it is natural to give the following deﬁnition: Deﬁnition 3.1. Let A ∈ B(2 ). We denote by σn (A) the Cesaro sum associ n n |k| . Then ated to Sn (A) := k=−n Ak , that is σn (A) = k=−n Ak 1 − n+1 we say that A is a continuous matrix if lim ||σn (A) − A||B(2 ) = 0.

n→∞

Let us denote by C(2 ) the vector space of all continuous matrices and consider on it the usual operator norm. Now recall that the space of all Schur multipliers M (2 ) is a commutative unital Banach algebra with respect to Schur product. Moreover we have Bennett’s theorem (see for instance Theorem 2.14): The Toeplitz matrix M = (cj−k )j,k , where (cn )n∈Z is a sequence of complex numbers, is a Schur multiplier if, and only if, there exists a (bounded, complex, Borel) measure μ on (the circle group) T with μ (n) = cn

for n = 0, ±1, ±2, . . . .

Moreover, we then have ||M ||M (2 ) = ||μ||. We also mention the following well-known fact (see Theorem 2.13): The Toeplitz matrix M represents a linear and bounded operator on 2 if and only if there exists a function f ∈ L∞ (T) with Fourier coeﬃcients f(n) = mn for all n ∈ Z. Moreover, we have ||M ||B(2 ) = ||f ||L∞ (T) .

3.2

Some properties of the space C(2 )

First of all let us observe the following fact: Remark 3.2. By Fejer’s theorem (A) we have that a Toeplitz matrix T = def ikθ (tk )k∈Z ∈ C(2 ) if and only if fT (θ) = ∈ C(T), and in this k∈Z tk e way we can see that the notion of a continuous matrix may be regarded as an analogue of that of a continuous function.

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Now let C∞ denote the space of all matrices deﬁning compact operators. Proposition 3.3. C(2 ) is a proper closed ideal of B(2 ) with respect to Schur multiplication which, in its turn, contains C∞ properly. Proof.

We have:

||σn (A)||B(2 )

$ n $ $ |k| $ $ $ =$ Ak 1 − $ $ n+1 $ k=−n

≤ ||Mn ||M (2 ) ||A||B(2 ) , B(2 )

where Mn is the n-band type Toeplitz matrix with the entries ⎧ |j − i| ⎨ 1− if |j − i| ≤ n, mij = n+1 ⎩0 otherwise. Hence C(2 ) is a closed subspace of B(2 ). Now we observe that for A, B ∈ C(2 ), σn (A ∗ B) = σn (A) ∗ B and then we have that, for A ∈ C(2 ), B ∈ B(2 ) ||A ∗ B − σn (A ∗ B)||B(2 ) = ||[A − σn (A)] ∗ B||B(2 ) ≤ ||A − σn (A)||B(2 ) · ||B||M (2 ) ≤ ||B||B(2 ) · ||A − σn (A)||B(2 ) . Here we have used the simple fact that ||B||M (2 ) = ||B ∗ E||M (2 ) ≤ ||B||B(2 ) · ||E||M (2 ) = ||B||B(2 ) , E = (Eij ) where Eij = 1 for all i, j ∈ N, and ||E||M (2 ) = 1. Hence, C(2 ) is a closed ideal of B(2 ) with respect to Schur multiplication. Next we note that C(2 ) is a proper ideal of B(2 ). Denoting by eij the matrix whose single non-zero entry is 1 on the ith row and on the jth column, we consider the matrix A = k∈N Ak , where Ak = ek+1,2k+1 , k ≥ 0, which belongs to B(2 ), since (AA∗ )1/2 = I (I is the identity matrix). Moreover, $ $ n $ $ 1 k $ $ ||σn (A)−A||B(2 ) = $ Ak + kAk $ = max ∨1 = 1 $ k≤n n + 1 $ n+1 k>n

for all n and, thus, A ∈ / C(2 ). Now let A ∈ C∞ . Denoting by Pn (A)(i, j) =

k=0

B(2 )

aij i, j ≤ n 0 otherwise,

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we have that ||Pn (A) − A||B(2 ) → 0, as n → ∞. But, by Bennett’s theorem, we have that for k > n : $ $ n $ || $ $ $ (Pn (A)) ||Pn (A) − σk (Pn (A))||B(2 ) = $ $ $ k + 1$ =−n

$ $ n $ || iθ $ $ $ e $ ≤$ $ $ k+1 =−n

B(2 )

· ||Pn (A)||B(2 ) → 0 as k → ∞. M (T)

Hence, Pn (A) ∈ C(2 ) for all n ∈ N and, consequently, C∞ ⊂ C(2 ). Since it is easy to see and well-known that a Toeplitz matrix does not represent a compact operator, by the previous remark it follows that C∞ is a proper subspace of C(2 ). The proof is complete. 3.3

Another characterization of the space C(2 ) and related results

We will give another characterization of the space C(2 ) by using continuous vector-valued functions but ﬁrst we note the following simple fact: Remark 3.4. Consider the function fA : T → B(2 ) given by fA (t) = A ∗ ei(j−k)t j,k≥0 . Then ||fA (θ)||B(2 ) = ||A||B(2 ) for all θ ∈ T. Indeed, by Bennett’s multiplier theorem, we have that ||fA (θ)||B(2 ) ≤ ||A||B(2 ) ||δ−θ || = ||A||B(2 ) , where δθ ∈ M the Dirac point mass at θ ∈ T. Similarly, since (T) denote A = fA (θ) ∗ ei(j−k)t j,k≥0 , we have that ||A||B(2 ) ≤ ||fA (θ)||B(2 ) . An easy consequence of this remark is that fA is continuous on T if and only if it is continuous at one single point. Now we ask ourselves how the matrix A should be in order that the function fA shall be continous. The answer to this question is as follows: Theorem 3.5. Let A be an inﬁnite matrix. Then fA is a B(2 )-valued continuous function if and only if A ∈ C(2 ), with equality of the corresponding norms.

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Proof.

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By Remark 3.4 it follows that ||σn (fA ) − fA ||C(T,B(2 )) = ||σn (A) − A||B(2 ) .

Now reasoning as in the proof of Fejer’s result (A) (see for instance [41]) we get that for a continuous function fA : T → B(2 ) it follows that ||σn (A) − A||B(2 ) → 0, as n → ∞. Thus A ∈ C(2 ). The converse implication follows easily from Remark 3.4.

Now we shall study the following question: What can we say about subspaces of M (2 ) in connection with the multiplier property? The following theorem gives a justiﬁcation of introducing C(2 ) and also a partial answer to the above question. It is the matriceal analogue of Theorem 11.10, Chap. IV-[96]. Theorem 3.6. The Toeplitz matrix M = (mk )k∈Z is a Schur multiplier from B(2 ) into C(2 ) iﬀ mk eikθ ∈ L1 (T). k∈Z ∞ If we identify f ∈ L (T) with its corresponding Toeplitz operator Tf = f(j − k) in B(2 ), then it is straightforward to see that

Proof.

j,k≥0

a Schur multiplier M = (mj−k )j,k≥0 mapping B(2 ) into C(2 ) induces a ∞ Fourier multiplier sequence m = {mn }∞ n=−∞ mapping L (T) into C(T), which is known to correspond, in the manner indicated in the statement above, to a function from L1 (T) (see [96]). The converse follows also by the same lines. Guided by [11] we propose for matrices a similar notion to that of Lebesgue integrable functions. Deﬁnition 3.7. We say that an inﬁnite matrix A is an integrable matrix if σn (A) → A as n → ∞ in the norm of M (2 ). The space of all such matrices, endowed with the norm induced by M (2 ), is denoted by L1 (2 ). Of course L1 (2 ) is a Banach space. Remark 3.8. If A ∈ L1 (2 ), then it follows that A ∗ B ∈ C(2 ) for all B ∈ B(2 ).

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Indeed, for B ∈ B(2 ), in view of the fact that ||σn (A ∗ B) − A ∗ B||B(2 ) ≤ ||σn (A) − A||M (2 ) · ||B||B(2 ) → 0, as n → ∞, we ﬁnd that A ∗ B ∈ C(2 ). Now it is clear that L1 (2 ) is a closed ideal of M (2 ) with respect to the Schur product. We have the following analogue of the Riemann-Lebesgue Lemma: Lemma 3.9. Let M ∈ L1 (2 ). Then lim ||Mk ||L1 (2 ) = 0.

|k|→∞

Proof. It is clear from Deﬁnition 3.7 that, for any > 0, there is a number n() such that, for |k| ≥ n(), it follows that ||Mk ||L1 (2 ) ≤ and the proof is complete. Remark 3.10. It is easy to see that for a diagonal matrix Ak , k ∈ Z, we have that ||Ak ||B(2 ) = ||Ak ||M (2 ) .

Thus, in Lemma 3.9 we can take ||Mk ||B(2 ) instead of ||Mk ||L1 (2 ) . In view of Theorem 3.5 and Remark 3.8 it is natural to ask: If A is a Schur multiplier which maps B(2 ) into C(2 ) does it follow that A ∈ L1 (2 )? The answer to the above question is negative. In fact, we have that Example 3.11. Let A be the following matrix: ⎛ ⎞ 1 1 1 ... ⎜ ⎟ A = ⎝0 0 0 ... ⎠. .. .. .. . . . . . . The matrix A is a Schur multiplier which maps B(2 ) into C(2 ) but it does not belong to L1 (2 ). In fact, A is a Schur multiplier with the property that A ∗ B ∈ C(2 ) for all B ∈ B(2 ) since the matrix A ∗ B has rank 1 and therefore represents a compact operator and consequently it belongs to C(2 ). Moreover, A does not belong to L1 (2 ) by Lemma 3.9.

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Therefore the Banach space (B(2 ), C(2 )) of all multipliers from B(2 ) into C(2 ) is diﬀerent from both M (2 ) and L1 (2 ). Thus, it seems that this space deserves to be studied in more detail. On the other hand the space (C(2 ), C(2 )) of all inﬁnite matrices A such that A ∗ B ∈ C(2 ) for all B ∈ C(2 ) can be described easily. More precisely we have: Theorem 3.12. (C(2 ), C(2 )) is exactly the space M (2 ) of all Schur multipliers. Proof. Since σn (A ∗ M ) = σn (A) ∗ M it follows easily that M ∈ (C(2 ), C(2 )) if M ∈ M (2 ) and A ∈ C(2 ). Conversely, assuming that M ∈ (C(2 ), C(2 )), we have for A ∈ B(2 ) that ||M ∗ σn (A)||B(2 ) ≤ C||σn (A)||B(2 ) . Moreover, σn (A) → A in the weak topology of operators in B(2 ), that is < σn (A)x, y > → < Ax, y > for all x, y ∈ 2, where < ·, · > is the scalar product in 2 (use 2 -sequences with a ﬁnite number of nonzero components and a standard approximation argument). This yields that ||M ∗ A||B(2 ) ≤ C||A||B(2 ) , that is M is a Schur multiplier. The proof is complete. Next we give a characterization of an integrable matrix in the spirit of Theorem 3.5: Theorem 3.13. Let A ∈ M (2 ) and fA (θ) = A ∗ (ei(j−k) )j,k≥0 for θ ∈ T. Then fA is a pointwise well-deﬁned function fA : T → M (2 ) such that ||fA (θ)||M (2 ) = ||A||M (2 ) for all θ ∈ T. Furthermore, fA ∈ C(T, M (2 )), that is fA is continuous, if and only if A ∈ L1 (2 ). Proof. For μ ∈ M (T) let us introduce the notation Tμ for the Toeplitz matrix with symbol μ, that is μ(j − k))j,k≥0 , where μ (n) = e−int dμ(t) Tμ = ( is the nth Fourier coeﬃcient of μ. Note that fA (θ) = A ∗ Tδ0 , where δ0 ∈ M (T) denotes the unit point mass at θ ∈ T. By Bennett’s multiplier theorem (see [11]) we obtain that ||fA (θ)||M (2 ) ≤ ||A||M (2 ) . Similarly, since A = fA (θ) ∗ Tδ0 , we have that ||A||M (2 ) ≤ ||fA (θ)||M (2 ) . Assume next that A ∈ L1 (2 ). We then have that ||σN (fA )(θ) − fA (θ)||M (2 ) = ||σn (A) − A||M (2 ) → 0.

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Thus, σn (fA ) → fA uniformly and we obtain that fA ∈ C(T, M (2 )). Assume now that fA ∈ C(T, M (2 )). We consider then the M (2 )-valued integral 1 fN (θ) = fA (θ − t)KN (t)dt, θ ∈ T, 2π T where KN is the N th Fejer kernel. It is straightforward to see that fN − fA converges in C(T, M (2 )) as N → ∞. An easy computation yields that fN (θ) =

N n=−N

1−

|n| N +1

inθ f( , A (n)e

where f( A (n) is the nth Fourier coeﬃcient of fA . To compute the Fourier coeﬃcient f( A (n) we need only to observe that the operation M → mjk of taking the (j, k)th entry is a bounded linear functional on M (2 ). By this we clearly have that f( A (n) = An . Summing up, we have shown that lim

N →∞

N n=−N

|n| 1− N +1

An einθ = fA (θ)

in C(T, M (2 )). For θ = 0 this yields σN (A) → A in M (2 ). The proof is complete. We also remark that f is continuous on T if and only if it is continuous at one single point. This is clear by the ﬁrst assertion of the above theorem. The next remark is an easy consequence of Fejer’s theory. Remark 3.14. Let A be a Toeplitz matrix. Then A ∈ L1 (2 ) if and only if it represents a function from L1 (T). Now we recall the following well-known result (see [41], Chapter 2). A function F on T belongs to L∞ (T) if and only if sup ||σn (f )||L∞ (T) < ∞. n

We have the following matrix-version of the previous result: Proposition 3.15. Let A be an inﬁnite matrix. Then A belongs to B(2 ) if and only if sup ||σn (A)||B(2 ) < ∞. n

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Proof. Assume that supn ||σn (A)||B(2 ) < ∞. Then, by reasoning as in the proof of Theorem 3.12, we get easily that A ∈ B(2 ). The converse implication can be proved by using the same arguments as in the proof of Proposition 3.3. Proposition 3.16. A ∈ M (2 ) if and only if sup ||σn (A)||M (2 ) < ∞. n

Proof.

Assume that sup ||σn (A)||M (2 ) < ∞ n

and ﬁx an arbitrary B ∈ B(2 ). We then have that ||σn (B ∗ A)||B(2 ) ≤ ||B||B(2 ) sup ||σn (A)||M (2 ) < ∞. n

It follows that σn (B ∗ A) → B ∗ A in the weak topology of operators in B(2 ). (See the proof of Theorem 3.12.) In particular, B ∗ A ∈ B(2 ). Since B is arbitrary this means that A is a Schur multiplier. Conversely, let A ∈ M (2 ). Then σn (A) = A ∗ σn (M ), where M = (mi )i∈Z with mi = 1. Thus, by using Bennett’s theorem, we ﬁnd that ||σn (A)||M (2 ) ≤ ||A||M (2 ) · ||σn (M )||M (2 ) ≤ ||A||M (2 ) . The proof is complete.

3.4

A matrix version for functions of bounded variation

Following [75] we introduce now a matrix version of functions with bounded variation and prove the analogue of classical Jordan’s theorem on trigonometric series. We denote by A the matrix k∈Z kAk . Deﬁnition 3.17. As in the Fourier series framework we say that a matrix A is a matrix of bounded variation if A ∈ M (2 ). The space of all matrices A of bounded variation is denoted by BV (2 ) and it is a Banach space endowed with the norm ||A||BV (2 ) = ||A0 ||B(2 ) + ||A ||M (2 ) .

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Finally, we say that a matrix A is absolutely continuous if A ∈ L1 (2 ) and A0 ∈ B(2 ). Let us remark that by Lemma 3.9 it follows that ||Ak ||B(2 ) = o k1 for an absolutely continuous matrix A. Moreover, it is easy to see that for a matrix M ∈ M (2 ) we have that ||Mk ||M (2 ) ≤ ||M ||M (2 ) , for all k ∈ Z. On the other hand, it is possible that ||Mk ||M (2 ) → 0, as |k| → ∞. For instance if M coincides with E, the matrix having only 1 as entries, then obviously lim ||Mk ||M (2 ) = 1.

|k|→∞

Next we introduce another interesting subspace of B(2 ) by U (2 ) = {A ∈ C(2 ); such that ||Sn (A) − A||B(2 ) → 0 as n → ∞} endowed with the norm ||A||U (2 ) = sup ||Sn (A)||B(2 ) . n

Obviously, U (2 ) ⊂ C(2 ). Using the well-known example of Du Bois Raymond (see [96]) there exists a Toeplitz matrix from C(2 ) which does not belong to U (2 ), so that U (2 ) is in fact a proper subspace of C(2 ). But there are more sophisticated such examples of matrices. For instance we can adapt an example found by Fejer to the framework of inﬁnite matrices which are not Toeplitz matrices. We sketch in what follows this example: For any integers n, μ and for all x ∈ R we put (see [96]-page 168) n sin kx . Q(x, μ, n) = − cos(μ + n)x k k=1

Since the partial sums of the series sin x + 12 sin 2x + . . . are less than a constant C in absolute value, we have that |Q| ≤ C, for every x, μ, n. If we denote by Q(μ, n) the inﬁnite Toeplitz matrix associated to the periodical function Q(x, μ, n), then we get that ||Q(μ, n)||B(2 ) ≤ C. Now, for every μ, n, we choose the decreasing sequences aμ,n = {alμ,n }l≥1 , such that there exist 0 < a < b < ∞, a, b independent of all indices μ, n, with a ≤ alμ,n ≤ b, for all l ≥ 1. Then we consider the matrix [aμ,n ] deﬁned in [18] as ⎞ ⎛ 1 aμ,n a1μ,n . . . . . . ⎜ a1μ,n a2μ,n a2μ,n . . . ⎟ ⎟ ⎜ [aμ,n ] = ⎜ a1 a2 a3 . . . ⎟ . ⎠ ⎝ μ,n μ,n μ,n .. .. .. .. . . . .

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∞ Since l=1 |alμ,n − al+1 μ,n | < ∞, for all μ, n, we get that [aμ,n ] ∈ M (2 ), (see [18]), and ||[aμ,n ]||M (2 ) ≤ b − a, for all μ, n. Let {nk }, {μk } be sets of integers which we shall deﬁne in a moment, and let αk > 0, α1 + α2 + · · · < ∞. We deﬁne the matrices [aμk ,nk ] as described above. Then the series ∞ αk [aμk ,nk ] ∗ Q(x, μk , nk ) k=1

converges uniformly to a continuous matrix, which we denote by G ∈ C(2 ). If μk + 2nk < μk+1 (k = 1, 2, . . . ), then Q(x, μk , nk ) and Q(x, μl , nl ) do not overlap for n = l. 3 If αk = k −2 , μk = nk = 2k , then the continuous matrix G deﬁned above has an expansion G = k∈Z Gk which does not converge in the operator norm. Indeed, the sequence aμk ,nk = {(alμk ,nk )−1 }l≥1 is increasing and also bounded, and, therefore, [aμk ,nk ] ∈ M (2 ), and its norm in M (2 ) is less than a−1 − b−1 . Thus, √ ) * αk (log nk )/ 2 < a−1 − b−1 ||Sμk +nk (G) − Sμk −1 (G)||B(2 ) , which obviously implies that the sequence Sn (G) does not converge in the operator norm. Next we give the matrix analogue of a well-known Jordan’s theorem (see [75]). Theorem 3.18. Let A ∈ C(2 ) ∩ BV (2 ). Then A ∈ U (2 ). Proof.

Let A ∈ C(2 ). Then it follows that σn (A) → A. Now let σn,k (A) =

Sn (A) + Sn+1 (A) + · · · + Sn+k−1 (A) k

(n + k)σn+k−1 (A) − nσn−1 (A) k n n = 1+ σn+k−1 (A) − σn−1 (A). k k =

Note that σn,k (A) = Sn (A) +

n+k−1 ν+1

1−

ν−n k

It follows by the ﬁrst relation that ||σn,k (A) − A||B(2 )

(Aν + A−ν ).

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n n n n = || 1 + σn+k−1 (A) − 1 + A − σn−1 (A) + A||B(2 ) k k k k n n ≤ 1+ ||σn+k−1 (A) − A||B(2 ) + ||σn−1 (A) − A||B(2 ) → 0. k k Thus, if A ∈ C(2 ), that is σν (A) → A, as ν → ∞, and A ∈ BV (2 ), then it follows that |k| ||Ak ||B(2 ) ≤ C for all k ∈ Z and, by the second relation above, we get that ||σn,k (A) − Sn (A)||B(2 ) ≤

n+k−1

||Aν ||B(2 ) ≤ C

|ν|+1

n+k−1 n+1

1 k−1 ≤C . ν n

n n 1 Now, if > 0 and k = [n] + 1, then C k−1 n ≤ C. Since k < n < is bounded, it follows that ||σn (A) − Sn (A)||B(2 ) → 0 and, consequently,

lim sup ||Sn (A) − A||B(2 ) ≤ C. n

Since can be arbitrarily small it follows that ||Sn (A) − A||B(2 ) → 0, as n → ∞. The proof is complete. 3.5

Approximation of inﬁnite matrices by matriceal Haar polynomials

The main goal of this section is to formulate and prove an extension of the approximation theorem of continuous functions by Haar functions (see Theorem A in the subsection 3.5.1) to the case with inﬁnite matrices (see Theorem C in the subsection 3.5.1). The extension to the matriceal framework is based on one side on the fact that periodical functions belonging to L∞ (T) may be identiﬁed one-to-one with Toeplitz matrices from B(2 ) (see Theorem 0 in the subsection 3.5.1), and on the other side on some notions given below; for instance we mention: ms - an unital commutative subalgebra of ∞ , C(2 ) the matriceal analogue of the space of all continuous periodical functions C(T), the matriceal Haar polynomials, etc. After some introductories considerations in subsection 3.5.1, we present some results concerning the ms space, which is important for this generalization, in subsection 3.5.2, and the proof of the main theorem, denoted as Theorem C, is considered in the third subsection.

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Introduction

The classical form of Haar’s theorem. Let T be the one-dimensional torus identiﬁed with the interval [0, 2π). Now we consider the Haar L2 (T)-normalized functions hk given by h0 (t) = 1 for t ∈ T and, for n = 2k + m, k ≥ 0 and m ∈ {0, . . . , 2k − 1}, by ⎧ (k+1) 2k/2 , t ∈ Δ2m , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (k+1) hn (t) = −2k/2 , t ∈ Δ2m+1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (k) 0 , t ∈ T \ Δm , (k)

where Δm = [ 2mk · 2π, m+1 · 2π). 2k We can now state the following well-known theorem of approximation of continuous functions on T (i.e. periodical continuous functions on [0, 2π]) by means of Haar functions (periodically extended on R) due to Haar (see [37]). Theorem A. If f is a continuous function on T (i.e. if f ∈ C(T)) and if > 0, then there exists a Haar polynomial of degree n() ∈ N Sn (f ) =

n−1

αk hk ,

αk ∈ C,

k=0

such that ||f − Sn (f )||L∞ (T) < .

Translation of Theorem A to a matriceal framework. The following result as well as thereafter remark constitute the starting point of the whole theory presented here (see [18]). Theorem 0. A Toeplitz matrix A = (ak )+∞ k=−∞ belongs to B(2 ) if and ∞ only if there exists a unique function fA ∈ L (T) whose Fourier coeﬃcients 2π 1 −int f( dt are equal to an , for n ∈ Z and, moreover, A (n) = 2π 0 f (t)e ||A||B(2 ) = ||fA ||L∞ (T) .

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In order to develop the theory we consider in the previous result two different ”geometric” directions to be followed. Model 1: Diagonal matrix. As we noted already in the introductory chapter, for an infinite matrix A = (aij ), and an integer k, possibly negative, we denote by Ak the matrix whose entries a0i,j are given by ai,j if j − i = k, 0 ai,j = 0 otherwise. Then Ak will be called the kth-diagonal matrix associated to A. In the preceding theorem we remark that there is a one-to-one correspondence between Ak and fbA (k) for A ∈ B(`2 ) and f ∈ L∞ (T). Consequently, we may imagine (Ak )k∈Z , as the ”matriceal Fourier coefficients” associated to the matrix A. Model 2: Corner matrix. In the sequel we use another notation, which is more appropriate for our aims. For the entries of the matrix A, we put al,l+k , k ≥ 0, l = 1, 2, 3 . . . , l ak = al−k,l , k < 0, l = 1, 2, 3, . . . , and denote A sometimes as A = (alk )l≥1, k∈Z . ∗ Let A(l) = (bm k )k∈Z, m≥1 , where l ∈ N , be the matrix given by l ak if m = l, bm k = 0 if m 6= l.

We call the matrix A(l) , the lth-corner matrix associated to A. Now, if for any corner-matrix A(l) = (bm k )k∈Z, m≥1 we associate a distribution on T, denoted by fl such that blk = fbl (k), we get, in case A ∈ T ∩ B(`2 ), that fl = f ∈ L∞ (T), for all l ∈ N∗ . Using the models. a) Model 1. In this case we recall that Ak plays the role of the “kth Fourier coefficient of the matrix A.” Theorem 0 allows us to write the formula [T ∩ B(`2 )]∗ = L∞ (T), where by [H]∗ we denote the image of the space H of matrices by the correspondence A → fA .

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Remark 3.19. For brevity we write in what follows equations like the previous one in the following manner: T ∩ B(2 ) = L∞ (T), C(2 ) ∩ T = C(T).

b) Model 2. We can identify the matrix A = (A(l) )l∈N∗ with its sequence of associated distributions f = (fl )l∈N∗ , writing this fact as A = Af . By Theorem 0 we have the following correspondences: f ∈ L∞ (T) if and only if Af ∈ T ∩ B(2 )

f = (f, f, f, . . . )

g ∈ L∞ (T) if and only if Ag ∈ T ∩ B(2 )

g = (g, g, g, . . . ).

Then, of course, it follows that f g ∈ L∞ (T) if and only if Afg ∈ T ∩ B(2 ) where fg = (f g, f g, f g, . . . ). We recall that the matrix A = (aij ) is said to be of n-band type if aij = 0 for |i − j| > n. Having these notions in mind we introduce a commutative product of inﬁnite matrices: Deﬁnition 3.20. Let A = Af and B = Ag be two inﬁnite matrices of ﬁnite band type. We introduce now the commutative product given by AB := Afg .

Remark 3.21. (1) We mention that in the previous deﬁnition since A = Af , and B = Ag are inﬁnite matrices of ﬁnite band type it yields that f and g are trigonometric polynomials, and we may consider the product f g. 2) This product can be deﬁned also for all matrices A, B ∈ B(2 ), but AB does then not in general belong to B(2 ) as can be easily seen. 3) Of course, if Af , Ag ∈ T ∩B(2 ), then it follows that Af Ag = Afg ∈ T ∩ B(2 ).

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We end the presentation of this model by considering an important particular case: Let α = (α1 , α2 , α3 , . . . ) be a sequence of complex numbers and B = Af ∈ B(2 ), where f = (f1 , f2 , . . . ). Considering α as a sequence of constant functions on T, we get, by Deﬁnition 3.20, that Aα B = Aαf , where αf = (α1 f1 , α2 f2 , . . . ). For brevity we denote Aα B by α B. In what follows it will be important to know more about the sequences α satisfying the condition B ∈ B(2 ) ⇒ α B ∈ B(2 ). Actually, the entire next subsection will be devoted to this question, but for the moment, for understanding its implications we will rewrite the operation under a diﬀerent form. We associate to any sequence α = (α1 , α2 , . . . ), the matrix [α] whose entries [α]lk are equal to αl , for l ≥ 1 and k ∈ Z. Then it is clear that α B = [α] ∗ B. Deﬁnition 3.22. We say that the sequence α ∈ ms if it has the property that α B ∈ B(2 )

∀ B ∈ B(2 ),

or, equivalently, [α] ∈ M (2 ). On ms we consider the norm ||α||ms := ||[α]||M (2 ) . Then ms is a unital commutative Banach algebra with respect to usual multiplication of sequences. Remark 3.23. Any constant complex sequence α = (α, α, . . . ) belongs to ms. In order to get an extension of Haar’s theorem we have to ﬁnd the appropriate analogues in the matrix context. They are summarized below: The function case

The matrix case

norm || · ||B(2 ) 1. norm || · ||L∞ (T) 2. space C(T) space C(2 ) 3. multiplication of a function by a scalar multiplication .

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The correspondence given by 3. becomes more transparent if we remark the sequence (α, α, . . . ), that, for α ∈ C and for f ∈ L∞ (T), denoting by α and by f the constant sequence (f, f, . . . ), we get that α Af = [α] ∗ Af = Aαf . Denoting by Hk the Toeplitz matrix associated like in Theorem 0 to the Haar function hk , for k = 0, 1, 2, . . . and by Sn (f ) the constant sequence (Sn (f ), Sn (f ), . . . ), where Sn (f ) = n−1 k=0 αk hk , for f ∈ C(T), αk ∈ C, and k ∈ {0, n−1}, we get the following translation of Theorem A in the Toeplitz matrices setting: Theorem B. Let A = Af ∈ C(2 ) be a Toeplitz matrix and let > 0. Then there is a matriceal polynomial given by ASn (f ) =

n−1 k=0

αk Hk =

n−1

α k Hk

k=0

such that ||A − ASn (f ) ||B(2 ) < , where α k = (αk , αk , . . . ). Now it is natural to ask ourselves about the existence of a class of matrices larger than C(2 ) ∩ T such that Theorem B still holds. The aim of our next Theorem is to give an answer to this question. More precisely, we prove the following theorem also formulated in our preface: Theorem C. Let A = (alk )l≥1, k∈Z be a matrix belonging to C(2 ) such that all sequences ak = (alk )l≥1 , k ∈ Z belong to the class ms. Then, for any > 0 there is an n ∈ N∗ and sequences αk ∈ ms, k ∈ {0, . . . , n − 1} such that ||A −

n−1

αk Hk ||B(2 ) < .

k=0

It is also worthwhile to mention the following open problem: Open problem. Does Theorem C still hold if the matrix A satisﬁes only the condition A ∈ C(2 )? If not, what is the best version of Theorem C in this case?

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3.5.2

About the space ms

As we remarked in the previous subsection (see also the statement of Theorem C) the space ms plays an important role for our theory and, consequently, it is desirable to know more facts about it. In this context, we again note that any constant sequence belongs to ms (see Remark 3.23). Our primary goal here is to prove that this algebra is far more rich than that; this richness will quantify the level of extension of the theorem of Haar in the matrix case, since in the functions case, corresponding to Toeplitz matrices, (see Theorem B) the algebra ms is reduced to exactly the constant sequences. Here is an outlook for this subsection: We give some suﬃcient conditions for a sequence to belong to ms, following two complementary ways: The ﬁrst one is based on deﬁning a particular algebra pms and showing that pms is intimately connected with ms. (See Proposition 3.25.) As a consequence we derive properties for ms displaying some necessary and some suﬃcient conditions for a sequence in order to belong to pms; (see Theorem 3.26) the second approach (Theorem 3.28) is involved with the structure of ms rather than of pms. For an inﬁnite matrix A = (aij )i≥1, j≥1 , we deﬁne its upper triangular projection PT (A) as follows:

ai,j if i ≤ j, PT (A) := 0 otherwise. Deﬁnition 3.24. A sequence b = (bn )n≥1 belongs to pms if and only if B := {b} = PT ([b]) ∈ M (2 ). Then pms endowed with the norm ||b|| = ||{b}||M (2 ) becomes a Banach algebra with respect to usual product of sequences. Proposition 3.25. Let b = (bn )n≥1 be a sequence of complex numbers. Then 1. b ∈ pms ⇒ b ∈ ms (so pms ⊂ ms.) 2. if we write (b1 , b2 , . . . , bn , . . . ) = (b1 , 0, b3 , 0, . . . ) +(0, b2 , 0, b4 , . . . ), or, equivalently, b = b10 + b20 , denoting by b1 = (b1 , b3 , . . . , b2n−1 , . . . ) and by b2 = (b2 , b4 , . . . ), we have that bi ∈ pms ⇔ bi0 ∈ ms for i ∈ {1, 2} and so bi ∈ pms, i ∈ {1, 2} ⇒ b ∈ ms. The proof is obvious, so we leave out the details.

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We now pass to a study of the algebra pms. We introduce a new method for estimating the norm in the space B(2 ). We associate to every sequence x = (xj )j≥1 from 2 (N) the function ∞ h(t) = j=1 xj e2πijt ∈ H02 ([0, 1]), where H02 ([0, 1]) consists of all functions 1 h : [0, 1] → C from the Hardy space H 2 such that 0 h(t)dt = 0. ∞ If A = (akj ) ∈ B(2 ), we deﬁne Lk (t) := j=1 akj e2πijt ∈ H 2 ([0, 1]) . It follows that ∞ 1 1 AB(2 ) = sup ( | Lk (t)h (s − t) dt|2 ) 2 < ∞ for any s. (3.1)

h 2 ≤1 k=1

0

See Chapter 2 for a proof of this formula. Theorem 3.26. Let b = (bn )n≥1 be a sequence of complex numbers. 1) If (in )n≥1 is a strictly increasing sequence of natural numbers with i1 = 0, and zin := maxin 0 such that ||b||ms = ||B||M (2 ) ≤ R inf (||(zin )n≥1 ||2 + ||(zin ln(in+1 − in ))n ||∞ ) . (in )

n+p 2 2 2) If b ∈ pms, then supn≥1;p≥1 lnn n k=p |bk | < ∞. 3) If (|bk |) k≥1 is a decreasing sequence, then b ∈ pms if and only if |bk | = O ln1k . Proof. 1) Let A ∈ B(2 ) and x ∈ 2 (N). By using the relation (3.1) for B A it follows that there exists R1 > 0 such that 2 1 ∞ 2 2 |bk | Lk (t) (h − Sk−1 (h) (−t)) dt , (B A) x2 ≤ R1 k=1

0

where Sk (h) is the Fourier partial sum of order k (i.e. if Dk is the Dirichlet kernel, then Sk (h)(t) = (h Dk ) (t) is the convolution of h and Dk ). Therefore, we have that 2 1 2 ∞ 1 2 2 |bk | Lk (t)Sk−1 (h)(−t)dt + Lk (t)h(−t)dt ||(BA)x||2 ≤ 2 0

k=1

≤2

∞ k=1

|bk |

≤2

1

2

∞ k=1

0

0

2 ∞ 2 Lk (t)Sk−1 (h)(−t)dt + 2||b||∞

2 |bk |

k=1

1 0

0

1

2 Lk (t)h(−t)dt

2 Lk (t)Sk−1 (h)(−t)dt + 2||b||2∞ ||A||2B(2 ) ||h||22 .

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Let (in )n≥1 be a strictly increasing sequence of natural numbers such that i1 = 0. Then we have that ||b||2∞ ≤ || (zin )n≥1 ||22 and

|bk |

∞ k=1

=

n=1 k=in +1

≤2

in+1 ∞

+2

in+1 ∞

∞

≤2

2

1

k=in +1

1 0

2 Lk (t)Sin (h) (−t) dt

2 Lk (t) (Sk−1 − Sin ) (h) (−t) dt

in+1

zi2n

n=1

0

k=in +1

n=1 ∞

1

in+1

zi2n

2 Lk (t)Sk−1 (h) (−t) dt

0

|bk |

0

|bk |

2

n=1 k=in +1

1

2

n=1 k=in +1

0

2 Lk (t)Sk−1 (h) (−t) dt

2 |bk |

in+1 ∞

1

2

1 0

2 Lk (t)Sin (h) (−t) dt +

2 Lk (t)(Sk − Sin ) (h) (−t) dt .

Moreover, by using the formula (3.1), we get that 2 in+1 1 2 Lk (t)Sin (h) (−t) dt ≤ AB(2 ) ||h||22 k=in +1

0

and

in+1

k=in +1

in+1 −in

∼

k=1

1 0

1 0

2 Lk (t)(Sk−1 − Sin ) (h) (−t) dt

2 Dk−1 (t){[Lk+in Sin+1 − Sin (h)](−t)e2πin it }dt

(3.2)

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≤ ×

X Z

in+1 −in k=1

0

1

sup

1≤k≤in+1 −in

||Dk−1 ||L1 (0,1)

2 |Dk−1 (t)| Lk+in ? Sin+1 − Sin (h) (−s) ds Z

≤ C log (in+1 − in )

!

1

sup

0 1≤k≤in+1 −in

|Dk−1 (s)|ds

in+1 −in

X

2 × | Lk+in ? (Sin+1 − Sin )(h) (·)|

k=1

L∞

≤ C 0 ||A||2B(`2 ) ||(Sin+1 − Sin )(h)||22 | ln(in+1 − in )|2 . Thus, using (3.2), we get that k(A ? B) xkB(`2 ) ≤ R kAkB(`2 ) ||h||2 (|| (zin )n≥1 ||2 + ||zin ln (in+1 − in ) ||∞ ). 2) Let B ∈ M (`2 ). Taking A ∈ T ∩ B(`2 ) such that alj = 1j for all j ∈ Z \ {0} and for all ˜ ? A˜ ∈ B(`2 ), where l ∈ N∗ and al0 = 0 for all l ∈ N, we obtain that B   1 1   · · · 1 b1 b1 b1 b1 · · ·   2 3   1  b2 b2 b2 b2 · · ·  1     1 · · ·   ˜ :=  · · · · · · · · · · · · · · ·  and A˜ :=  2 B . 2      bn bn bn bn · · ·  1 1   1 ···  3 2 ··· ··· ··· ··· ··· ··· ··· ··· ··· 1 if k ∈ {p, . . . , n + p} Letting xnp = (xk )k≥1 with xk = , where p, n ∈ 0 otherwise ∗ N are fixed, we get that n+p

X

˜ ˜ n 2 2 ln2 (n + 1) |bk | ≤ C B ? A xp ≤ C(n + 1). 2

k=p

Hence, n+p ln2 (n + 1) X |bk |2 < ∞. n+1 n≥1;p≥1

sup

k=p

3) Let (|bk |)k≥1be a decreasing sequence. Then, according to 2), we get that |bn | = O ln1n .

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Conversely, deﬁning r = (rn )n≥1 with rn = have that

1 ln(n+1)

for all n ≥ 1, we

||B||M (2 ) = ||{b}||M (2 ) ≤ C||{r}||M (2 ) . By choosing in = 2n for all n ≥ 2 and i1 = 0, it follows, by 1), that zin = r2n +1 ∼ n1 . Consequently, $ $ $ ,$ $ 1 $ 1 $ $ $ $ $ $ ln 2n <∞ ||{r}||M (2 ) ≤ R $ $ +$ $ $ n n≥1 $ $ n $ n≥1 2

∞

that is B ∈ M (2 ). The proof is complete.

Observe that results like Theorem 7.1 or Theorem 8.6 [11] cannot be applied in our situation. Remark 3.27. From the previous results we deduce that and that {(bn )n≥1

2 (N) ⊂ ms ⊂ ∞ (N) | |bn | = O ln1n } ⊂ ms, with proper inclusions.

Next we change the view and derive another set of suﬃcients conditions in order that b ∈ ms. These results use the estimate of the absolute value of diﬀerences of sequence’s terms rather than the absolute value of the terms themselves. Theorem 3.28. Let b = (bn )n≥1 be a sequence of complex numbers. n 2 |b − b | (1) If supn≥1 < ∞, then b ∈ ms, j n j=1 ∞ (2) If ||b||BV (N) = |b1 | + n=1 |bn+1 − bn | < ∞, then b ∈ ms. Proof.

1. We use the following result from [11]

A matrix M ∈ M (2 ) iﬀ there exists a P ∈ B(2 , ∞ ) and Q ∈ B(l1 , 2 ) such that M = PQ

and ||M ||M (2 ) ≤ ||P ||2,∞ ||Q|1,2 .

We recall that if Q = (qjk ), j ≥ 1, k ≥ 1 and P = (pjk ), j ≥ 1, k ≥ 1, then ⎛ ⎞ 12 ⎛ ⎞ 12 |qjk |2 ⎠ and ||P ||2,∞ = sup ⎝ |pjk |2 ⎠ . ||Q||1,2 = sup ⎝ k≥1

j≥1

j≥1

k≥1

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Let [b] = Bb + Cb , where

⎛

b1 ⎜ b1 ⎜ Bb = ⎜ b ⎝ 1 .. . and

b2 b2 b2 .. .

b3 b3 b3 .. .

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⎞ ··· ···⎟ ⎟ , ···⎟ ⎠ .. .

⎛

⎞ ··· ···⎟ ⎟ . ···⎟ ⎠ .. . 1 n 2 2 |b − b | < Bb ∈ M (2 ) ∀b ∈ ∞ , and, since ||Cb ||1,2 = supn n j=1 j ∞, by Bennett’s theorem it follows that Cb ∈ M (2 ). 2. If A ∈ B(2 ), (Lk )k≥1 , h ∈ H 2 ([0, 1]), x ∈ l2 (N) and hk = kDk are as ∞ before (3.1) and deﬁning f (t) = j=1 bj e2πijt (in the sense of distributions) we get that 0 b1 − b2 b1 − b3 ⎜0 0 b 2 − b3 ⎜ Cb = ⎜ 0 0 0 ⎝ .. .. .. . . .

∞ k=1

0

([b] A) x22 = 2 1 1 −2πikt f (t) (Lk hk ) (−t)dt + f (t) (Lk (h − hk )) (0)e dt ≤ 0

∞ 2 k=1

1 0

2 gk (s) (Lk h) (−s)ds + 2b2∞ A2B(2 ) x22 ,

where k k−1 gk (s) = j=1 fˆ(k) − fˆ(j) e2πijs = j=1 fˆ(j + 1) − fˆ(j) Dj (s). But 2 ∞ 1 gk (s) (Lk h) (−s)ds ≤ k=1

0

⎛ ⎞ ∞ ∞ ∞ ˆ ˆ 2 ⎝ |(hj Lk ) (0)| ≤ f (j + 1) − fˆ(j)⎠ f (j + 1) − fˆ(j) j=1

j=1

k=2

⎛ ⎞2 ∞ ˆ ⎝ f (j + 1) − fˆ(j)⎠ h22 A2B(2 ) , j=1

so that ([b] A) x2 ≤ CAB(2 ) x2 [b] ∈ M (2 ).

∞ j=1

|bj+1 − bj | + b∞ , that is

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Extension of Haar’s theorem

As we announced this section is dedicated to prove the generalized Haar theorem - see Theorem C from subsection 3.5.1. We start our exposition by introducing a useful vector space E(2 ). After that, we deﬁne the notion of generalized scalar product for matrices which allows us to give a more useful form for E(2 ) and also to see some similarities with what happens in the function case. Proposition 3.29 is useful to identify the constraints in the deﬁnition of E(2 ) and also to point out some of the diﬃculties in this theory. Finally, we deﬁne the space Cr (2 ) and prove that this space admits a Schauder type decomposition (see Proposition 3.30, Theorem 3.31). In this more general frame Theorem C will follow as a Corollary. We deﬁne the vector space E(2 ) as follows: n E(2 ) := {A ∈ B(2 ); A = k=0 αk Hk , such that αk Hk ∈ B(2 ) for all 0 ≤ k ≤ n, n ∈ N}, where αk ∈ ∞ , and αk Hk = [αk ]Hk . We introduce a generalized scalar product of matrices < A, B > for A = Af and B = Bg , where f = (f1 , f2 , . . . ) and g = (g1 , g2 , . . . ), in the following way: < A, B > = (< f1 , g1 >, < f2 , g2 >, . . . ). We say that a family (Φk )k∈N is an orthonormal system if the following orthogonality relations hold: Φk , Φl = 0 ∈ ∞ for k = l and Φk , Φk = 1 ∈ ∞ for all k ∈ N∗ . By the orthogonality of the system (Hk )k we deduce that A ∈ E(2 ) implies that A = nl=1 < A, Hk > Hl ∈ E (2 ) . Therefore, n E (2 ) = {A ∈ B (2 ) ; A = l=1 A, Hl Hl , such that A, Hl Hl ∈ B (2 ) for all l ≤ n, n ∈ N∗ }. Proposition 3.29. 1) There is A ∈ B (2 ) such that A, H1 ∈ ∞ and / B (2 ) . A, H1 H1 ∈ 2) If 0 ] ∈ M (2 ), which, in turn, implies that < A, Hk > Hk ∈ B(2 ) for any k ∈ N∗ . ∗ l Proof. 1) Let A = A1 with a12k−1 = 1, a2k 1 = 0 for k ∈ N and ak = 0, if

k = 1 and k ∈ N . Then A, H1 = (x1 , 0, x1 , 0, . . .) ∈ ∞ , where x1 is some constant.

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2i Hence A, H1 H1 = x1 B, where π ⎛ 0 1 0 13 ⎜ ⎜ −1 0 0 0 ⎜ ⎜ ⎜ 0 0 0 1 ⎜ ⎜ ⎜ B = ⎜ − 13 0 −1 0 ⎜ ⎜ ⎜ 0 0 0 0 ⎜ ⎜ 1 ⎜ − 5 0 − 13 0 ⎝ .. . . . . . . . . . .

0

1 5

0

0

0

1 3

0

0

0

1

−1 0 .. .. . .

57

⎞ ... .. ⎟ .⎟ ⎟ .. ⎟ .⎟ ⎟ .. ⎟ .⎟ ⎟. ⎟ .. ⎟ .⎟ ⎟ .. ⎟ .⎟ ⎠ .. .

But, ||I − PT (B)||B(2 ) is inﬁnite and ||I − PT (B)||B(2 ) ≤ ||B||B(2 ) , where I is the unit for the usual non-commutative multiplication of inﬁnite matrices. This result is not surprising, since, by using Proposition 3.25 and Theorem 3.26, we obtain that (x1 , 0, x1 , 0, . . . ) ∈ ms ⇔ x1 = 0. 2) Let p ≤ 2. By [94], it yields that any A ∈ B(2 ) belongs to Sp if and p only if, for any orthonormal basis (ek ) in 2 , we have that k Aek < ∞, p p ∞ ∞ ∞ 2 2 2 2 hence ∞ < ∞, j=1 < ∞. k=1 j=1 |akj | k=1 |akj | Thus, by using Cauchy-Schwarz inequality and the above inequalities, we get that < A, Hk > pp ≤ CHk pB(2 ) ApSp < ∞ for some constant C > 0. By Remark 3.27 it follows that [< A, Hk >] ∈ M (2 ). The last implication is now obvious. The proof is complete. Observe also that there exists an A ∈ B (2 ) such that A, Hk Hk ∈ B (2 ) , for all k ∈ N, but for a k0 ∈ N, we ﬁnd that A, Hk0 ∈ / ms. Indeed A = A0 = (an )n≥1 ∈ ∞ \ ms gives the answer to the above problem for k0 = 0. Therefore, in the deﬁnition of E(2 ), we prefer the weaker condition < A, Hk > Hk ∈ B(2 ) for all k rather than < A, Hk > ∈ ms for all k. On the ms-module E(2 ) we consider the norm m |||A||| := sup < A, Hk > Hk . m≤n k=0

B(2 )

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Since E(2 ) ∩ T can be identiﬁed with Ed ([0, 1]), the space of all dyadic step functions, whose completion with respect to the supremum norm is equal to the space of all countable piecewise continuous functions with discontinuities at dyadic points of [0, 1]. This space is denoted by Cr ([0, 1]), and we call Cr (2 ) the completion of (E(2 ), ||| · |||). In what follows we give some known classes of matrices which are embedded in Cr (2 ) . Examples. 1) Obviously all Toeplitz matrices, associated to functions from Cr ([0, 1]) belong to Cr (2 ) . 2) The Hilbert-Schmidt matrices A = alj , j ∈ Z, l ≥ 1, with AHS = 1 √ ∞ ∞ l 2 2 < ∞ belong to Cr (2 ) and ACr (2 ) ≤ 2 AHS . j=1 l=1 aj ∨ ∞ 2πijt 2πijt = ∞ a e . We denote by g (t) = g (−t) and Pl j=−∞ j j=l aj e Then, by the Fubini theorem and the Cauchy-Schwarz inequality, we get that 2 ∞ 1 2 PT (Sn (A))B(2 ) = sup Sn (fl ) (Pl g) (−t) dt

g H 2 ([0,1]) ≤1 l=1

=

sup

∞

g H 2 ([0,1]) ≤1 l=1 2

≤ AHS ·

1 0

0

2 ∨ fl Sn Pl g (−t) dt

sup

g H 2 ([0,1]) ≤1

$ $2 $ ∨$ 2 $Pl g $ 2 = AHS . L

√ Hence, ACr (2 ) ≤ 2 AHS . 3) Let A be a diagonal matrix having as non-zero entries the elements of the sequence α = (αi )i≥1 ∈ ms. Then A ∈ Cr (2 ) and ACr (2 ) ≤ αms . The proof is straightforward using the trivial observations that ms is an algebra with respect to usual multiplication and Cr (2 ) is a ms-module with |||α X||| ≤ αms · |||X|||. ∞ 4) If A = alj j∈Z,l≥1 is such that j=−∞ aj ms < ∞, where aj := l ∞ aj l≥1 , then ACr (2 ) ≤ j=−∞ aj ms and A ∈ Cr (2 ) . This statement follows easily from 3). 5) If A is the main diagonal matrix having as non-zero entries the elements aj with (aj )j ∈ ∞ , then A ∈ Cr (2 ) and AB(2 ) = ACr (2 ) . (Note that (aj )j may not belong to ms.) Proposition 3.30. If the sequence of matrices (An )n≥1 is a Cauchy sequence of E (2 ) with respect to norm |||·|||, then An , Hk Hk converges to

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some αk Hk in this norm. Moreover, αk Hk ∈ B (2 ) and An , Hk → αk n in ∞ . Proof. Step I. We ﬁrst prove that for all k ∈ N and A ∈ B (2 ) . (3.3) A, Hk ∞ ≤ 2 AB(2 ) If A = Af , where f = (f1 , f2 , . . . ), and Ql A is the matrix with entries

l aj k = l, j ∈ Z [Ql A]kj = , 0 k = l. then, by the Cauchy-Schwarz inequality, it follows that ⎛ ⎞ 12 ∞ l 2 a ⎠ || A, Hk ||∞ = (fl , hk )l ∞ ≤ sup fl L2 = sup ⎝ j l∈N∗

l∈N∗

√ ≤ 2 sup Ql AB(2 ) ≤ 2 AB(2 ) .

j=−∞

l∈N∗

n

Step II. Let now (A )n≥1 be a Cauchy sequence in E (2 ) . Then, for a ﬁxed k ∈ N, we have that An , Hk → αk in ∞ . n

Indeed, using (3.3) and the fact that AB(2 ) ≤ |||A|||, the statement follows by Step I. Step III. (An , Hk Hk )n≥1 is a Cauchy sequence of E(2 ) for all k and for a Cauchy sequence (Ann≥1 ) in E(2 ). Hence An , Hk Hk → B k ∈ Cr (2 ) in the norm ||| · |||. Thus, by (3.3), it follows n $ . /$ that lim $An , Hk − B k , Hk $∞ = 0, and by Step II it follows that n / . αk = B k , H k . . / Step IV. If we show that B k = B k , Hk Hk , then Proposition 3.30 is proved. But by Step III we have that An , Hk Hk → B k in B (2 ) . n

Then the entries of the matrices An , Hk Hk converge with respect to n k to the corresponding to . . k entry / of the matrix B . Moreover, according / Step n I, A , Hk → B , Hk in ∞ and, hence, it follows that B k , Hk Hk n

= B k . The proof is complete.

We use Proposition 3.30 in order to prove the existence of some kind of Schauder basis in Cr (2 ) given by the sequence (Hk )k≥0 . More speciﬁcally, we have the following result: Theorem 3.31. Let A ∈ Cr (2 ) . Then we have the decomposition ∞ A= A, Hk Hk , k=0

in the norm ||| · |||.

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Proof. Let A ∈ Cr (2 ) . Then there exists a Cauchy sequence An ∈ E (2 ) such that A = limAn . n

By Proposition 3.30 we obtain that limn→∞ ||| An , Hk Hk − αk Hk ||| = 0 for all k ≥ 0. Let ε > 0. Therefore, there exists nε ≥ 0 such that for all n ≥ nε , and k > j, we get that |||

k

A , Hi Hi −

i=j

k i=j

n

k

αi Hi ||| ≤ lim sup ||| m→∞

i=j

Am , Hi Hi ||| + lim

m→∞

k

k

An , Hi Hi −

(3.4)

i=j

||| Am , Hi Hi − αi Hi ||| ≤ ε.

i=j

By the orthogonality relations satisﬁed by the sequence (Hk )k and using k (3.4), we ﬁnd that there exists a number lε such that ||| i=j αi Hi ||| < ε for all k > j > lε . ∞ Thus, i=0 αi Hi = B ∈ Cr (2 ) . But, by taking j = 0 and k ≥ k(n) max (k (n) , lε ) , where i=0 An , Hi Hi = An , in (3.4), we ﬁnd that, for k(n) all ε > 0, and for all n ≥ nε , |||An − i=0 αi Hi ||| < ε. ∞ Thus A = B = i=0 αi Hi and, using the orthogonality relations satisﬁed by (Hk )k and the fact that the operator A → A, Hi : Cr (2 ) → ∞ ∞ is continuous, we conclude that A = i=0 < A, Hi > Hi the series being convergent in Cr (2 ). The proof is complete. In particular, we get the following extension of Haar’s theorem for matrices: Corollary 3.32. Let A ∈ Cr (2 ) . Then A = norm of B (2 ) .

∞ k=0

A, Hk Hk , in the

Of course there exists A ∈ C (2 )\Cr (2 ) , for instance A is the diagonal matrix A1 given by the sequence (an )n≥1 , where a2n−1 = 1 and a2n = 0 for all n = 1, 2, 3, . . . Proof of Theorem C. Let A be an inﬁnite matrix as in Theorem C and let > 0. Since A ∈ C(2 ) there is k ∈ N such that ||σk (A) − A||B(2 ) < . 2 Then, by hypothesis and by Example 4, it follows that σk (A) ∈ Cr (2 ), and n−1 consequently, by Theorem 3.31, there is a Haar polynomial i=0 αi Hi n−1 such that ||σk (A) − i=0 αi Hi ||B(2 ) < . The proof is complete. 2

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3.6

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61

Lipschitz spaces of matrices; a characterization

Let 1 ≤ p < ∞ and let A ∈ Cp . We deﬁne

2π 1/p ||A(x + h) − A(x)||pCp dx , ωp (δ) = ωp (δ; A) := sup 0
0

where ωp (δ) is called the p-modulus of continuity of A. If ωp (δ) ≤ M δ α , where 0 < α < 1, then, we write A ∈ Lip(α, p ; 2 ) and we say that A satisﬁes a Lipschitz condition in the p-metric. Now we indicate a relation between the diﬀerentiability properties of the matrix A and the behaviour of its ”Fourier coeﬃcients” Ak . Theorem 3.33. Let A ∈ C2 . If A ∈ Lip(α, 2 ; 2 ) then

||Ak ||2C2 ≤ C1

|k|≥n

(0 < α < 1), 1 n2α

(n ≥ 1),

(3.5)

(3.6)

and conversely. Proof.

Since ∞ πh πh kπh A(x + ) − A(x − ) ∼ 2i Ak , eikx sin 4 4 4 k=−∞

then 1 2π

0

2π

∞ kπh πh 2 πh ) − A(x − )|| dx = 4 . (3.7) ||A(x + ||Ak ||2C2 sin2 4 4 C2 4 k=−∞

If A ∈ Lip(α, 2; 2 ) (0 < α < 1), then, by (3.7), it follows that ∞ kπh ||Ak ||2C2 sin2 4 ≤ 2Ch2α , 4 k=−∞

where C does not depend on h. Thus, ||Ak ||2C2 ≤ Ch2α , 1 2 h ≤|k|< h

or, equivalently 2r h

r+1 ≤|k|< 2 h

||Ak ||2C2 ≤ C

h2α 22rα

(r = 0, 1, 2, . . . ).

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Adding these inequalities we ﬁnd that 4α ||Ak ||2C2 ≤ C α h2α , 4 − 1 1 |k|≥ h

and, consequently, (3.5) implies (3.6). ) * Assume now (3.6). Put in (3.6) n = h1 + 1. Then, in view of (3.7), it follows that 2π π 2 h2 2 πh 2 1 πh ) − A(x − )||C2 dx ≤ 4C1 h2α + 4 k , ||A(x + ||Ak ||2C2 2π 0 4 4 16 1 |k|≤ h

and we must only estimate Ih =

h2 k 2 ||Ak ||2C2 .

|k|≤n−1

Let γk =

∞

{||Am ||2C2 + ||A−m ||2C2 }

(k > 0),

m=k

such that γk ≤ C1

1 . k 2α

Represent now Ih as Ih = h2

n−1

k 2 (γk − γk+1 ).

k=1

Thus, Ih ≤ h2 {γ1 + γ2 (22 − 1) + γ3 (32 − 22 ) + · · · + γn−1 [(n − 1)2 − (n − 2)2 ]} ≤ 2h2

n−1

kγk ≤ 2C1 h2

k=1

n−1 k=1

1 k 2α−1

≤ C h2 (n − 1)2−2α ≤ C h2α ,

where C does not depend on h. Hence, 0 π3 πh α 8πC1 + ≤h C, ω2 2 2 and the proof is complete.

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We note that 2π ||A(x) − min Br

0

n−1 r=−n+1

Br eirx ||2C2 dx = 2π

||Ak ||2C2 .

|k|≥n

This formula represents the square of the best approximation in the 2metric of the matrix A with respect to band type matrices of the order < n. (2) If we denote this last expression with En [A], then the previous Theorem becomes: If ω2 (δ; A) ≤ Cδ α (0 < α < 1), then for every positive integer n 1 En(2) ≤ C α , n and conversely.

Notes In this chapter we investigate some spaces of inﬁnite matrices from the harmonic analysis point of view. Namely, in Section 3.2 we introduce the space of all inﬁnite matrices corresponding to linear bounded operators such that they are approximable in the operator norm by ﬁnite type band matrices. For such matrices, which are extensions of periodical continuous functions on the torus, we develop a Fejer theory for this situation. In Section 3.3 we introduce and study a space, which is an analogue of the classical space L1 (T). For the matrices belonging to this space a version of Riemann-Lebesgue Lemma holds. In Section 3.4 we brieﬂy investigate the matrix version of functions of bounded variation and we prove an analogue of the well-known Jordan’s Theorem about the uniform convergence of partial sums in the Fourier development of a continuous function. Section 3.5 is dedicated to some problems concerning matriceal approximation and it extends the well-known Haar Theorem about the development of an essentially bounded function in a series of Haar functions. In order to give this theorem we introduce an interesting space of sequences, denoted by ms, and we study some matrices, which are extensions of usual scalars, from the point of view of Schur multipliers. Finally, in Section 3.6 we investigate a class of matrices from the point of view of approximation with partial sums of its diagonals.

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Chapter 4

Matrix versions of Hardy spaces

4.1

First properties of matriceal Hardy space

The results from this section were communicated to us by V. Lie. We introduce a matrix version of the Hardy space, which will coincide with the classical one on the class of all Toeplitz matrices T . Let A = (ajk )j≥1; k≥1 be an inﬁnite matrix. We associate with A the matriceal periodical distribution (function) LA (x, t) on [0, 1] × [0, 1] deﬁned by LA (x, t) :=

∞ ∞

akj e2πijx e2πikt .

k=1 j=1

The above relation may be rewritten as LA (x, t) =

∞

2πikt LA = k (x)e

∞

CjA (t)e2πijx ,

j=1

k=1

LA k

where is the distribution (function) associated with row k whereas CjA is the distribution associated with column j. Because we work only with upper triangular matrices it is convenient to deﬁne A −2πikx . LA k (x) := Lk (x)e

Using these notations we have that LA (x, t) =

∞

2πik(x+t) LA . k (x)e

k=1

We remark that if A is an upper triangular Toeplitz matrix, then LA k := ∞ 2πik(x+t) A A L for all k, and LA (x, t) = L (x) k=1 e . 65

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The importance of matriceal distributions in what follows is stressed by the following equalities: 1 1 LC (t − μ, v)LD (μ, s − v)dμdv, LC∗D (t, s) = 0

0

LCD (x, t) =

1 0

LD (x, s)LC (−s, t)ds.

For an inﬁnite matrix A we denote by A the matrix whose matriceal distribution is given by ∞ LA (x, t) := LA (x)e2πik(x+t) . k k=1

We remark that the above deﬁnition coincides with the following implication: Ak , then A = kAk , if A = k∈Z

k∈Z

where Ak is the kth diagonal of A. Remark 4.1. If we consider the distributions LA k as acting on the torus, that is, if A 2πix ) LA k (x) = Lk (e and if we work only with upper triangular matrices A, then we may regard LA k as being a limit (in the space of distributions) at the border of an analytic distribution, that is 2πix 2πix) lim LA ) = LA . k (re k (e r→1

In this way we arrive at the following notation: Given an upper triangular matrix A we say that A(r) (0 < r < 1) is the analytic extension of A, if ∞ 2πix 2πik(x+t) LA(r) (x, t) = )e . LA k (re k=1

Now we extend in the framework of matrices the deﬁnition of classical Lebesgue spaces Lp , for 1 ≤ p ≤ 2. Deﬁnition 4.2. For 1 ≤ p ≤ 2 we deﬁne the spaces Lpr (2 ) as Lpr (2 ) := {A | ||A||Lpr (2 ) < ∞}, where

⎛ ||A||Lpr (2 ) =

sup ||x||2 (N) ≤1

⎝

1 0

∞ k=1

p/2 2 2 |LA k (s)| |xk |

⎞1/p ds⎠

.

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The above deﬁnition reﬂects the behaviour of the matrix A with respect to its rows; of course we may also deﬁne the spaces corresponding to the columns of A, namely Lpc (2 ) := {A | ||A||Lpc (2 ) < ∞}, where ⎛ p/2 ⎞1/p 1 ∞ ||A||Lpc (2 ) = sup ⎝ |CkA (s)|2 |xk |2 ds⎠ . ||x||2 (N) ≤1

0

k=1

Remark. Let A0 be the main diagonal submatrix of a matrix A ∈ L (2 ). Since ||A0 ||A∈L2 (2 ) = supk |ak0 |, it follows that L2 (2 ) is not isomorphic to a Hilbert space. This remark will be of special interest in Chapter 6. 2

For simplicity in what follows we write Lp (2 ) instead of LA r (2 ). Deﬁnition 4.3. We deﬁne the matriceal Hardy space H p (2 ) of index p, 1 ≤ p ≤ 2, in the following way: H p (2 ) := {A | A upper triangular ; A ∈ Lp (2 )}. Here

⎛

||A||H p (2 ) :=

sup ||x||2 (N) ≤1

⎝

0

1

∞

p/2 2 2 |LA k (s)| |xk |

⎞1/p ds⎠

.

k=1

An interesting property of the space H 1 (2 ) is the following HardyLittlewood type inequality: Proposition 4.4. Let A ∈ H 1 (2 ). Then we have that ⎧ ⎞2 ⎫1/2 ⎛ 1/2 1 ∞ ⎪ ⎪ ⎪ ⎪ 2 A 2πiθ 2 ⎬ ⎨ 1 r )| dθ ⎟ j=1 |xj | |(Lj ) (se 0 ⎜ sup ds (1 − r) ⎝ ⎠ dr ⎪ 1−s ||x||≤1 ⎪ 0 ⎪ ⎪ ⎭ ⎩ 0 ≤ ||A||H 1 (2 ) . Proof. Let x ∈ 2 (N) be given, so that ||x||2 ≤ 1, and let g ∈ L2 ([0, 1]) be a ﬁxed positive function such that ||g||2 ≤ 1. Using Khintchine’s inequality we get that, for all s ∈ [0, 1], 1 ∞ 2πiθ 2 1/2 ( |xj |2 |(LA )| ) dθ ∼ j ) (se 0

j=1

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1

0

1

0

|

∞

2πiθ xj j (ω)(LA )|dθdω. j ) (se

j=1

Here j (ω) stands for the jth Rademacher function. By duality, it is enough to prove that 1 r 1 1 ∞ 2πiθ √ | j=1 xj j (ω)(LA )|dθdω j ) (se 0 0 dsdr 1 − rg(r) 1−s 0 0 ! I

≤ ||A||H 1 (2 ) . Applying Fubini’s theorem we ﬁnd that I := r 1 ∞ 1 1 A ) (se2πiθ )|dθ √ | x (ω)( L j j j j=1 0 ds)dr dω ≤ 1 − rg(r)( 1−s 0 0 0 ⎛ 2 ⎞1/2 1 ∞ 1 1 2πiθ r | j=1 xj j (ω)(LA )|dθ j ) (se 0 ⎝ (1 − r) ds dr⎠ dω. 1−s 0 0 0 We get, by using Cauchy-Schwarz inequality, that 1 2 1 r 2πiθ |f (se )|dθ 0 (1 − r) ds dr ≤ 1−s 0 0 r 1 r 2 1 ds 2πiθ (1 − r) |f (se )|dθ ds dr ≤ 2 0 0 (1 − s) 0 0 2 1 r 1 2πiθ |f (se )|dθ ds dr ≤ (by Fubini’s theorem) ≤ 0

1 0

0

(1 − r)

0

0

1

|f (re

2πiθ

2 dr ≤ (by inequality (HL) on page 4)

)|dθ

≤ C 2 ||f ||2H 1 . ∞ 2πiθ ), we get from Hence, denoting by f (se2πiθ ) = j=1 xj j (ω)(LA j )(se the inequalities above that 1 1 ∞ 2πiθ | xj j (ω)(LA )|dθdω ∼ (by Fubini’s theorem and I≤C j )(e 0

0

j=1

Khintchine’s inequality ) ∼

1 0

⎛ ⎞1/2 ∞ 2πiθ 2 ⎠ ⎝ |xj |2 |(LA )| dθ j )(e j=1

≤ ||A||H 1 (2 ) ||x||2 . The proof is complete.

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4.2

Hardy-Schatten spaces

In 1983 A. Shields proved an interesting inequality which holds in Schatten class C1 (see [84]). This inequality is similar to the following well-known inequality of Hardy and Littlewood (see e.g. [96]):

∞

1/p (n + 1)

p−2

|an |

p

≤ C(p)||f ||H p ,

0 < p ≤ 2,

n=0

which holds for all functions f (t) = n≥0 an eint belonging to the Hardy spaces H p (T). (See [23].) The similarity between functions and inﬁnite matrices was remarked for the ﬁrst time by J. Arazy in [1] and A. Shields exploited this crucial idea further in [84]. It is natural to consider the Schatten class of order p as the similar notion of the Lebesgue space Lp (T), 0 < p < ∞ (see [84]). We note that the above analogy is not perfect since for 0 < p1 < p2 < ∞, it follows that Lp2 (T) ⊂ Lp1 (T) contrarily to the known inclusion Cp1 ⊂ Cp2 . Moreover, the analogue of the Riesz projection P0 (f ) := n≥0 an eint , where f is a trigonometric polynomial, is the triangular projection PT (A) = (aij )i,j≥1 , where

aij if 0 ≤ j − i ≤ k , A = (aij )i,j≥1 and aij = 0 if |i − j| > k aij = 0 otherwise for some ﬁxed k ∈ N. Therefore, the analogue of the Hardy space H p (T), 0 < p < ∞ is the space Tp := {A| A upper triangular matrix; A ∈ Cp }, with the norm ||A||Tp = ||A||Cp . On the other hand, if f (t) = an eint , then it follows that ||f ||H p = |an |, and, thus, the corresponding object to |an | would be ||An ||Cp = ||(ai,i+n )i ||p , where An = (ai,i+n )i≥1 , n ∈ N. Thus we may expect that the following inequalities hold: ∞ (n + 1)p−2 ||An ||pCp ≤ K(p)||A||pTp (4.1) n=0

for all A ∈ Tp , 1 ≤ p ≤ 2, or, equivalently: ∞ ∞ p−2 p (n + 1) |ai,i+n | ≤ K(p)||A||pTp , 1 ≤ p ≤ 2. n=0

i=1

(4.2)

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For p = 1 the inequality (4.2) holds with K(1) = π, that is ∞ ∞ −1 (n + 1) |ai,i+n | ≤ π||A||T1 , A ∈ T1 n=0

i=1

which was proved by Shields in 1983 (see [84]). We discuss this fact further in the next section. For another, more general, proof see [15]-Thm. 2.2-a) and also Section 4.4. Now we brieﬂy describe the main content of the section. First we consider the case 1 < p ≤ 2 and prove (only for 1 < p < 2) a weaker result then (4.1). However, we conjecture that (4.1) holds also in general in the case 1 < p < 2. Of course, (4.1) is easy to prove in the case p = 2 with K(2) = 1. Moreover, it is possible to state and prove a result dual to the ﬁrst inequality. In order to state and prove the result we have to recall some notations from the paper [53]. Let A be an upper triangular matrix and let (Ak )k≥0 be the sequence of its diagonal matrices. We denote by Tp (2R ) the completion of the space of all ﬁnite sequences (Ak )nk=0 with respect to the norm n A∗k Ak )1/2 ||Cp , where 1 ≤ p ≤ 2. ||(Ak )nk=0 ||Tp (2R ) := ||( k=0

Here A∗k is the adjoint matrix of Ak . It is clear that Tp (2R ) is a space of upper triangular matrices. Similarly, Tp (2C ) is the completion of the space of all ﬁnite sequences (Ak )nk=0 with respect to the norm n Ak A∗k )1/2 ||Cp . ||(Ak )nk=0 ||Tp (2C ) := ||( k=0

Tp (2C ) is a space of upper triangular matrices too. Now let us denote by Tp (2R ) + Tp (2C ) the space of all upper triangular matrices A such that there exist A ∈ Tp (2R ) and A ∈ Tp (2C ) with A = A + A . We introduce on this space the norm 1/2 1/2 ||A||Tp (2R )+Tp (2C ) := inf {||( A∗ ||Cp +||( Ak A∗ ||Cp }, k Ak ) k ) Ak =Ak +Ak

1 ≤ p ≤ 2. Let us remark that ||(

∞ k=0

⎡

A∗k Ak )1/2 ||Cp = ⎣

k≥0

j ∞ j=1

i=1

k≥0

p/2 ⎤1/p ⎦ , for 1 ≤ p ≤ 2, |aij |2

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and, similarly, that ⎡ ||(

∞

⎢ Ak A∗k )1/2 ||Cp = ⎣

∞

⎛ ⎝

i=1

k=0

so that ||A||Tp (2R )+Tp (2C )

∞

⎞p/2 ⎤1/p ⎥ |aij |2 ⎠ ⎦ ,

j=i

⎧⎡ j p/2 ⎤1/p ⎪ ∞ ⎨ ⎦ ⎣ = inf |aij |2 ⎪ Ak =Ak +A ,k≥0 k ⎩ j=1 i=1

(4.3)

⎫ ⎛ ⎞p/2 ⎤1/p ⎪ ⎪ ∞ ∞ ⎬ ⎢ ⎝ 2 ⎠ ⎥ +⎣ |aij | . ⎦ ⎪ ⎪ i=1 j=i ⎭ ⎡

Moreover, the relation (I.13)-[53] implies that ⎞1/2 ⎛ ⎝ ||Ak ||2Cp ⎠ ≤ ||A||Tp (2R )+Tp (2C ) .

(4.4)

k≥0

We also note that

⎞1/p

⎛

||An ||Tp (2R )+Tp (2C ) = ||An ||Cp = ⎝

|ai,i+n |p ⎠

, 1 ≤ p ≤ 2.

(4.5)

i≥1

Now we state and prove the following result: Theorem 4.5. Let 1 < p < 2 and A ∈ Tp (2R ) + Tp (2C ). Then there exists a positive constant K(p) such that ∞

(n + 1)p−2 ||An ||pCp ≤ K(p)||A||Tp (2R )+Tp (2C ) .

(4.6)

n=0

Proof. We follow the idea of the proof in [23] pp. 95-97. n Let A be an upper triangular matrix such that A = k=0 Ak , and let μ be the measure on N deﬁned by 1 μ(n) = , n = 0, 1, 2, . . . (n + 1)2 int Let us consider A(t) = ∞ = A ∗ Et , where t ∈ T, ∗ means n=0 An e the Schur product of matrices and Et = (ekj )k,j≥1 is the Toeplitz matrix, where ekj = ei(j−k)t , for all j, k ≥ 1. Therefore, ||A(t)||Cp ≤ ||A||Tp .

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Now denote (n + 1)An by A˜n and ﬁx s > 0. Then, A(t) = Φs (t) + Ψs (t), t ∈ T, where

A(t) if ||A(t)||Tp > s Φs (t) = , 0 if ||A(t)||Tp ≤ s and Ψs (t) = A(t) − Φs (t), t ∈ T. It is clear that [A(t)]n = [Φs (t)]n + [Ψs (t)]n , n = 0, 1, 2, . . . ; and, therefore, An ∗ [Et ]n = [Φs ]n ∗ [Et ]n + [Ψs ]n ∗ [Et ]n , where [Φs ]n = An and [Ψs ]n = 0 if ||A||Tp = supt∈T ||A(t)||Tp > s. The same holds for Ψs . Hence, (n + 1)p−2 ||An ||pCp = ||A˜n ||pCp μ(n) (4.7) n≥0

⎛ ≤ 2p ⎝

n≥0

˜ s ]n ||p μ(n) + ||[Φ Cp

n≥0

⎞ ˜ s ]n ||p μ(n)⎠ . ||[Ψ Cp

n≥0

˜ s ]n ||C > s} and β(s) = μ{n; ||[Ψ ˜ s ]n ||C > Put now α(s) = μ{n; ||[Φ p p s} := μ(Es ). Then ||[Ψs ]n ||2Cp . (4.8) s2 β(s) ≤ n≥0

˜ s ]n ||C > s}, by Fs . We denote the set {n ≥ 0; ||[Φ p Since ||An ||Cp ≤ ||A||Cp , n ∈ N, (see [34]) we have that μ(Fs ) = μ{n; (n + 1)||[Φs ]n ||Cp > s} ≤ μ{n; (n + 1)||Φs ||Cp > s} = s {n;||Φs ||Cp > n+1 }

1 ≤4 (n + 1)2

where n0 = min{n; ||Φs ||Cp > Thus,

s {n;||Φs ||Cp > n+1 }

n+1 n

s n+1 }.

α(s) = μ(Fs ) ≤

4||Φs ||Cp , s > 0. s

According to (4.9) we have that ∞ ∞ p ˜ s ]n ||p μ(n) = − ||[Φ s dα(s) = p Cp 0

n=0

∞

4p 0

sp−1

4 1 , dx ≤ 2 x n0 + 1

||Φs ||Cp ds ≤ 4p s

0

||A||Cp

∞ 0

(4.9)

sp−1 α(s)ds ≤

||A||Cp sp−2 ds ≤

4p ||A||pTp , p−1

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for all t ∈ T. Moreover, by (4.8) and (4.4), it follows that ∞ ∞ ∞ ∞ p−1 p−3 ˜ s ]n ||p μ(n) = p ||[Ψ s β(s)ds ≤ p s ||[Ψs ]n ||2Cp ds = Cp 0

n=0

p

∞

∞

||A||Cp n=0

0

||An ||2Cp sp−3 ds ≤

n=0

∞ 2p p−2 ||An ||2Cp ≤ ||A||C p 2−p n=0

p ||A||Tp−2 ||A||2Tp (2 )+Tp (2 ) . p R C 2−p By using (4.7) we get that ∞

: ; 2 (n+1)p−2 ||An ||pCp ≤ K(p) ||A||pTp + ||A||Tp−2 ||A|| 2 )+T (2 ) . (4.10) ( T p p p R

n=0

C

But in [38] it is proved that ||A||Cp ≤ ||A||Tp (2R )+Tp (2C ) and, consequently, we get the following inequality of Hardy-Littlewood type:

(n + 1)p−2 ||An ||pCp ≤ K(p)||A||pTp (2 )+Tp (2 ) , 1 < p < 2. R

n≥0

C

(4.11)

Since Tp (2R )+Tp (2C ) is the completion of the space of all ﬁnite sequences Theorem 4.5 is proved. Remark 4.6. Theorem 4.5 is weaker than the following strong version of the Hardy-Littlewood inequality: the inequality (n + 1)p−2 ||An ||pCp ≤ K(p)||A||pTp , (4.12) n≥0

holds for all upper triangular matrices A. Indeed in [38] it is proved that there is an upper triangular matrix A ∈ Cp such that A ∈ Tp (2R ) + Tp (2C ), 1 < p < 2. Then it is natural to raise the following: Question: Does Theorem 4.5 hold for 1 =

∞ k=0

tr (Ak Bk∗ ),

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where A =

Ak , B =

Bk , we have the following: Theorem 4.7. Let 2 ≤ q < ∞ and A = k≥0 Ak such that n≥0 (n + 1)q−2 ||An ||qCq < ∞. Then A ∈ Tq (2R ) ∩ Tq (2C ) and ⎛ ⎞ j ∞ ∞ ∞ |ai,j |2 )q/2 )1/q , ( ( |ai,j |2 )q/2 )1/q ⎠ ||A||Tq (2R )∩Tq (2C ) := max ⎝( ( k≥0

k≥0

i=1 j=i

⎛

≤ C(q) ⎝

j=1 i=1

⎞1/q (n + 1)q−2 ||An ||qCq ⎠

.

n≥0

Proof. Let p = q/(q − 1) and G = nk=0 Gk be a ﬁnite type band matrix with ||G||Tp (2R )+Tp (2C ) ≤ 1. Let Sn (A) = nk=0 Ak . Then n n ∞ | < G, Sn (A) > | = | tr (Gk A∗k )| ≤ | gi,k+i ai,k+i | ≤ (by H¨ older’s k=0 i=1

k=0

inequality) ≤

n k=0

(

|gi,k+i |p )1/p (

i

|ai,k+i |q )1/q =

i

n

||Gk ||Cp ||Ak ||Cq

k=0

n ≤ (again by H¨older’s inequality) ≤ ( ||Gk ||pCp (k + 1)p−2 )1/p k=0

(

n

||Ak ||qCq (k + 1)q−2 )1/q ≤ (by Theorem 4.5)

k=0

≤ C(p)||G||Tp (2R )+Tp (2C )

n

1/q ||Ak ||qCq (k

+ 1)

q−2

k=0

≤ C(p)

n

1/q ||Ak ||qCq (k

+ 1)

q−2

.

k=0

Hence, ||Sn (A)||Tq (2R )∩Tq (2C ) = ≤ C(p)

n

sup ||G||T

2 2 ≤1 p (R )+Tp (C )

||Ak ||qCq (k

+ 1)q−2

| < G, Sn (A) > | 1/q ,

k=0

for all n and the proof is complete.

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Matrix versions of Hardy spaces

Now we discuss an inequality of Hausdorﬀ-Young type. Theorem 4.8. (Hausdorﬀ-Young’s inequality) For 1 ≤ p ≤ ∞, let q be the conjugate index, with p1 + 1q = 1. 1/q ∞ q ≤ ||A||Tp . (i) If 1 ≤ p ≤ 2, then A ∈ Tp implies that n=0 ||An ||Tp (ii) If 2 ≤ p ≤ ∞, then {||An ||Tp } ∈ q implies that ||A||Tp ≤

∞

1/q ||An ||qTp

.

n=0

Proof.

In case (ii), for p = q = 2, if

∞ ∞

2 1/2 < n=0 l=1 |al,n+l | ∞ ∞ 2 1/2 , n=0 l=1 |al,n+l |

∞, then

in other clearly A = (aij ) ∈ T2 and ||A||T2 ≤ words the map T ({An }n≥0 ) = A has norm less than 1 from the n≥0 n space 2 (2 ) into T2 . Here 2 (2 ) means the space of all matrices (aij )i,j ∞ ∞ such that n=1 i=1 |ai,i+n |2 < ∞. If p = ∞ it follows that q = 1 and the map T has the norm less than 1 ∞ from 1 (∞ ) into T∞ , since, clearly, ||A||T∞ ≤ n=0 ||An ||T∞ . Using complex interpolation (see [19]), we get the conclusion. Case (i) follows by duality, since p (q )∗ = q (p ), and (Tp )∗ = Tq .

4.3

An analogue of the Hardy inequality in T1

In this section we present an important inequality due to A. Shields [84]. In fact, the paper of Shields containing this inequality was the starting point of the material described in the present book. Let us state the analogue of the inequality of Hardy, Littlewood and Fej´er. In the next section we give also a more general useful inequality, which will be proved using diﬀerent methods. Theorem 4.9. Let M ∈ C1 have the upper triangular form with respect to the orthonormal basis {en } (n = 1, 2, . . . ) of 2 . Then k ∞ |M (j, k)| ≤ π||M ||T1 , 1+k−j j=1 k=1

with equality only when M = 0.

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It is easy to observe that another form of the above inequality is as follows: ∞ 1 ||Mk ||T1 ≤ π||M ||T1 . k+1 k=0

This inequality is similar with the classical inequality of Hardy [23]. In order to prove Theorem 4.9 we require three lemmas. Throughout our discussion the orthonormal basis {en }, n = 1, 2, . . . will be ﬁxed; upper triangularity will always be with respect to this basis. Lemma 4.10. Let R denote either the space B(2 ), with the weak operator topology, or any of the Banach spaces Cp (1 < p < ∞) with its weak topology. If {An }, {B n } ⊆ R, with An → A and B n → B weakly, and if each B n has the upper triangular form, then An B n → AB weakly. Proof. We write as usual in this book operators as matrices. One can easily verify that if {An } ⊂ R, then An → A weakly if and only if {||An ||R } is a bounded sequence, and An (i, j) → A(i, j) for all i, j. Thus to complete the proof we must show that a) ||An B n ||R are bounded, and b) dn (i, j) → d(i, j) for all i, j. Here we let dn and d denote the matrix entries of An B n and of AB. Concerning the ﬁrst point we recall that in Cp we have that ||An B n ||Tp ≤ ||An B n ||Cp/2 ≤ ||An ||Cp ||B n ||Cp (see [34], Chap. III, (7.4) and (7.5)). Thus {||An B n ||R } is a bounded sequence. For the second point we note that

dn (i, j) =

j

An (i, k)B n (k, j)

k=1 n

since B is upper triangular. A similar equation holds for d(i, j), and, thus, for each ﬁxed choice of i, j, it yields that dn (i, j) → d(i, j). The proof is complete. Lemma 4.11. Let P ∈ C1 with ||P ||T1 = 1, be a positive semi-deﬁnite operator. Then there exists B ∈ C2 with ||B||C2 = 1 such that B has the upper triangular form and P = B ∗ B.

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Proof. Let us denote by En the subspace span{e1 , ..., en }, for all n ∈ N. We ﬁrst prove the lemma under the additional assumption that P is oneto-one on each of the subspaces En (n ≥ 1). Then P 1/2 is also one-toone on each of these spaces. Let Fn = P 1/2 En . Then F1 ⊂ F2 ⊂ . . . , and dim Fn for all n. Hence, there is an orthonormal set {fk } such that Fn = span{f1 , . . . , fn }. Deﬁne an operator V by: V fn = en (n ≥ 1), and V = 0 on the orthogonal complement of the span of {fn }. Then V is a partial isometry. Let B = V P 1/2 . Then BEn ⊂ En and, thus, B has the upper triangular form. Moreover, B ∗ B = P 1/2 V ∗ V P 1/2 = P, since V ∗ V is the projection onto the span of {fn }, which contains the range of P 1/2 . Finally, ||B||T2 ≤ ||V || ||P 1/2 ||T2 = 1, and, therefore, 1 = ||P ||T1 ≤ ||B ∗ ||T2 ||T ||T2 ≤ 1, which completes the proof in this case. Now suppose that P is not one-to-one. Let S be a ﬁxed positive operator from C1 with trivial kernel. Let P n = (P + n−1 S)dn , where dn = ||P + n n−1 S||−1 has norm one, and P n → P, in C1 . In view of the T1 . Then P result proved above, there is a sequence {B n } of operators in C2 , having the upper triangular form, with P n = (B n )∗ B n , ||B n ||T2 = 1, for all n. By passing to a subsequence we may assume that {B n } is weakly convergent in C2 : B n → B for some B in the unit ball of C2 . The limit operator B must have the upper triangular form and, by Lemma 4.10, we have that B ∗ B = P. From this we have that ||B||T2 ≥ 1 and, hence, the norm must be equal to unity. The proof is complete. Lemma 4.12. Let M ∈ C1 have the upper triangular form with ||M ||T1 = 1. Then there exist upper triangular operators A, B ∈ C2 with M = AB and ||A||T2 = ||B||T2 = 1. Proof. We ﬁrst prove the lemma with the additional assumption that M is one-to-one on each of the spaces En ; this is equivalent to requiring that all diagonal matrix entries are diﬀerent from 0; < M ej , ej > = 0 for all j. Let M = U P be the polar decomposition of M. Then P = (M ∗ M )1/2 is a positive operator of norm one in C1 and U maps the range of P isometrically onto the range of M. Since ||P f || = ||M f || for all f we see that P has the same kernel as M. Therefore, P is one-to-one on each of the spaces En . By Lemma 3.3, P = B ∗ B, where B is an upper triangular operator of norm one in C2 . We see that B must be one-to-one on each of the spaces En . Now let A = U B ∗ . Then A is in the unit ball of C2 , and AB = M. From this we see that ||A||C2 = 1. To show that A has the upper triangular form we must show that it maps each space En into itself. Since B is one-to-one

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on En and En is ﬁnite-dimensional we have that En = BEn . Hence, AEn = ABEn = M En = En . Now suppose that M is not one-to-one on each of the spaces En , that is, some diagonal matrix entries are 0. Let S be a matrix from C1 , with non-zero diagonal entries precisely in those places where M has a zero. Let n M n = (M + n−1 S)dn , where dn = ||M + n−1 S||−1 C1 . Then M satisﬁes all the conditions of the lemma and, in addition, is one-to-one on each of the spaces En . By what was proved above there are upper triangular operators An , B n in the unit ball of C2 , Ball(C2 ), with M n = An B n . By passing to a subsequence we may assume that the sequences {An } and {B n } are weakly convergent in C2 : An → A, B n → B, where A, B ∈ Ball(C2 ). By Lemma 4.10 we have that An B n → AB, and so M = AB. This completes the proof since weak convergence preserves the upper triangular form. Proof of Theorem 4.9. Without loss of generality we may assume that ||M ||T1 = 1. Then by Lemma 4.12 there are upper triangular operators A, B of norm one in C2 such that M = AB. Let mij , aij , bij denote the matrix entries of M, A, B, respectively. The following summations are written with each variable going from 1 to ∞. Because of the upper triangularity, however, the terms are equal to zero if j > k, or if j > r, or if r > k. Thus, we really have that 1 ≤ j ≤ r ≤ k < ∞. We use the boundedness of the second Hilbert matrix, that is the matrix with entries (n − m)−1 when n = m, and 0 when n = m (n, m = 1, 2, . . . ); then we use the CauchyBuniakovsky-Schwarz inequality and obvious estimates to obtain that j,k

|ajr brk | |ajr brk | |mjk | ≤ = 1+k−j 1+k−j 1+k−j r r j,k

≤π

⎛

≤ π⎝

⎞1/2 |ajr |2 ⎠

j

r

⎝

r

⎛

j,k

j

1/2 |brk |2

k

⎞1/2 |ajr |2 ⎠

r

1/2 |brk |2

= π.

k

We have strict inequality because the bound π for the second Hilbert matrix is not attended. The proof is complete.

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4.4

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The Hardy inequality for matrix-valued analytic functions

Here we present another proof of the previous inequality of Shields. In fact we present a more general inequality for vector-valued analytic functions emphasizing the special case of matrix Banach spaces. All the results of this section are due to O. Blasco and A. Pelczynski [15]. We recall that if f = j≥0 aj eijt is an analytic trigonometric polynomial, then π |aj |(j + 1)−1 ≤ C1 |f (t)|dt, −π

j≥0

⎛ ⎝

k≥0

⎞1/2 |a2k |

2⎠

≤ C2

π −π

|f (t)|dt,

where C1 and C2 are numerical constants independent of f (cf. [23]). The ﬁrst fact is called the Hardy inequality; the second is a particular case of a theorem of Paley, where (2k ) is replaced by any sequence (nk ) of positive integers with inf k nk+1 /nk > 1. It is also known that both of these inequalities are false if analytic trigonometric polynomials are replaced by arbitrary trigonometric polynomials. In what follows we are interested in ﬁnding under which additional conditions on a Banach space X the inequalities remain true if the Fourier coeﬃcients aj ’s are elements of X and absolute values are everywhere replaced by norms. We remark that it is known that in that general setting for arbitrary Banach spaces the inequalities are false (see for instance [15]). It appears that the validity of X-valued versions of these inequalities depends on geometric properties of X. We are specially interested in the case when X is some matrix space, for instance if X = C1 , the Banach space of all trace class matrices. p The main idea of the proofs is to use vector-valued Hardy spaces HX and to consider and use some operators induced by bounded multipliers from H 1 into 1 . 4.4.1

p Vector-valued Hardy spaces HX

All Banach spaces are considered to be taken over the complex number ﬁeld C. Given a Banach space X and p ∈ [1, ∞) (respectively p = ∞) we denote

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by LpX the space of all X-valued 2π-periodic functions on the real line R, which are Bochner absolutely integrable in the pth power (respectively essentially bounded) under the norm " #1/p π −1 p ||f (t)|| dt for 1 ≤ p < ∞ ||f ||p = (2π) −π

(respectively ||f ||∞ = ess supt∈R ||f (t)||). Given f ∈ L1X and an integer j, the jth Fourier coeﬃcient of f is deﬁned by π −1 f (j) = (2π) e−ijt f (t)dt. −π

If for some nonnegative integer n, f(j) = 0 for |j| > n, then f is called an X-valued trigonometric polynomial of degree ≤ n; if, moreover, f(j) = 0 for j < 0, then f is called an X-valued analytic trigonometric polynomial. p is deﬁned to be the closure of all Given p ∈ [1, ∞) the Hardy space HX X-valued analytic trigonometric polynomials under the norm || · ||p ; or in other words p = {f ∈ LpX : f(j) = 0 for j < 0}. HX 4.4.2

(H p − q )-multipliers and induced operators for vector-valued functions

Let m = (mj )j≥0 be a complex sequence and let X be a Banach space. Deﬁne the operator mX from X-valued analytic trigonometric polynomials into the eventually zero X-valued sequences by mX (f ) = (mj f(j))j≥0 . We call mX the operator induced by the multiplier m. The operator mX is said to be (p, q)-bounded provided that there exists a constant K = K(m, X) such that, for every X-valued analytic trigonometric polynomial f, ⎛ ⎞1/q ∞ ⎝ ||mj f(j)||q ⎠ ≤ K||f ||p . j=0

If mX is (p, q)-bounded, then it can be uniquely extended to an operator p into the Banach space (q )X , where (also denoted by mX ) from HX 1/q ||xj ||q < ∞}. (q )X = {(xj ) ⊂ X : ||(xj )||q = We call m an (H p − q )-multiplier if, for X = C, mC is (p, q)-bounded. Deﬁnition 4.13. A Banach space X is of (H 1 − 1 )-Fourier type provided that, for every (H 1 − 1 )-multiplier m, the induced multiplier mX is (1, 1)bounded.

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Recall the following beautiful description of (H 1 − 1 )-multipliers given by Ch. Feﬀerman in an unpublished manuscript: Theorem 4.14. A scalar sequence m = (mj )j≥0 is an (H 1 − 1 ) multiplier if and only if ⎛ ⎞2 ⎞1/2 ⎛ (k+1)s ∞ ⎜ ⎟ ⎝ ρ(m) = ⎝|m0 |2 + |m1 |2 + sup |mj |⎠ ⎠ < ∞. s≥1

k=1

j=ks+1

Following the lines of the paper [88] we give a proof of the above theorem. Let us denote by Λ the lattice of all integers of R and by Qα the interval {x ∈ R : α − /2 ≤ x < α + /2}, where α ∈ Λ and > 0. Now we state the following theorem belonging to Sledd and Stegenga [88]: Theorem 4.15. Let μ be a positive Borel measure on R \ {0}. Then |f|dμ < ∞ (4.13) sup ||f ||H 1 (R) ≤1

if and only if sup >0

μ(Qα )2

1/2

< ∞.

(4.14)

Moreover, the corresponding suprema are equivalent. Corollary 4.16. Let {mα }α∈Λ be nonnegative numbers and deﬁne a mea sure on R \ {0} by μ = α =0 mα δα , where δα is the point mass at x = α. Then sup |f(α)|mα < ∞ (4.15) ||f ||H 1 (T) ≤1 α =0

if and only if μ satisﬁes condition (4.14). It is easy to see that Corollary 4.16 is nothing else than Theorem 4.14. Now we proceed to the proof of Theorem 4.15. Proof. We recall that an atom a(x) corresponding to an interval Q is a measurable function supported on Q which has zero mean and is bounded by |Q|−1 (| · | meaning the Lebesgue measure). By a fundamental result of R. Coifman [21] we may take as a deﬁnition of a function f of H 1 (R) the equality f = i λi ai , where i |λi | < ∞ and ||f ||H 1 (R) = inf{ i |λi |}, for all {λi } as before.

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Thus, the suﬃciency of condition (4.14) follows if there is a c < ∞ with | a|dμ ≤ c (4.16) for all atoms a. Part of the proof of (4.16) is straightforward. If a is an atom corresponding to an interval of length δ, then it is easy to see that | a(y)| ≤ c|y|δ for y ∈ Q0 , where = δ −1 . Here c is a constant nondepending of a. Now it is clear that (4.14) implies |x|dμ(x) ≤ c −1 Q0

and, hence, (4.16) will follow from R\Q0

| a|dμ ≤ c

(4.17)

where is related to a as above. This result is now easily seen to be a consequence of condition (4.14) and the following theorem Theorem 4.17. There is a constant c < ∞ such that if a(x) is an atom corresponding to an interval with the length 2δ and = δ −1 , then sup | a|2 ≤ c. α

Qα

Proof. It suﬃces to assume that a is smooth and supported in the interval [−δ/2, δ/2]. Fix an interval I of length and assume that f is continuously diﬀer entiable on I. It is elementary to see that supI |f − b| ≤ I |f |, where b is the average |I|−1 I f. Hence, " # 1 |f |2 + |f |2 . sup |f |2 ≤ 2 I I I Normalizing the Fourier transform so that ||f ||2 = ||f||2 , we obtain that # " ∞ ∞ 1 2 2 2 sup | a| ≤ 2 | a| + | a| −∞ −∞ α Qα

δ/2 1 δ/2 2 2 =2 |a| + |2πixa| dx −δ/2 −δ/2 from which Theorem 4.17 follows.

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The following Lemma is useful for our purposes. Lemma 4.18. Let g ∈ L2 (R) and assume that g = 0 on |y| ≤ 1. If f = gχ [−1,1] , then f ∈ H 1 and ||f ||H 1 ≤ c||g||2 . (Here χ[−1,1] is the characteristic function for the unit interval centered at the origin.) Proof. Assume that g is a C ∞ -function with compact support in |y| > 1. Then f is the convolution g ∗ χ[−1,1] and, hence, is a rapidly decreasing

function, which vanishes in a neighborhood of the origin. Thus f is in H 1 . If u ∈ BM O(R) and b is its average over [−1, 1], then, by the Schwarz inequality, 1/2

|u − b|2 dx ≤ c||g||2 ||u||BM O . | f u| = | f (u − b)| ≤ c||g||2 1 + |x|2

The ﬁrst inequality is a well-known estimate for χ [−1,1] and the second a slight extension of inequality (1.2) in [32]. Now use the duality. The proof of Theorem 4.15 is complete once we establish the necessity of the condition (4.14). However, if (4.13) holds with supremum A, then from Lemma 4.18 we deduce that (4.18) [μ(y + [−1, 1])]2 dy ≤ cA2 . Hence, there is an M < ∞, δ > 0, for which |α|≥M μ(Qδα ) ≤ cA2 , where c is an absolute constant. But then a dilation argument gives this inequality for all δ > 0 and (4.14) now follows in an elementary way. Thus, also the proof of Theorem 4.15 is complete. Proof of Corollary 4.16. The space H01 (T) is the subspace of H 1 (T) consisting of functions with zero mean. Given f ∈ H 1 (R), we deﬁne P f (x) := f (x + α). α∈Λ 1

Since f ∈ L (R) we have that P f ∈ L1 (T) and, by the Poisson summation formula (see [90]), it follows that f(α) = (P f )(α) for α ∈ Λ. The proof of the corollary is an immediate consequence of the following theorem: Theorem 4.19. It yields that P (H 1 (R)) = H01 (T). Proof. Let φ be a nonnegative rapidly decreasing function for which φ has support contained in the open interval (−1, 1) and φ(0) = 1. Put ϕ (x) = 2πiαx ϕ(x) for 0 < < 1. For a polynomial F (x) = aα e let f = F ∗ ϕ .

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Claim. lim→0 ||f ||H 1 (R) ≤ ||F ||H 1 (T) . We start with the easily derived fact that lim

→0

R

g|ϕ | =

gdx

(4.19)

T

for all continuous functions g on T. Observe that ||ϕ ||1 = 1. Let S, R denote the Riesz projections on T, R. Then, by (4.19), we obtain that lim sup[||f ||1 + ||Rf ||1 ] ≤ ||F ||H 1 (T) + lim sup ||Rf − (SF )(ϕ )||1 →0

→0

so that we must show that the second term on the right hand side is zero. Since F is a polynomial it suﬃces to ﬁx α ∈ Λ with α = 0, put (y − α) h (y − α) = (y/|y| − α|α|)ϕ and show that lim→0 || h ||1 = 0. Now ϕ is supported in [−, ]. Thus, we may assume that h (y) = m(y)ϕ (y), where m is smooth, all derivatives up to order 2 are bounded by a constant, and |m(y)| ≤ c|y|. The conditions on m imply that ||Dh ||1 ≤ c, where D = d2 /dy 2 . Hence, | h (x)| ≤ c|x|−2 . Clearly, lim→0 ||h ||1 = h ||∞ = 0 and, thus, the above estimate implies that 0 so that lim→0 || lim→0 || h ||1 = 0. This proves the claim. To complete the proof we ﬁx F in H01 (T) and note that there are poly nomials Fn ∈ H01 (T) with ||Fn ||H 1 < ∞ and F = Fn . Using the above ||fn ||H 1 (R) < ∞ and P fn = Fn . Thus, we ﬁnd that fn ∈ H 1 (R) with P f = F where f = fn is a function in H 1 (R). The proof is complete. By summing up we note that also the proof of Theorem 4.14 is complete. In the sequel F M means the Banach space of all scalar sequences satisfying the above relation equipped with the norm ρ(·). Now we present a dual description of (H 1 − 1 ) Fourier type spaces. Proposition 4.20. For every Banach space X the following statements are equivalent: (i) X is an (H 1 − 1 ) Fourier type space; 1 , (ii) there is C > 0 such that, for every m ∈ F M and f ∈ HX ||mj f (j)|| ≤ Cρ(m)||f ||1 ; j≥0

(iii) there is C > 0 such that for every eventually zero sequence (x∗j )j≥0 of elements of X ∗ there is an X ∗ -valued trigonometric polynomial g ∗ such that g∗ = x∗j for j ≥ 0; ||g ∗ ||∞ ≤ Cρ((||x∗j ||)j≥0 ).

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Proof.

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(i) ⇒ (ii). Put 1 ||mj f(j)||; f ∈ HX ; ||f ||1 = 1}. ρX (m) = sup{ j≥0

Using the Baire category argument we get that ρX (·) is a bounded norm on F M. 1 the linear functional (ii) ⇒ (iii). Let x∗j = 0 for j ≥ N. We deﬁne on HX N ∗ ∗ ∗ 1 φ0 by φ0 (f ) = j=0 xj (fj ) for f ∈ HX . It follows from (ii) that ||φ∗0 || ≤ Cρ((||x∗j ||)j≥0 ). Let φ∗ be a norm-preserving extension of φ∗0 onto L1X . Let V be the N th de la Vall`e Poussin kernel, i.e., V (j) = 1 for |j| ≤ N, V (j) = 0 for |j| ≥ 2N and V (j) linear for −2N ≤ j ≤ −N and for N ≤ j ≤ 2N. It is well known that ||V ||1 ≤ 2. Deﬁne, for j = 0, ±1, ±2, . . . , yj∗ ∈ X ∗ by yj∗ (x) = V (j)φ∗ (xej ) for x ∈ X, where ej (t) = eijt . Put g ∗ (t) = |j|≤2N yj∗ eijt . Then (denoting by a ∗ b the convolution of the functions a and b) < f, g ∗ > = φ∗ (V ∗ f )

for f ∈ L1X .

Thus, using the inequality ||V ||1 ≤ 2, we get that ||g ∗ ||∞ ≤ ||φ∗ || ||V ||1 ≤ 2Cρ((||x∗j ||)j≥0 ). 1 On the other hand, taking into account that x · ej ∈ HX for j ≥ 0 and x ∈ X, we get that, for 0 ≤ j ≤ N,

g∗ (j)(x) = yj∗ (x) = V (j)ϕ∗ (xej ) = V (j)ϕ∗0 (xej ) = x∗j (x). Thus, g∗ (j) = x∗j for 0 ≤ j ≤ N. (iii) ⇒ (i). Let m ∈ F M and let f be an X-valued analytic trigonometric polynomial of degree N. For j = 0, 1, . . . , N pick yj∗ ∈ X so that ||yj∗ || = 1 and yj∗ (f(j)) = ||f(j)||. Put x∗j = |mj |yj∗ for 0 ≤ j ≤ N and x∗j = 0 for j > N. Obviously, ρ((||x∗j ||j≥0 )) ≤ ρ(m). We have that ∞

||mj f(j)|| = < f, g ∗ > ≤ ||g∗ ||∞ ||f ||1 ≤ Cρ((||x∗j ||j≥0 ))||f ||1 ≤

j=0

Cρ(m)||f ||1 . Hence mX is (1, 1)-bounded. The proof is complete.

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Now we recall that a Banach space Y is crudely ﬁnitely representable in a Banach space X if there is K ≥ 1 such that for every ﬁnite-dimensional subspace E of Y there is a linear operator u : E → X such that ||e|| ≤ ||u(e)|| ≤ K||e|| for e ∈ E. As a useful example consider for 1 ≤ p < ∞ p (D) of all X-valued analytic functions on the unit disk D = the space HX p and {z ∈ C; |z| < 1} such that for each 0 < r < 1 the function Fr ∈ HX p it |||F |||p = sup{||Fr ||p : 0 < r < 1} < ∞, where Fr (t) = F (re ). Clearly HX p isometrically embeds into HX (D). Conversely, it is not hard to verify that p p HX (D) is crudely ﬁnitely representable in HX . Deﬁnition 4.13 clearly yields the following: Corollary 4.21. Every Banach space crudely ﬁnitely representable in a space of (H 1 − 1 )-Fourier type is an (H 1 − 1 )-Fourier type space. Let (μ, Ω) be a measure space and let X be a Banach space. By L1X (μ) we denote the space of X-valued Bochner μ integrable functions on Ω. Proposition 4.22. If X is an (H 1 − 1 )-Fourier type space, then so is L1X (μ). Proof. It is enough to show that (1 )X is of (H 1 − 1 )-Fourier type, because for every measure space (μ, Ω), L1X (μ) is ﬁnitely representable in (1 )X . Let f = (fk ) be an (1 )X -valued analytic trigonometric polynomial. Then obviously each of the coordinates fk is an X-valued analytic trigonometric polynomial. Hence, by the hypothesis on X there is C > 0, such that, for every m ∈ F M, ∞ ||mj fk (j)||X ≤ Cρ(m)||fk ||X (k = 0, 1, . . . ). j=0

Summing over all k, we get that ∞ ∞ ∞ ∞ ||mj f(j)||(1 )X = ||mj fk (j)||X ≤ Cρ(m)||fk ||X j=0

j=0 k=0

= Cρ(m)||f ||(1 )X .

k=0

Another obvious but useful consequence of Deﬁnition 4.13 is the following: Corollary 4.23. Assume that a Banach space X satisﬁes the following condition:

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1 (∗) there is C > 0 such that for every f ∈ HX there is a complex valued 1 function ϕ ∈ H such that

||f(j)|| ≤ |ϕ(j)| for j = 0, 1, . . . ;

||ϕ||1 ≤ C||f ||1 .

(4.20)

Then X is a (H 1 − 1 )-Fourier type space. Theorem 4.24. The space C1 satisﬁes (∗) with C = 1 + , for all > 0, and, hence, it is a (H 1 − 1 )-Fourier type space. The proof of Theorem 4.24 is a consequence of the following result: Theorem 4.25. (The noncommutative factorization theorem [45].) Let > 0. For every f ∈ HC1 1 there are g and h in HC2 2 such that f = g · h,

(1 + )||f ||C1 ,1 ≥ ||g||C2 ,2 ||h||C2 ,2 ≥ ||f ||C1 ,1 .

Here f = g · h means that f (t) = g(t) · h(t) for t ∈ R, i.e. at each point t the matrix f (t) is the product of the matrix h(t) with the matrix g(t). Moreover, 1/r π ||f (t)||rCp dt for 1 ≤ p < ∞ and 1 ≤ r < ∞. ||f ||Cp ,r := (2π)−1 −π

Remark: In fact Theorem 4.25 holds also for = 0. This strong version is due to Sarason [85]. We need only the weaker -version of Sarason’s theorem. In fact, we need only a much weaker version of Corollary 4.23 (see [84]), namely the following: Corollary 4.26. Assume that the analytic matrix Banach space X satisﬁes the following condition: (∗) there is C > 0 such that for every A ∈ X there is a complex valued function ϕ ∈ H 1 such that for j = 0, 1, . . . ; ||Aj || ≤ |ϕ(j)|

||ϕ||1 ≤ C||A||X .

(4.21)

Then X has the following property: If there is a constant K > 0 such that, of complex numbers (mj ) andf ∈ H 1 we have that for a sequence ∞ ∞ j=0 |mj f j| ≤ K||f ||H 1 , then j=0 ||mj Aj ||X ≤ K||A||X . Next we present a proof of Theorem 4.24 by using Sarason’s factorization theorem as in the Haagerup and Pisier’s proof (see [45]):

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Proof of Theorem 4.24. Let f ∈ HC1 1 and g, h ∈ HC2 2 satisfy the relation of Theorem 4.25. Put j ∞ ϕ(t) = || g (k)||C2 || h(j − k)||C2 eijt . j=0 k=0

By using the inequality ||A · B||C1 ≤ ||A||C2 ||B||C2 and that f = g · h, we get that |ϕ(j)| ≥

j

|| g (k) · h(j − k)||C1 ≥ ||

k=0

j

g(k) · h(j − k)||C1 = ||f(j)||C1 .

k=0

On the other hand, note that ϕ = G · H, where ∞ ∞ G= || g (k)||C2 eijt , H = || h(j)||C2 eijt . j=0

j=0

Next we observe that ||G||2 = ||g||C2 ,2 and ||H||2 = ||h||C2 ,2 . Now using the Schwarz inequality and Theorem 4.25 we get that ||ϕ||1 ≤ ||G||2 ||H||2 = ||g||C2 ,2 ||h||C2 ,2 ≤ (1 + )||f ||C1 ,1 . The proof is complete. Corollary 4.27. The dual of B(2 ) has (H 1 − 1 )-Fourier type. Proof. We have C1∗ = B(2 ). But it is well-known by the Local Reﬂexivity Principle [54] that the second dual of any Banach space is ﬁnitely representable in the space. Hence, the proof of the Corollary follows. Now we give the proof of Theorem 4.25 as given in [45]. Proof of Theorem 4.25. First we recall the well-known fact (see [35]) 2 may be isometrically identiﬁed that the projective tensor product 2 ⊗ with C1 . Here 2 ⊗2 is the completion of algebraic tensor product 2 ⊗ 2 under the norm ∞ ||u|| = inf{ ||xi || · ||yi ||; u = xi ⊗ yi }. i

i=1

Then, in the framework of tensor products the statement of Theorem 4.25 can be reformulated as follows: , there are sequences (gk ) and (hk ) in H22 Let > 0. For any f ∈ H1 ⊗ 2 2 such that ∞ gk (z) ⊗ hk (z) (4.22) ∀ z ∈ D f (z) = k=1

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and ||f ||H 1

2 ⊗ 2

≤

∞

||gk ||H2 ||hk ||H2 . 2

k=1

2

(4.23)

Indeed, let (en ) denote the canonical basis of 2 . Let us denote by (gij (z)) and (hij (z)) the coeﬃcients of the matrices g(z) and h(z) relative to the basis (ei ⊗ ej ), and similarly for f. We have, by the ﬁrst relation in Theorem 4.25, that gik (z)hkj (z) fij (z) = k

and, hence, f (z) =

fij (z)ei ⊗ ej =

gk (z) ⊗ hk (z),

k

where gk (z) =

gik ei and hk (z) =

i

hkj ej .

j

This proves that (4.22) and (4.23) follow from the relations in Theorem 4.25. (The converse direction is also easy.) Let us prove the relations (4.22) and (4.23). We denote by P the linear subspace of HC1 1 formed by all the polynomials with coeﬃcients in 2 ⊗ 2 . Moreover, we denote by || ||1 the norm in HC1 1 , and by || ||2 the norm in H22 . Clearly, for every f in P there are polynomials with coeﬃcients in 2 gi , hi such that ∀z ∈ D

f (z) =

n

gi (z) ⊗ hi (z).

i=1

We introduce a norm on P by n ||gi ||2 ||hi ||2 }, ||f || = inf{ i

where the inﬁmum runs over all possible representations. Note that we obviously have that, ||f ||1 ≤ ||f || and || || is indeed a norm on P. The main point of the proof of Theorem 4.25 is to check that actually this ”new” norm ||f || coincides with ||f ||1 . Using duality, we will show that

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this follows rather directly from known results in the theory of vectorial Hankel operators due to S. Parott [70]. To explain this more precisely, we need to identify the dual spaces to P equipped with the norms || ||1 and || ||. Let us denote by Λ the space of all sequences a = (an )n≥0 with an ∈ B(2 ) such that the Hankel matrix Ha with entries (Ha )ij = ai+j (i ≥ 2 . By deﬁnition, 0, j ≥ 0) deﬁnes a bounded operator on 2 (2 ) = 2 ⊗ we set ||a|| = ||Ha ||. (For a deﬁnition of a Hankel matrix and for some its properties see [65] and also Chapter 5.) Let us denote by X (resp. X1 ) the normed space obtained by equipping P with the norm || || (resp. || ||1 ). We may introduce a duality between P and Λ as follows. Let (fn ) denote the Taylor coeﬃcients of an element f in P. Then, for all a in Λ, we deﬁne ∞ < an , fn > . < a, f > := n=0

(Note that this sum is ﬁnite.) With this duality, we have that ||a||X ∗ = sup < a, g ⊗ h > = sup

(aij gj , hi ) = ||Ha ||,

ij

where each of the above suprema runs over all g, h in P such that ||g||2 ≤ 1 and ||h||2 ≤ 1. (Of course we have that ||g||2 = ( ||gj ||2 )1/2 .) This shows that Λ can be naturally identiﬁed isometrically with the dual of X. the space of all sequences α = (αn )n∈Z Similarly, let us denote by Λ with αn ∈ X ∗ ⊂ B(2 ) such that the matrix Tα deﬁned by (Tα )ij = αi+j

∀i, j ∈ Z

(4.24)

deﬁnes a bounded operator on 2 (Z, 2 ). By deﬁnition, we set ||α||Λ := ||Tα ||. isometrically. Here again it is simple to check that L1 (T, C1 )∗ = Λ 2 (2 ) Equivalently, this means that the natural mapping from L2 (2 )⊗L into L1 (C1 ) is a metric surjection. This can be viewed as a consequence 1 and the fact that every scalar function of the identity L1 (C1 ) = L1 ⊗C with L1 -norm 1 is the product of two functions with L2 -norm 1. Let us now return to our original problem to show that X coincides with X1 , or simply that ||f || ≤ ||f ||1 for all f in P. To prove that it suﬃces to show that every a in the unit ball of X ∗ deﬁnes an element in the unit ball of X1∗∗ .

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Equivalently, it is enough to show that for any a = (an )n≥0 in the unit ball which of Λ = X ∗ , there is an α = (αn )n∈Z in the unit ball of L1 (C1 )∗ = Λ, is such that < a, f > = < α, f > for all f in P. Clearly this means that αn = an for all n ≥ 0. We have thus reduced our problem to the fact that every Hankel matrix with coeﬃcients in B(2 ) can be completed to a matrix with coeﬃcients in B(2 ) of the form (4.24) and of the same norm. This is precisely what Parrott shows in [70]. In fact, he gives an explicit inductive construction of the coeﬃcients α−1 , α−2 , etc. which can be added to the sequence a = (an )n≥0 in order to form an extended sequence with the desired property ||Tα || = ||Ha ||. This allows us to conclude that X and X1 are identical. Since their completions must be also identical, we obtain the proof of the theorem. Now we wish to prove Parrott’s result, which we used previously. In order to do this we state and prove some technical results concerning the completing matrix contractions. These results belong also to Parrott, but we follow the presentation from Peller’s monograph [65]. Let H, K be Hilbert spaces, A a bounded linear operator on H, B a bounded linear operator from K to H, and C a bounded linear operator from H to K. The problem is to ﬁnd out under which conditions there exists a bounded linear operator Z on K such that the operator AB QZ = (4.25) CZ on H ⊕ K is a contraction, that is, ||QZ || ≤ 1. It is easy to see that if the problem is solvable, then the operators A and A B (4.26) C from H to H ⊕ K and from H ⊕ K to H, respectively, are contractions. It turns out that the converse is also true. Theorem 4.28. Let H, K be Hilbert spaces and let, A : H → H, B : K → H, and C : H → K bounded linear operators. Then there is an operator Z : K → K for which the operator QZ deﬁned by (4.25) is a contraction on H ⊕ K if and only if A || || ≤ 1 and || A B || ≤ 1. (4.27) C

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Next we describe all operators Z on K for which QZ is a contraction. In order to be able to state this description we need some preliminaries. Lemma 4.29. Let H, H1 , and H2 be Hilbert spaces, and let T : H1 → H and R : H2 → H be bounded linear operators. Then T T ∗ ≤ RR∗ if and only if there exists a contraction Q : H1 → H2 such that T = RQ. Proof.

Suppose that T = RQ and ||Q|| ≤ 1. We have that (T T ∗ x, x) = (RQQ∗ R∗ x, x) = (Q∗ R∗ x, Q∗ R∗ x) = ||Q∗ R∗ x||2 ≤ ||R∗ x||2 = (RR∗ x, x).

Conversely, assume that T T ∗ ≤ RR∗ . We deﬁne the operator L on Range R∗ as follows: LR∗ x = T ∗ x,

x ∈ H2 .

The inequality T T ∗ ≤ RR∗ implies that L is well-deﬁned on Range R∗ and ||L|| ≤ 1 on Range R∗ . We can extend L by continuity to the closure clos Range R∗ and put L|KerR = L|(Range R∗ )⊥ = 0. Set Q = L∗ . Clearly T = RQ.

For a contraction A : H1 → H2 the defect operator DA is deﬁned on H1 by DA = (I − A∗ A)1/2 . It is also convenient besides DA to consider other operators DA : H1 → such that D∗ DA = I − A∗ A, where H is a Hilbert space. In this case H A DA = V DA for some isometry V deﬁned on clos Range DA . be Hilbert spaces, A : H → H, B : K → H Lemma 4.30. Let H, K, H be an operator such linear operators such that ||A|| ≤ 1. Let DA∗ : H → H ∗ ∗ that DA DA∗ = I − AA . Then || A B || ≤ 1 (4.28) ∗ if and only if B = DA ∗ K for a contraction K : K → H.

Proof.

It is easy to see that (4.28) is equivalent to the fact that A∗ ≤ IH , AB B∗

∗ D A∗ . which means that AA∗ + BB ∗ ≤ I or, which is the same, BB ∗ ≤ DA ∗ By Lemma 4.29, this is equivalent to the fact that B = DA∗ K for some The proof is complete. contraction K : K → H.

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Remark. The conclusion of Lemma 4.30 is valid if DA∗ = DA∗ = (I − AA∗ )1/2 . It is easy to see that we can choose a contraction K : K → H such that Range K ⊂ clos Range(I − AA∗ ) and B = DA∗ K. Clearly, such a contraction K is unique. be Hilbert spaces and let A : H → H, C : H → Lemma 4.31. Let H, K, H be an operator K be linear operators such that ||A|| ≤ 1. Let DA : H → H ∗ ∗ such that DA DA = I − A A. Then $ $ $ A $ $ $ (4.29) $ C $≤1 → K. if and only if C = LDA for some contraction L : H Proof. The result follows from Lemma 4.30 since (4.29) is equivalent to the inequality || A∗ C ∗ || ≤ 1. Remark. As in Lemma 4.30 we can take DA = DA = (I − A∗ A)1/2 . Clearly, one can ﬁnd a contraction L : H → K such that L|(Range(I − A∗ A))⊥ = 0 and C = LDA . As in Lemma 4.30 it is easy to see that such a contraction L is unique. Now we are in a position to state the description of those operators Z : K → K for which the operator QZ deﬁned by (4.25) is a contraction. As we have already observed, the operators in (4.26) are contractions. Therefore (see the Remarks after Lemmas 4.30 and 4.31) there exist unique contractions K : K → H and L : H → K such that RangeK ⊂ clos Range(I − AA∗ ), L|(Range(I − A∗ A))⊥ = 0,

B = DA∗ K, C = LDA .

(4.30) (4.31)

Theorem 4.32. Let H, K be Hilbert spaces, A : H → H, B : K → H, and C : H → K bounded linear operators satisfying (4.27). Let K : K → H and L : H → K be the operators satisfying (4.30) and (4.31). If Z : K → K is a bounded linear operator, then the operator QZ , deﬁned by (4.25), is a contraction if and only if Z admits a representation Z = −LA∗ K + DL∗ M DK , where M is a contraction on K.

(4.32)

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Note that we may always assume that M |(RangeDK )⊥ = 0

and

RangeM ⊂ clos RangeDL∗ .

(4.33)

If these two conditions are satisﬁed, then Z determines M uniquely, and so the contractions M satisfying (4.33) parametrize the solutions Z. It is easy to see that Theorem 4.28 follows from Theorem 4.32. Indeed, we can always take M = 0. To prove Theorem 4.32, we need one more lemma. Lemma 4.33. Let A, B be as above and let K : K → H be an operator satisfying (4.30). Then the operator DA −A∗ K D(AB) = (4.34) 0 DK satisﬁes ∗ A ∗ D(AB) = IH⊕K − D(AB) AB . ∗ B Proof.

We have that ∗ ∗ DA −A∗ K DA −A∗ K A + AB ∗ B 0 DK 0 DK 0 DA DA −A∗ K A∗ = + AB ∗ ∗ −K A DK 0 DK B 2 A∗ A A∗ B −DA A∗ K DA + = 2 −K ∗ ADA K ∗ AA∗ K + DK B∗A B∗B IH − A∗ A −DA A∗ K = 2 − K ∗ D A∗ D A∗ K −K ∗ ADA K ∗ K + DK ∗ A A A∗ B + . B∗A B∗B

Let us show that DA A∗ = A∗ DA∗ . We have that (I − A∗ A)A∗ = A∗ (I − AA∗ ). It follows that φ(I − A∗ A)A∗ = A∗ φ(I − AA∗ ) for any polynomial φ, so the same equality holds for any continuous function φ. If we take φ(t) = t1/2 , t ≥ 0, we obtain that DA A∗ = A∗ DA∗ . Similarly, DA∗ A = ADA . Consequently, ∗ A ∗ D(AB) D(AB) + AB ∗ B IH − A∗ A −A∗ B A∗ A A∗ B IH 0 = + = . I − B∗B −B ∗ A B∗A B∗B 0 IH

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Proof of Theorem 4.32. Suppose that ||QZ || ≤ 1. By Lemma 4.29 C Z = X Y D(AB) , where X Y is a contraction from H ⊕ K to K and D(AB) is deﬁned by (4.34). Then C = XDA . Let P be the orthogonal projection from H onto = XP. Clearly, C = XD A and, by the Remark clos Range(I −A∗ A). Put X after Lemma 4.31, we have that X = L. It is easy to see that (4.35) C Z = L Y D(AB) . Clearly,

LY

= XY

P 0 0 IH

,

which proves that L Y is a contraction. Then, by Lemma 4.30, the operator Y admits a representation Y = DL∗ M for a contraction M on K. Formula (4.32) follows now immediately from (4.35). Suppose that Z satisﬁes (4.32), where M is a contraction on K. Then it is easy to see that ⎞⎛ ⎞ ⎛ A DL∗ 0 I0 AB I0 0 ⎝ DA −A∗ 0 ⎠ ⎝ 0 K ⎠ . (4.36) = CZ 0 L DL∗ 0 0 M 0 DK The result follows from the fact that all factors on the right-hand side of (4.36) are contractions, which is a consequence of the following lemma: Lemma 4.34. Let T be a contraction on a Hilbert space. Then the operator T DT ∗ DT −T ∗ is unitary. Proof. We have ∗ ∗ T DT ∗ T T + DT2 T ∗ DT ∗ − DT T ∗ T DT ∗ = . DT −T ∗ DT −T ∗ DT ∗ T − T DT DT2 ∗ + T T ∗ It has been shown in the proof of Lemma 4.33 that T ∗ DT ∗ = DT T ∗ and T DT = DT ∗ T. Thus, ∗ T DT ∗ T DT ∗ I0 = . DT −T ∗ DT −T ∗ 0I Similarly, ∗ T DT ∗ T DT ∗ I0 = . 0I DT −T ∗ DT −T ∗ The proof is complete.

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Remark. The same results are valid if A is an operator from H1 to H2 , B is an operator from K1 to K2 , C is an operator from H1 to K2 , and Z is an operator from K1 to K2 , where H1 , H2 , K1 , K2 are Hilbert spaces. The proof given above works also in this more general situation. Remark. It is clear that if we replace in Theorem 4.28 H by K and conversely we get that the matrix with operators as its entries Z C BA C is a contraction on K ⊕ H if and only if B A and are contractions. A Now let us present the Parrott’s argument to complete the proof of Theorem 4.25 (see [70]): Consider the block Hankel matrix (that is a matrix having operators as its entries) ΓΩ = {Ωj+k }j,k≥0 , where Ω = {Ωj }j≥0 is a sequence of bounded linear operators from H to K, for two separable Hilbert spaces H, K, and the matrix ΓΩ is of the form: ⎞ ⎛ Ω 0 Ω1 Ω2 Ω3 . . . ⎜ Ω 1 Ω2 Ω3 Ω4 . . . ⎟ ⎟ ⎜ ⎟ ⎜ . ⎜ Ω2 Ω3 Ω4 .. . . . ⎟ ⎟. ⎜ ⎟ ⎜ ⎟ ⎜ Ω Ω ... ⎠ ⎝ 3 4 .. .. . . Ω of the form Now we wish to construct matrix Γ ⎛ a block Hankel ⎞ .. .. .. . . ...⎟ ⎜. ⎜Ω Ω Ω ...⎟ −2 −1 0 ⎟ ⎜ ⎟ ⎜ Ω Ω Ω ⎜ −1 0 1 ...⎟ ⎟ ⎜ ⎜ Ω 0 Ω1 Ω2 . . . ⎟ ⎟ ⎜ . ⎟ ⎜ ⎜ Ω1 Ω2 Ω3 .. ⎟ ⎠ ⎝ .. .. .. . . . Ω || = ||ΓΩ ||. such that ||Γ Ω inductively, ﬁrst constructing a Hankel matrix We construct Γ ⎞ ⎛ Z Ω 0 Ω1 Ω2 . . . ⎜ Ω 0 Ω1 Ω2 Ω3 . . . ⎟ ⎟ ⎜ ⎜ Ω 1 Ω2 Ω3 Ω4 . . . ⎟ ⎟ ⎜ Γ−1 = ⎜ ⎟ ⎜ Ω Ω Ω Ω ... ⎟ 2 3 4 5 ⎠ ⎝ .. .. .. .. . . . . . . .

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such that ||Γ−1 || = ||ΓΩ ||. This of course involves the choice of Ω−1 = Z. Given that this construction is always possible, successive iterations will Ω . To choose Ω−1 we write produce Ω−2 , Ω−3 , . . . , and Γ ⎛ ⎞ ⎞ ⎛ Ω0 Ω1 Ω2 Ω3 . . . ⎜ Ω1 ⎟ ⎜ Ω2 Ω3 Ω4 . . . ⎟ ⎜ ⎟ ⎟ ⎜ A = ⎜Ω Ω Ω ...⎟, B = ⎜Ω ⎟, ⎝ 2⎠ ⎠ ⎝ 3 4 5 .. .. .. .. . . . . . . . C = Ω0 Ω 1 Ω 2 . . . . The matrix Γ−1 can be identiﬁed with the matrix Z C . BA Clearly, || B A || = ||ΓΩ ||,

$ $ $ C $ $ $ $ A $ = ||ΓΩ ||.

By Theorem 4.28 and the last two remarks, there exists an operator Z such that $ $ $ Z C $ $ $ $ B A $ = ||ΓΩ ||. Now put Ω−1 = Z. We have ||Γ−1 || = ||ΓΩ ||. The proof of Theorem 4.25 is complete. 4.5

A characterization of the space T1

In the book of M. Pavlovi´c ([68] page 96) there is the following beautiful characterization of functions belonging to the Hardy space H 1: Pavlovi´ c Theorem For a function f, which is analytic in D, the following assertions are equivalent: a) f ∈ H 1 ; b) sup n

n 1 1 ||sj f || < ∞; an j=0 j + 1

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c) sup ||Pn f || < ∞. n

Here, for a function f analytic in D let Pn f =

n n 1 1 1 sj f, where an = (n = 0, 1, 2, . . . ) an j+1 j+1 j=0

j=0

and sj f are the partial sums of the Taylor series of f. An analogue of this result using the vector-valued Hardy inequality given in Section 4.4 is also true and is presented below. More precisely, we have the following result: Theorem 4.35. Let A ∈ B(2 ) be an upper triangular matrix. The following assertions are equivalent: a) A ∈ T1 ;

b) sup n

n 1 1 ||sj A|| < ∞; an j+1 j=0

c) sup ||Pn A|| < ∞. n

Here Pn A = and sj A =

n n 1 1 1 sj A, where an = (n = 0, 1, 2, . . . ) an j=0 j + 1 j + 1 j=0

j k=0

Ak .

Proof. Obviously b) ⇒ c). a) ⇒ b). Let A ∈ T1 , and for ﬁxed n ≥ 2, w ∈ D, and r = 1 − n1 < 1, deﬁne the matrix-valued function g(z) = (1 − rz)−1 [A ∗ C(rwz)] (|z| ≤ 1), 1 for where C(z) is the Toeplitz matrix corresponding to the function 1−z each z ∈ D. Then we have that

g(z) =

∞ k=0

k

k k

Ak r w z

∞ l=0

l l

rz

=

∞ k,l=0

Ak wk r k+l z k+l =

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m

Ak w

k

rm z m =

∞

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sm (A ∗ C(w))rm z m .

m=0

k=0

Hence g(m) = sm (A ∗ C(w))rm , m = 0, 1, 2, . . . It is well-known (and easy to see) that ||sm A||T1 ≤ C ln(m + 1)||A||T1

∀A ∈ T1 and m ∈ N,

(4.37)

where C > 0 is an absolute constant. Moreover, g ∈ HC1 1 since, by (4.37), we have that 1 C ln(m + 1) ||sm A||T1 ≤ ∀m ∈ N and |w| < 1. 1 − |w| 1 − |w|

||sm (A ∗ C(w))||T1 ≤ Therefore ∞

||sm (A ∗ C(w))||T1 r m ≤

C

∞

r m ln(m + 1) < ∞. 1 − |w|

m=0

m=0

We conclude that ∞ j=0

∞

1 1 ||sj (A ∗ C(w))||T1 rj = || g (j)||T1 j+1 j+1 j=0 ≤ (by Corollary 4.14 for X = T1 )

≤ C||g||HT1 = 1

||A ∗ C(rweit )||T1 |1 − reit |

for all t ∈ [0, 2π).

Since rj = (1− n1 )j ≥ c ∀0 ≤ j ≤ n, where c > 0 is an absolute constant, we have: 2π n 1 dt ||sj (A ∗ C(w))||T1 ≤ C ||g(reit )||T1 j+1 2π 0 j=0

2π

=C 0

||A ∗ C(rweit )||T1 dt . |1 − reit | 2π

Integrating this inequality over the circle |w| = 1 and since sj (A ∗ C(w)) = sj (A) ∗ C(w), we ﬁnd, using limw→eiθ ||sj (A) ∗ C(w)||T1 = ||sj A ∗ C(eiθ )||T1 ∀j, that n j=0

1 j+1

0

2π

||sj A∗C(eiθ )||T1

dθ π ≤ 2π c

2π 0

0

2π

||A ∗ C(rei(θ+t) )||T1 dt dθ |1 − reit | 2π 2π

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=

π c

2π 0

2π 0

||A ∗ Pr (t + θ)||T1

dθ 2π

dt 2π|1 − reit |

2π π 1 dt ≤ C||A||T1 ln n, ≤ ||A||T1 c 2π 0 |1 − reit | where Pr (t + θ) is the usual Poisson kernel on the unit circle and C > 0 is an absolute constant. However, denoting by Eθ the Toeplitz matrix corresponding to δθ (the Dirac measure concentrated in θ) we have that 2π 2π 1 1 iθ ||sj A ∗ C(e )||T1 dθ = ||sj A ∗ Eθ ||T1 dθ. 2π 0 2π 0 Since ||B||T1 = ||B ∗ Eθ ∗ E2π−θ ||T1 ≤ ||B ∗ Eθ ||T1 ||E2π−θ ||M (2 ) = ||B ∗ Eθ ||T1 ≤ ||B||T1 ||Eθ ||M (2 ) = ||B||T1 ∀θ ∈ [0, 2π] it follows that 2π n n 1 1 dθ ||sj A ∗ C(eiθ )||T1 ||sj A||T1 ≤ ≤ C||A||T1 ln n, j + 1 j + 1 2π 0 j=0 j=1 that is n 1 1 ||sj A||T1 ≤ C1 ||A||T1 an j=0 j + 1

and b) holds. c) ⇒ a) It is clear that if A is a ﬁnite matrix, then ||A||C1 ≤ sup ||Pn A||C1 . n

Now assume that A is any matrix such that supn ||Pn A||C1 < ∞. Let Em be the canonical projection which projects a matrix to its submatrix of order m at the left upper corner. Since Pn and Em commute, we ﬁnd that sup sup ||Pn Em A||C1 < ∞. m

n

By the preceding remark, we have that sup ||Em A||C1 ≤ sup sup ||Pn Em A||C1 < ∞; m

m

n

whence A ∈ C1 and ||A||C1 ≤ sup ||Pn A||C1 . n

This inequality holds without the assumption that A is upper triangular. The proof is complete.

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A simple consequence of Theorem 4.35 is the following: Corollary 4.36. If A ∈ T1 , then lim

n 1 1 ||A − sj A|| = 0 an j=0 j + 1

(4.38)

lim

n 1 1 ||sj A|| = ||A||. an j=0 j + 1

(4.39)

n

and, consequently,

n

Proof. Obviously (4.38) holds if A is a ﬁnite matrix. Since ﬁnite matrices are dense in T1 the proof of (4.38) follows. The second assertion follows immediately from (4.38). We remark that B. Smith [89] proved in 1983 the relation (4.39) for f ∈ H 1 instead of A ∈ T1 , and this in fact, motivated Pavlovi´c to give his theorem. As a consequence of this theorem we have that Corollary 4.37. If A ∈ T1 , then lim inf n→∞ ||A − sn A||T1 = 0. Of course the last three results are matrix versions of some theorems concerning the strong convergence in H 1 . (See [68].)

4.6

An extension of Shields’s inequality

In this section we present an extension of the matrix version of Shields’s inequality. The results concerning the functions and analytic measures on T were obtained by C. McGehee, L. Pigno and B. Smith in 1981 (see [63]). In particular, the above mentioned authors proved the Littlewood conjecture [40] from 1948. In what follows we denote by M (T) the usual convolution algebra of Borel measures on T. First we present the following generalization of the classical Hardy inequality: Theorem 4.38. There is a real number C > 0 such that, given any set ⊂ S, the following S = {n1 < n2 < . . . } ⊂ Z and μ ∈ M (T), with supp μ

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inequality holds: ∞ | μ(nk )| k=1

k

≤ C||μ||.

A direct consequence of Theorem 4.38 is the following result: ink θ Corollary 4.39. If p(θ) = N where {n1 < n2 < · · · < nk } ⊂ Z k=1 ck e and |ck | ≥ 1 for all k, then ||p||1 ≥

1 log N. C

The following lemma will be quite useful in the sequel. Lemma 4.40. Let h ∈ L2 (T) and suppose that 0 ≤ h ∈ L∞ (T) and supp h ⊂ Z. Then −h } ⊂ Z− = {n ∈ Z; n < 0} (a) e−h ∈ L∞ (T), supp {e and (b) ||e−h − 1||2 ≤ ||h||2 . Proof.

To prove (b) we notice that, for all z ∈ C with z ≥ 0, −z e − 1 ≤ 1. z

Therefore, since h ≥ 0, it follows that ||e−h − 1||2 ≤ ||h||2 .

Proof of Theorem 4.38. Let S = {n1 < n2 . . . } ⊂ Z and μ ∈ M (T) be given with supp μ ⊂ S. Put S0 = {n1 }, S1 = {n2 < n3 < n4 < n5 }, S2 = {n6 < · · · < n21 }; continuing in this fashion we obtain disjoint sets Sj satisfying: S=

∞ <

Sj

j=0

and card Sj = 4j

(j = 0, 1, 2, . . . ).

(4.40)

Deﬁne now trigonometric polynomials fj (j = 0, 1, 2, . . . ) by requiring that fj = 0

oﬀ Sj ;

(4.41)

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|fj (n)| = 4−j

Put |fj | =

∞

if n ∈ Sj ;

fj μ ≥ 0. inθ

−∞ cn e

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(4.42) (4.43)

and deﬁne

hj (θ) :=

−1 1 {c0 + 2 cn einθ }. 4 −∞

Moreover, observe that, via the conditions (4.40), (4.41), (4.42), we have that ||fj ||2 = 2−j , so that

√ 2 2 −j ||hj ||2 ≤ ||fj ||2 = 2 < 3 · 2−j−3 . 4 4 Let F0 = (1/5)f0 and deﬁne inductively,

(4.44)

√

(4.45)

1 Fj+1 = Fj e−hj+1 + fj+1 (j = 0, 1, 2, . . . ). 5 Notice that since hj = (1/4)|fj |, we have that e−hj ∈ L∞ (T) because |fj | ≤ 1 from conditions (4.40) through (4.42). Thus Fj ∈ L∞ (T) and we ﬁnd, according to (4.40) and the inequality e−x/4 + x/5 ≤ 1 for 0 ≤ x ≤ 1 that ||Fj ||∞ ≤ 1

(j = 0, 1, 2, . . . ).

Moreover, since supp hj ⊂ Z− , we have that supp{ehj } ⊂ Z−

(4.46)

via part (a) of Lemma 4.40. We now claim that, for any m, 1 1 |fj (n)| if n ∈ Sj and j ≤ m. |Fm (n) − fj (n)| ≤ 5 10 In order to prove the above assertion we ﬁrst observe that m m 1 1 F m = f0 e − 1 hk + f1 e − 2 hk + . . . 5 5

(4.47)

1 1 + fm−1 e−hm + fm ; 5 5 as a consequence of (4.41) and (4.46). Hence, if n ∈ Sj and j < m, then

: ; fj − m 1 j+1 hk − 1 (n) Fm (n) − fj (n) = e 5 5

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+

; fj+1 : − m hk j+2 e − 1 (n) 5

+··· +

* fm−1 ) −hm e − 1 (n). 5

It is now obvious from part (b) of Lemma 4.40 and the Cauchy-Schwarz inequality that m m 1 |Fm (n) − fj (n)| ≤ {||fj ||2 || hk ||2 + ||fj+1 ||2 || hk ||2 + . . . 5 j+1 j+2

+ ||fm−1 ||2 ||hm ||2 }. Thus, in view of the inequalities (4.44) and (4.45), we obtain that 1 3 −j−1 1 −j 1 (4 4 = |fj (n)|. + 4−j−2 + . . . ) = |Fm (n) − fj (n)| ≤ 5 10 10 10 Let j > 0 and suppose that nk ∈ Sj . Then 3k > 4j and, hence, |fj (nk )| = 4−j >

1 . 3k

(4.48)

Notice that 1 fj (nk ) μ(nk ) (4.49) 10 because of the inequalities (4.43) and (4.47). Moreover, as a consequence of the inequalities (4.43), (4.48) and (4.49), we see that if nk ∈ Sj , j ≤ m, then 1 | μ(nk )|. (Fm μ )(nk ) ≥ (4.50) 30 Put Bm = S0 ∪ S1 ∪ · · · ∪ Sm and assume for the moment that μ is a trigonometric polynomial on T. Then, on the one hand (Fm μ )(nk ) ≥

|(F ∗ μ)(0)| ≤ ||μ||1 because ||Fm ||∞ ≤ 1, while on the other hand μ(n)|. |(Fm ∗ μ)(0)| = | Fm (n) n∈Bm

Hence, the inequality (4.50) permits us to conclude that, for all trigonometric polynomials μ with supp μ ⊂ S, ∞ | μ(nk )| 1

k

≤ 30||μ||1 .

(4.51)

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A standard approximate identity argument implies the inequality (4.51) for all µ ∈ M (T) with supp µ b ⊂ S. The proof is complete. PN ink θ Proof of Corollary 4.39. Given p(θ) = k=1 ck e (n1 < n2 < · · · < nN ), it follows from Theorem 4.38 that N N X |b p(nk )| X |ck | = ≤ C||p||1 . k k k=1 k=1 PN Since |ck | ≥ 1 for all k we see that log N ≤ k=1 k1 ≤ C||p||1 . Obviously, we have also proved the following version of Corollary 4.39: Let µ ∈ M (T) and suppose for {n1 < n2 < · · · < nN } ⊂ Z− we have that µ b(nk ) = ck , where |ck | ≥ 1 for k = 1, 2, . . . , N and µ b(n) = 0 for all − other n ∈ Z . Then 1 ||µ|| ≥ log N. C

Corollary 4.41. Given S = {n1 < n2 < . . . } ⊂ Z, and {ck }∞ 1 ⊂ C such ∞ that |ck | ≤ 1/k, there is an F ∈ L (T), ||F ||∞ ≤ C, such that Fb (nk ) = ck for all k. Proof. Given S = {n1 < n2 < . . . } ⊂ Z, put L1S (T) = {g ∈ L1 (T) : supp b g ⊂ S}; define Λ(g) :=

∞ X

ck b g(nk )

1

(g ∈ L1S (T)),

where {ck }∞ 1 ⊂ C is fixed and |ck | ≤ 1/k for all k. It follows from Theorem 4.38 that Λ is a bounded linear functional on L1S (T) such that ||Λ|| ≤ C. Extend Λ to L1 (T) by the Hahn-Banach theorem. We have thus obtained an F ∈ L∞ (T), ||F ||∞ ≤ C, such that Fb (nk ) = ck

for all k.

We recall that Theorem 4.24 implies that C1 is of (H 1 − `1 )-Fourier type. On the other hand, by Theorem 4.38, it follows that the sequence {an }∞ 1 , where 1 if nk for the integers n1 < n2 < . . . , ak = k 0 otherwise is (1, 1)-bounded multiplier. Therefore we have the following:

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Corollary 4.42. (Generalized Shields’s inequality) There is a constant C > 1 such that given any set n1 < n2 < ... < nk ⊂ Z, and P∞ A = k=1 Ank ∈ C1 , we have that ∞ X ||Ank ||C1 ≤ C||A||C1 . k k=1

Also Corollary 4.42 implies the matrix version of Corollary 4.39, which is the positive answer to a Littlewood conjecture. Corollary 4.43. There is a constant C > 1 such that, given any set {n1 < PN n2 < ... < nN } ⊂ Z and matrix A = k=1 Ank with ||Ank ||C1 ≥ 1 for all k, ||A||C1 ≥ C log N. Notes The results of the first section were communicated to us by V. Lie. Proposition 4.4 is a Hardy-Littlewood type inequality and will be used in Chapter 8 to prove Theorem 8.6 which is the matrix version of a celebrated statement of M. Mateljevic and M.Pavlovic 1990 (see [60]). The space H p (`2 ) introduced in Definition 4.3 seems to be the right matrix version of classical Hardy space and can be used in many problems about matrix spaces suggested by the classical results in Hardy space theory and, more generally, in Harmonic Analysis. Starting from a very interesting inequality of A. Shields [84] in Section 4.2 we get some new inequalities for the Pisier and Lust-Picquard spaces Tp (`2R ) + Tp (`2C ) and Tq (`2R ) + Tq (`2C ). It remains an open question if these inequalities may be replaced by more simple and more interesting matrix version of the Hardy-Littlewood inequality X (n + 1)p−2 ||An ||pCp ≤ K(p)||A||pTp , n≥0

for all upper triangular matrices A ∈ Cp . The inequality of Shields is proved in Section 4.3 following the original proof from 1983. Another more general proof of this inequality, due to O. Blasco and A. Pelczynski [15], is given in Section 4.4. Here the spaces of (H 1 − `1 )-Fourier type are studied. We present also the proof due to W. T. Sledd and D. A. Stegenga [88] of Ch. Fefferman’s theorem about the (H 1 − `1 ) multipliers.

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Also the proof of Sarason’s noncommutative factorization theorem with the proof of U. Haagerup and G. Pisier is given. We mention the interesting and useful Parott’s result about the completion of a Hankel matrix with B(2 )-coeﬃcients. In an excellent monograph [68] dedicated to analytic functions on the disk M. Pavlovic characterized the analytic functions from H 1 by the condition n 1 sup ||sj f || < ∞. n an j=0

Inspired by this result we proved in [73] a characterization of upper triangular matrices from C1 . This characterization is stated and proved in Section 4.5. Finally, in Section 4.6 we present the proof of an extension of Shields’s inequality.

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Chapter 5

The matrix version of BM OA

First properties of BM OA(2 ) space

5.1

We introduce diﬀerent deﬁnitions for the matrix version of BM OA, which may be diﬀerent from each other but on the class of Toeplitz matrices, they all coincide with the classical BM OA. One of these deﬁnitions for the matrix version of the space BM OA, denoted for short by BM OA(2 ), is as follows: For λ ∈ D we put kλ (e2πiθ ) := 1−λe12πiθ and denote by Kλ the Toeplitz matrix associated with the function kλ (·). Deﬁnition 5.1. The space BM OA(2 ) is deﬁned by BM OA(2 ) := {A | A upper triangular ||A||BM OA(2 ) < ∞}, where

||A||2BM OA(2 ) := sup { λ∈D

0

1

||A (r)Kλr ||2L2 (2 ) (1 − r2 )(1 − |λ|2 )rdr}.

Here L2 (2 ) was introduced in Section 4.1. We remark that BM OA(2 ) is a Banach space endowed with the norm ||A0 ||B(2 ) + ||A||BM OA(2 ) . Our BM OA(2 ) is a proper subspace of BM OAC (L2 (2 )) introduced by Blasco in [13]. Let V M OA(2 ) be the subspace of BM OA(2 ) consisting of those matrices A such that 1 ||A (r)Kλr ||2L2 (2 ) (1 − r2 )(1 − |λ|2 )rdr} = 0. lim { λ→1

0

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We need the above deﬁnition of BM OA(2 ) and the following results in order to extend a nice theorem of Mateljevic and Pavlovic (see [60]). Proposition 5.2. If A ∈ L2r (2 ), and B ∈ B(2 ), then we have that ||AB||L2r (2 ) ≤ ||A||L2r (2 ) ||B||B(2 ) . Proof.

We note that

LAB (x, t) =

1 0

LB (x, s)LA (−s, t)ds,

which implies that LAB (x, t) =

∞

A LB k (x)Ck (t)

=

∞ ∞ j=1

k=1

LB k (x)ajk

e2πijt ,

k=1

and, hence, LAB j (x) =

∞

LB k (x)ajk ,

k=1

for all j ≥ 1. Of course, here A = (ajk )j≥1; k≥1 , and B = (bjk )j≥1; k≥1 . We deduce that 1 1/2 AB 2 ||AB||L2r (2 ) = sup |Lj (x)| dx = j≥1

sup

sup

j≥1 ||h||L2 ≤1

sup

sup

j≥1 ||h||L2 ≤1

∞

|

1 0

0

∞

|ajk |

LB k (x)ajk

h(x)dx| ≤

k=1

1/2 2

k=1

∞ | k=1

1 0

1/2 2 LB k (x)h(x)dx|

= ||A||L2r (2 ) ||B||B(2 ) and the proof is complete.

Remark 5.3. By reasoning as in the proof of Proposition 5.2 we can derive that ||AB||L2r (2 ) ≤ ||A||B(2 ) ||B||L2c (2 ) .

(5.1)

If the matrix B is a Toeplitz matrix, then ||B||L2r (2 ) = ||B||L2c (2 ) and, consequently, (5.1) is equivalent to that ||AB||L2r (2 ) ≤ ||A||B(2 ) ||B||L2r (2 ) for B a Toeplitz matrix, and we may delete the subscript r in the above formulas.

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111

We give now a suﬃcient condition to guarantee that an upper triangular matrix belongs to BM OA(2 ). Corollary 5.4. If A is a matrix of ﬁnite band type A = nk=0 Ak , then we have that 1 1/2 2 ||A (r)||B(2 ) (1 − r)dr . ||A||BM OA(2 ) ≤ 0

Proof. ||A||BM OA(2 ) ≤ (by Proposition 5.2 and Remark 5.3) ≤

C sup λ∈(0,1)

0

1

||A

(r)||2B(2 ) ||Kλr ||2L2 (2 ) (1

We note that ||Kλr ||2L2 (2 ) and since

(1−|λ|2 )r 1−|λr|2

1

= 0

2

− r)(1 − |λ| )rdr

1/2 .

1 dθ = , 2πiθ 2 1 − |λr|2 |1 − λre |

≤ 1 we are done.

See also Proposition 1.2-[13] where a rather strong result was proved for the Blasco space BM OAC (X) with an arbitrary Banach space X. Inspired by Proposition 1.3-[13] we give an example of a matrix belonging to BM OA(2 ). Let us denote ﬁrst by eij , for ﬁxed i ≥ 1, j ≥ 1, the matrix having 1 on the intersection of the ith row with the jth column and 0 otherwise. ∞ 1 en,2n . Then the computations done Example 5.5. Let A = k=1 n ln(n+1) in [13]-Proposition 1.3 give us that 1 dr < ∞. ||A||BM OA(2 ) ≤ C 1 2 0 (1 − r)(ln( 1−r ))

5.2

Another matrix version of BM O and matriceal Hankel operators

In the last thirty years a number of important research papers are devoted to Hankel operators. See for instance the very impressive monograph [65]. In that book there is a chapter dedicated to vectorial Hankel operators, which are important for applications and now we recall some notions and deﬁnitions from this chapter.

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Let us denote by H 2 (H) the Hardy class of functions with values in a separable Hilbert space H and by B(H, K) the space of bounded linear operators from H to K and by L∞ (B(H, K)) the space of bounded weakly measurable B(H, K)-valued functions. We recall that the Hardy class H 2 (H) is deﬁned as follows: H 2 (H) = {F ∈ L2 (H) : Fˆ (n) = 0, n < 0}, where L2 (H) is the space of weakly measurable H-valued functions F for which ||F (ζ)||2H dm(ζ) < ∞. ||F ||2L2 (H) = T

2 Put H− (H) = L2 (H) H 2 (H). Let K be another separable Hilbert space and let Φ be a function in L2s (B(H, K)) (see [65], chapter 2), i.e. ||Φ(ζ)x||2K dm(ζ) < ∞ for any x ∈ H. (5.2) T

ˆ Recall that for functions satisfying (5.2) the Fourier coeﬃcients Φ(n) ∈ B(H, K) are deﬁned by n ˆ Φ(n)x = ζ Φ(ζ)xdm(ζ), n ∈ Z, x ∈ H. T

Now we can deﬁne for such functions Φ the Hankel operator HΦ : 2 H 2 (H) → H− (K) on the set of polynomials in H 2 (H) by HΦ F = P− ΦF, F ∈ H 2 (H),

(5.3)

where we denote by P+ and P− the orthogonal projections onto H 2 (H) and 2 (K), respectively. H− We denote by C2 the set of all Hilbert-Schmidt matrices, equipped with the usual Hilbert-Schmidt norm || · ||C2 . Let T2 be the set of all upper triangular Hilbert-Schmidt matrices endowed with the norm induced by || · ||C2 . Of course T2 is a Hilbert space. If A ∈ T2 and k = 0, 1, 2, 3, . . . we denote by Ak the kth-diagonal of A. Then, formally, A = k∈Z Ak and we denote by P+ the triangular projection P+ A = k≥0 Ak . It is well-known that P+ : T2 → T2 is bounded with norm less than 1. Now put P− = I−P+ , where I is the identity on T2 . Of course ||P− || ≤ 1. Put ˆ H ∞ (B(T2 , T2 )) := {Φ ∈ L∞ (B(T2 , T2 )) : Φ(n) = 0, n < 0}.

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Then the following statement is known (see [65]): Vectorial Nehari Theorem. Let Φ be a B(T2 , T2 )-valued function that satiﬁes (5.2). Then the operator HΦ deﬁned by (5.3) extends to a bounded operator on H 2 (H) if and only if there exists a function Ψ in L∞ (B(T2 , T2 )) such that ˆ ˆ Ψ(n) = Φ(n), n < 0, and ||HΦ || = distL∞ (Ψ, H ∞ (B(T2 , T2 )). Let Φ be an inﬁnite matrix such that for all matrices A = nk=0 Ak , n ∈ N, we have that P− (ΦA) ∈ C2 . Examples of such matrices are either all matrices representing linear bounded operators on 2 , or matrices Φ such that P− Φ = 0. We deﬁne the matrix version of Hankel operator HΦ as follows: HΦ : T2 → (T2 )− := C2 T2 , on the dense subspace in T2 of all matrices A = n k=0 Ak , n ∈ N, such that ΦA ∈ C2 , by HΦ (A) = P− (ΦA). The matrix Φ is called the symbol of the Hankel operator HΦ . If Φ is an inﬁnite matrix and if we denote by Φ also the operator func tion given by Φ(ζ) = k∈Z eikζ Φk , then it is easy to verify that the matrix version of the Hankel operator HΦ deﬁned on T2 coincides with the restriction of Peller’s vectorial Hankel operator HΦ to the subspace of H 2 (T2 ) ikζ consisting of all matrix functions ∞ Ak , for some A ∈ T2 . k=0 e In 1985 S. Power [77] gave the matrix versions of Nehari’s and Hartman’s theorems. In what follows we present these results about matrix versions of Hankel operators HΦ with diﬀerent proofs. We denote as in [84], by A˜ the matrix whose entries a ˜kl are given by ⎧ ⎨ −iakl if k < l, a ˜kl = ia if k > l, ⎩ kl 0 if k = l, where akl are the entries of A. We recall that B(2 ) means the space of representing matrices of all linear bounded operators on 2 , equipped with the usual operator norm.

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Moreover, we denote by BM OF (2 ) the space of all matrices A such that ˜ equipped with there exist matrices Φ and Ψ from B(2 ) with A = Φ + Ψ, ˜ Of course the norm ||A||BM OF (2 ) := inf{||Φ||B(2 ) + ||Ψ||B(2 ) ; A = Φ + Ψ}. BM OF (2 ) is a Banach space. We also deﬁne BM OAF := BM OF (2 ) ∩ H ∞ (2 ), where H ∞ (2 ) is the set of all upper triangular matrices A ∈ B(2 ). Then the following holds [69] (for a more general result see [77], [59]): Theorem 5.6. (Matrix version of the Nehari theorem) Let Φ be an inﬁnite matrix such that ΦA ∈ C2 for all ﬁnite band-type matrices A. Then the following conditions are equivalent: (a) HΦ is a bounded linear operator on T2 . (b) There is Ψ ∈ B(2 ) such that Ψk = Φk for all k < 0. (c) P− Φ ∈ BM OF (2 ). Proof. (b)⇒(a). By simple computations we get that HΦ (A) = HΨ (A) for all A = nk=0 Ak , n ∈ N. Hence, ||HΦ (A)||C2 = ||HΨ (A)||C2 ≤ ||ΨA||C2 ≤ ||Ψ||B(2 ) ||A||T2 , that is ||HΦ || ≤ ||Ψ||B(2 ) < ∞. (a)⇒ (b). We note that HΦ (A) = P− (ΦA) = P− [P− (Φ)A] for all A = n k=0 Ak , n ∈ N. Assume that HΦ : T2 → (T2 )− is a bounded linear operator. Then ||HΦ (A)||(T2 )− =

sup ||B||(T2 )− ≤1

| < HΦ (A), B > | =

sup ||B||(T2 )− ≤1

sup ||B||(T2 )− ≤1

|tr HΦ (A)B ∗ | =

|tr P− [(P− Φ)A]B ∗ |.

(5.4)

But, obviously tr P− [(P− Φ)A]B ∗ = tr (P− Φ)AB ∗ for B ∈ (T2 )− and A ∈ T2 . Hence, (5.4) can be written as ||HΦ (A)||(T2 )− =

sup ||B||(T2 )− ≤1

|tr (P− Φ)(AB ∗ )|.

Since A, B ∗ ∈ T2 and every A ∈ T1 may be written as A = B1 B2 , where 1/2 B1 , B2 ∈ T2 and ||B1 ||T2 = ||B2 ||T2 = ||A||T1 (see [84]), we have that ∞ > M = ||HΦ || =

sup ||A||(T2 ) ≤1, ||B||(T2 )− ≤1

|tr (P− Φ)AB ∗ | =

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sup ||A||(T1 ) ≤1; A0 =0

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|tr (P− Φ)A| = ||P− Φ||(T1,0 )∗ ,

j≥1

∞ > ||HΦ || =

Aj }. Using the Hahn-Banach theorem we inf

P− Φ−Ψ∈(T1,0 )⊥

||Ψ||B(2 ) .

But (T1,0 )⊥ = {Ψ ∈ B(2 )| tr(ΨB) = 0 for all B ∈ T1,0 } = {Ψ ∈ B(2 )| P− Ψ = 0} and, hence, there is Ψ ∈ B(2 ) such that P− Φ = P− Ψ. Thus (b) holds. (b) ⇒ (c). P− Φ = P− Ψ for Ψ ∈ B(2 ). But ˜ + P+ Ψ − Ψ0 = −iΨ ˜ + Ψ − P− Ψ − Ψ0 P− Ψ = −iΨ ˜ + 1 Ψ − 1 Ψ0 , that is P− Ψ ∈ BM OF (2 ). and, thus, P− Ψ = − 12 iΨ 2 2 (c) ⇒ (b). Let P− Φ ∈ BM OF (2 ). Then there exist A, B ∈ B(2 ) with ˜ But P− Φ = A + B. ˜ = P− (A)+iP− B = P− (A+iB) = P− Ψ, Ψ ∈ B(2 ), P− Φ = P− (A)+P− (B) and the implication follows. The proof is complete.

Corollary 5.7. It yields that ||HΦ || = inf{||Φ − F ||B(2 ) ; F ∈ H ∞ (2 )} = distB(2 ) (Φ, H ∞ (2 )). Proof.

It follows directly from the equality ||HΦ || = inf{||Ψ||B(2 ) ; Ψ − P− Φ ∈ (T1,0 )⊥ }.

We intend to give a matrix version of Hartman’s Theorem (see [65]) about compacity of Hankel operators. We need the following lemma: Lemma 5.8. Let Φ ∈ C∞ , the space of all compact operators on 2 . Then distB(2 ) (Φ, H ∞ (2 )) = distB(2 ) (Φ, T∞ ), where T∞ = {A ∈ C∞ | A upper triangular}.

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Proof. The inequality distB(2 ) (Φ, H ∞ (2 )) ≤ distB(2 ) (Φ, T∞ ) is trivial. Conversely, let Φ ∈ C∞ , H ∈ H ∞ (2 ) and Pn be the projection on B(2 ), given by Pn A = (aij )i,j≥1 , where

aij if 1 ≤ 1, j ≤ n , n = 1, 2, 3, . . . . aij = 0 otherwise Then, since Φ is a compact operator, for each > 0 there exists n0 = n0 () such that ||Φ − Pn0 Φ|| < . Therefore, ||Φ − H||B(2 ) ≥ ||Pn0 Φ − Pn0 H||B(2 ) ≥ ||Φ − Pn0 H||B(2 ) − ≥ distB(2 ) (Φ, T∞ ) − and, since is arbitrary, we have that distB(2 ) (Φ, H ∞ (2 )) ≥ distB(2 ) (Φ, T∞ ). The proof is complete.

Theorem 5.9. H ∞ (2 ) + C∞ is a closed Schur subalgebra of B(2 ). Proof. Lemma 5.8 shows that the canonical inclusion C∞ /T∞ → B(2 )/H ∞ (2 ) is an isometry and, therefore, C∞ /T∞ is a closed subspace of B(2 )/H ∞ (2 ). Let ρ : B(2 ) → B(2 )/H ∞ (2 ) be the canonical map. It follows that ∞ H (2 ) + C∞ = ρ−1 (C∞ /T∞ ) is a closed subspace of B(2 ). Now we should show that H ∞ (2 ) + C∞ is a Schur subalgebra of B(2 ). But, in fact, (A1 + B1 ) ∗ (A2 + B2 ) = (A1 ∗ A2 + B1 ∗ A2 + A1 ∗ B2 ) + B1 ∗ B2 , for all A1 , A2 ∈ H ∞ (2 ) and B1 , B2 ∈ C∞ . The ﬁrst expression on the right hand side of the above equality belongs to H ∞ (2 ) by [11]. On the other hand B1 ∗B2 ∈ C∞ , since limn Pn Bi = Bi , i = 1, 2, again by [11]. Now let us denote by S the linear and bounded operator on C2 given ⎛ 1 ⎞ ⎛ 1 ⎞ a−1 a10 a11 . . . a0 a11 a12 . . . ⎜ 1 ⎟ ⎜ 1 ⎟ ⎜ a−2 a2−1 a20 . . . ⎟ ⎜ a−1 a20 a21 . . . ⎟ ⎜ ⎟ ⎜ ⎟ by S(A) = ⎜ ⎟ for A = ⎜ ⎟. ⎜ a1 . . . . . . . . . ⎟ ⎜ a1 a 2 . . . . . . ⎟ ⎝ −3 ⎠ ⎝ −2 −1 ⎠ .. .. .. .. .. .. .. .. . . . . . . . . Then we have the following: Lemma 5.10. Let K : T2 → (T2 )− be a compact operator and S : T2 → T2 . Then lim ||KS n || = 0. n→∞

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Proof. It is enough to prove the lemma for rank one operators K. Let K(A) = < A, B > C, where B ∈ T2 , C ∈ (T2 )− . Then we have that KS n (A) = < S n (A), B > C = < A, S ∗ n (B) > C, where ⎛ 1 1 1 ⎛ 1 1 1 ⎞ ⎞ a1 a 2 a 3 . . . a0 a1 a2 . . . ⎜ ⎜ ⎟ ⎟ ⎜ 0 a21 a22 . . . ⎟ ⎜ 0 a20 a21 . . . ⎟ ⎜ ⎜ ⎟ ⎟ ∗ S (A) = ⎜ ⎟ for A = ⎜ ⎟ ∈ T2 . ⎜ 0 0 ... ... ⎟ ⎜ 0 0 ... ... ⎟ ⎝ ⎝ ⎠ ⎠ .. . . . . . . .. . . . . . . . . . . . . . . Hence ||KS n || = ||S ∗ n B||T2 ||C||T2 → 0.

Lemma 5.11. Let Φ ∈ C∞ . Then HΦ : T2 → (T2 )− is a compact operator. Proof. Let F = eij , for ﬁxed i, j ≥ 1 such that i − j ≥ 1. Then HF (A) = i−j−1 j ak ei,k+1 , where A has the entries (ajk )∞ j=1 on the kth-diagonal, for k=0 k ≥ 0. Therefore HF is a compact operator and, consequently, it is easy to see that HPn Φ is a compact operator too, for all n ≥ 0. By Theorem 5.6, and using the fact that Φ is a compact operator, we have that ||HΦ − HPn Φ || ≤ ||Φ − Pn Φ||B(2 ) → 0. Consequently, HΦ is a compact operator.

Now we are ready to state the matrix version of the Hartman theorem (see [69], [77]): Theorem 5.12. Let Φ ∈ B(2 ). The following conditions are equivalent: (i) HΦ is a compact operator. (ii) Φ ∈ H ∞ (2 ) + C∞ . (iii) There exists Ψ ∈ C∞ with HΦ = HΨ . Proof. Obviously (ii)⇒ (iii). Since (iii) implies that Φk = Ψk for all k < 0, the assertion (i) follows in the same way as (b)⇒ (a) in Theorem 5.6. (i) ⇒ (ii). We have that ||HΦ S n || = ||HS n Φ || = (by Corollary 5.7) = distB(2 ) (Φ, (S −1 )n H ∞ (2 )). Then, for all > 0 and for all n ≥ 0 there exist B n ∈ H ∞ (2 ) and a −n matrix An of ﬁnite band type such that An = j=−1 (An )j ; An ∈ B(2 ) and, by using Theorem 5.6, ||HΦ S n ||B(2 ) ≥ ||Φ − An − B n ||B(2 ) − ≥ ||HΦ − HAn || −

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≥ ||Pm HΦ − Pm HAn || − for all m ≥ 1. Hence, ||HΦ S n || ≥ ||HΦ − Pm HAn || − ||HΦ − Pm HΦ || − for all m, n. But Pm HAn = HPm (An ) , and, therefore, ||HΦ S n || ≥ ||HΦ − HPm (An ) || − ||HΦ − Pm HΦ || −

∀m, n.

Moreover, by hypothesis, HΦ is a compact operator, and, consequently, there exist m0 and C ∈ H ∞ (2 ) such that ||HΦ − Pm0 HΦ || < , and ||HΦ S n || ≥ ||HΦ − HPm (An ) || − 2 ≥ (by Corollary 5.7) ≥ ||Φ − Pm0 (An ) − C||B(2 ) − 3, for all n. By Lemma 5.10 we get that ≥ distB(2 ) (Φ, H ∞ (2 ) + C∞ ) − 3. Therefore distB(2 ) (Φ, H ∞ (2 ) + C∞ ) = 0 and, by Theorem 5.9, it fol lows that Φ ∈ H ∞ (2 ) + C∞ . The proof is complete. If we denote by ˜ A, B ∈ C∞ } V M OF (2 ) := {Φ| Φ = A + B, equipped with the norm induced by that of BM OF (2 ), then, as in the proof (b) ⇒ (c) of Theorem 5.6, we can derive the following statement: Theorem 5.13. Let Φ be an inﬁnite matrix. Then HΦ is a compact operator if and only if P− Φ ∈ V M OF (2 ). BM OAF (2 ) is the space of all upper triangular matrices A ∈ BM OF (2 ). This space is of interest because the proof of Theorem 5.6- (b) ⇒ (c), Hahn-Banach theorem and the well-known fact [34] that C1∗ = B(2 ) via the bilinear map < A, B > = tr AB ∗

for A ∈ B(2 ) and B ∈ C1

show us that T1∗ = BM OAF (2 ) by the previous bilinear map. See also Theorem 2.3 [77].

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5.3

119

Nuclear Hankel operators and the space M1,2

We shall here derive a suﬃcient condition to guarantee that the Hankel operator HA introduced in Section 5.2, where A is an upper triangular matrix, is nuclear. Moreover, if the matrix A has only a ﬁnite number of rows this condition is also necessary. Let 1 ≤ p < ∞. We denote by Sp the Schatten class of all operators from T2 into T2 − . We denote by Mp,2 the space of all upper triangular inﬁnite matrices A = (alk )k≥0, l≥1 such that ⎛ ⎜ ||A||p,2 := ⎝

∞ ∞ l=1 j=0

⎛ ⎝

∞

⎞p/2 ⎞1/p ⎟ |aln |2 ⎠ ⎠ < ∞.

(5.5)

n=j

Our next result reads: Theorem 5.14. Let A be an upper triangular inﬁnite matrix such that A ∈ M1, 2 . Then the Hankel operator HA∗ : T2 → T2 − is nuclear and, moreover, ||HA∗ ||S1 ≤ ||A||M1, 2 . Proof.

We easily see that

HA∗ (B) =

∞ ∞

⎛ ⎝

l=0 k=1

where B ∈ T2 , Akl =

∞

⎞1/2 |akj |2 ⎠

· < Elk , B > Akl ,

(5.6)

j=l+1

∞ j=l+1

k+l akj E−j+l

∞ j=l+1

|akj |2

−1/2

, the series

(5.6) converges in the norm of C2 and where j ∈ Z, and k ≥ 1, is the inﬁnite matrix having as entries 1 on the kth place on the jth-diagonal, and 0 otherwise. Consequently, ⎛ ⎞1/2 ∞ ∞ ∞ ⎝ |akj |2 ⎠ · Elk ⊗ Akl HA∗ = Ejk ,

l=0 k=1

j=l+1

is a nuclear operator for A = A ∈ M1, 2 and ||HA∗ ||S1 ≤ ||A||M1, 2 . The proof is complete.

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Notes In Section 5.1 we introduce the matrix versions of the classical spaces BM O and BM OA, BM O(2 ) and BM OA(2 ). These spaces are proper subspaces of that introduced previously by O. Blasco in [13]. These deﬁnitions were invented also by V. Lie and he kindly communicated his results to us. We also mention a kind of H¨ older’s inequality (Proposition 5.2). These results will be used in Chapter 8. Another kind of matrix version of a BM O space is introduced in Section 5.2. This space, denoted by BM OF (2 ), is useful in the study of the matrix version of Hankel operator HΦ and in a more general context of nest algebras it was introduced by S. Power in [77]. Theorem 5.14 seems to be new. It is also easy to observe that a converse of this theorem holds for matrices A with a ﬁnite number of rows.

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Chapter 6

Matrix version of Bergman spaces

6.1

Schatten class version of Bergman spaces

We intend now to introduce the Schatten class version of Bergman spaces. In order to do this we apply Proposition 1.1 to obtain that, for a w∗ measurable B(2 )-valued function f, the function t → ||f (t)||B(2 ) is Lebesgue measurable on D. We introduce also the following matrix spaces: L∞ (D, 2 ) := {r → A(r) being a w∗ -measurable function on [0, 1) | ess sup ||A(r)||B(2 ) := ||A(r)||L∞ (D,2 ) < ∞}, 0≤r<1

∞ (D, ), is the subspace of L∞ (D, ) consisting of all strong measurable = L 2 2 functions on [0, 1) and

:= L∞ a (D, 2 ) := {A inﬁnite analytic matrix | ||A||L∞ a (D,2 ) sup ||C(r) ∗ A||B(2 ) = ||A(r)||L∞ (D,2 ) < ∞}.

0≤r<1

Deﬁnition 6.1. Let 1 ≤ p < ∞. We denote by Lp (D, 2 ) the space of all strong measurable Cp -valued functions deﬁned on [0, 1), such that 1/p 1 ||A(r)||Lp (D,2 ) := 2 0 ||A(r)||pCp rdr < ∞, where Cp is the Schatp ˜ ten class of order p, and we deﬁne La (D, 2 ) as the space of all func-

tions A(r) := A ∗ C(r), where A is an upper triangular matrix with ||A||Lp (D,2 ) < ∞. Here of course C(r) means the Toeplitz matrix associated to the Cauchy 1 , for 0 ≤ r < 1 and ∗ stands for Schur product. kernel 1−r p ˜ a (D, 2 ) is a subspace of Lp (D, 2 ). L By Lpa (D, 2 ) we mean the space of all upper triangular matrices such that ||A(·)||Lp (D,2 ) < ∞. 121

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˜ pa (D, 2 ) and Lpa (D, 2 ) and call Lpa (D, 2 ) the BergmanWe identify L Schatten classes. We intend to introduce the concept of Bergman projection and we prove ﬁrst some introductory results: Lemma 6.2. Let A be an upper triangular matrix belonging either to Cp , 1 ≤ p ≤ ∞, or to B(2 ). Then the function r → A(r) is a continuous function on [0, 1] taking values in Cp , 1 ≤ p ≤ ∞, or in B(2 ), too. Proof.

If A ∈ B(2 ), then we have, when rn → r ∈ [0, 1], that ||(C(rn ) − C(r)) ∗ A||B(2 ) ≤ (by Theorem 2.14) ≤

|rn − r| −→ 0. ||A||B(2 ) n→∞ |1 − rn | |1 − r|

Using the duality and interpolation between Cp we have that lim ||(C(rn ) − C(r)) ∗ A||Cp = 0,

n→∞

1 ≤ p ≤ ∞,

if A ∈ Cp . By Lemma 6.2 we have:

Corollary 6.3. Let 1 ≤ p ≤ ∞ and let A be an upper triangular matrix. If A ∈ Cp (respectively if A ∈ B(2 )), then sup ||C(r) ∗ A||Cp = sup ||C(r) ∗ A||Cp

0≤r<1

0≤r≤1

and similarly with || ||B(2 ) instead of || ||Cp . Proof.

We have to show that, for 1 ≤ p ≤ ∞, ||C(1) ∗ A||Cp ≤ sup ||C(r) ∗ A||Cp , 0≤r<1

(respectively the similar inequality for || ||B(2 ) .) According to Lemma 6.2 we have that ||A||Cp = ||C(1) ∗ A||Cp = lim ||C(r) ∗ A||Cp ≤ sup ||C(r) ∗ A||Cp r→1

r<1

and similar for B(2 ).

∞ ∞ In the sequel we denote L∞ a (D, 2 ) by H (D, 2 ), or simply by H (2 ).

Corollary 6.4. H ∞ (2 ) is a Banach subspace in B(2 ).

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Proof.

We have that ||A||L∞ = sup ||C(r) ∗ A||B(2 ) = (by Corollary 6.3) a (D,2 ) 0≤r<1

= sup ||C(r) ∗ A||B(2 ) 0≤r≤1

= sup r∈[0,1] ||B||C

1 ≤1

≤ ||A||B(2 )

,B

|tr(C(r) ∗ A)B| sup a lower triangular matrix sup

||B||C1 ≤1,rank(B)<∞,B

lower triangular

||C(r) ∗ B||C1

≤ (by Lemma 6.2) ≤ ||A||B(2 ) . On the other hand ||A||L∞ = sup ||C(r) ∗ A||B(2 ) ≥ ||A||B(2 ) , a (D,2 ) 0≤r≤1

∼ ||A||B(2 ) . and, consequently, ||A||L∞ a (D,2 )

˜ p (D, 2 ) is a closed subProposition 6.5. Let 1 ≤ p < ∞. Then L a p p space in L (D, 2 ), and, thus, La (D, 2 ) may be identiﬁed by the map A → A(r), r ∈ [0, 1), with a closed subspace of Lp (D, 2 ). Consequently, the Bergman-Schatten space Lpa (D, 2 ) is a Banach space. Proof. Let An ∈ Lpa (D, 2 ). If An (r) → A(r) ∈ Lp (D, 2 ), then (An (r))n is a Cauchy sequence in Lp (D, 2 ) and, hence ||C(r) ∗ (An − Am )||Cp −→ 0 a.e. with respect to the Lebesgue measure on [0, 1]. Consequently, n,m→∞ by Lemma 6.2, it follows that limn,m→∞ ||C(r) ∗ (An − Am )||Cp = 0 for all r ∈ [0, 1], that is the sequence (C(r) ∗ An )n is a Cauchy sequence in Cp for all r ∈ [0, 1], which in turn implies that limn→∞ (C(r)∗An )(i, j) = A(r)(i, j) for all i, j ∈ N and for all 0 ≤ r ≤ 1. We conclude that A(r) is an upper triangular matrix for all r ≤ 1 and since (An ∗ C(r))(i, j) = anij r j−i it follows that there is limn→∞ anij = aij for all i, j ∈ N and A(r) = C(r) ∗ A, where A = (aij )i,j . Thus A ∈ Lpa (D, 2 ). Remark. We note that in the above proof we have that limr→1 ||A − A(r)||Lpa (D,2 ) = 0. It is also clear that the matrices of ﬁnite-band type are dense in Lpa (D, 2 ). See also [14]-Proposition 2.3 for related results. There are some general results about Bergman-Schatten classes essentially dues to O. Blasco [14].

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Proposition 6.6. Let 1 ≤ p < ∞ and ⎛ 1 1 a0 a1 ⎜ ⎜ 0 a20 ⎜ A=⎜ ⎜0 0 ⎝ .. .. . .

⎞ a12 . . . . ⎟ a21 . . ⎟ ⎟ . ⎟ a30 . . ⎟ ⎠ .. . . . .

be an upper triangular inﬁnite matrix Then A ∈ Lpa (D, 2 ) if and only if ⎛ 1 1 1 ⎞ a1 a2 a3 . . . ⎜ ⎟ ⎜ 0 a21 a22 . . . ⎟ ⎜ ⎟ B := ⎜ ⎟ ∈ Lpa (D, 2 ). ⎜ 0 0 a3 . . . ⎟ 1 ⎝ ⎠ .. .. .. . . . . . . Moreover, there exists a positive constant K < 1 such that K(||A0 ||Cp + ||B||Lpa (D,2 ) ) ≤ ||A||Lpa (D,2 ) ≤ ||A0 ||Cp + ||B||Lpa (D,2 ) . Since A = A0 + S(B), where S is the unilateral shift, that ⎛ 1 1 1 ⎛ 1 1 1 ⎞ ⎞ 0 c0 c1 c2 . . . c0 c1 c2 . . . ⎜ ⎜ ⎟ ⎟ ⎜ 0 0 c20 c21 . . . ⎟ ⎜ 0 c20 c21 . . . ⎟ ⎜ ⎜ ⎟ ⎟ is S(C) := ⎜ ⎟ for C = ⎜ ⎟ , we ﬁnd that ⎜ 0 0 0 c3 . . . ⎟ ⎜ 0 0 c3 . . . ⎟ 0 0 ⎝ ⎝ ⎠ ⎠ .. .. .. .. . . .. .. .. . . . . . . . . . . . ||A||Lpa (D,2 ) ≤ ||A0 ||Cp + ||B||Lpa (D,2 ) . To get the other inequality, note 1 2π that An = (n + 1) 0 0 A(reit )r n e−int πr dtdr for n ≥ 0. This implies the ∞ estimate ||An ||Cp ≤ (n+1)||A||Lpa (D,2 ) . Hence, since B(r) = n=0 An+1 rn , we obtain that 1 ( ||B(r)||pCp 2dr)1/p Proof.

0

≤(

0

1/2

||B(r)||pCp 2rdr)1/p + (

1 1/2

||B(r)||pCp 2rdr)1/p

1 ∞ (n + 1) p ≤ 1/p ( )||A||La (D,2 ) + 2( (||A0 ||Cp + ||A(r)||Cp )p rdr)1/p . n 2 4 1/2 n=0 1

This gives that ||B||Lpa (D,2 ) ≤ L||A||Lpa (D,2 ) , and taking K = 1/(L+1), the proof is complete.

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Another interesting result about Bergman-Schatten classes is as follows [14]: Theorem 6.7. Let A be an upper triangular matrix, n ∈ N, 1 ≤ p < ∞. Then A ∈ Lpa (D, 2 ) if and only if the function r → (1 − r2 )n A(n) (r) ∈ Lp (D, 2 ). Proof. We prove that for any upper triangular matrix A and k ≥ 0, the function (1 − r2 )k A(r) belongs to Lp (D, 2 ) if and only if (1 − r2 )k+1 A (r) also does. Then a recurrent argument gives the statement. Note that (1 − r2 )k+1 A (r) ∈ Lp (D, 2 ) if and only if 1 (1 − r2 )pk+p ||rA (r)||pCp 2dr < ∞. 0

∞ n We denote B(r) = rA (r) = n=0 nAn r and observe that for each 2 r < 1 we have that B(r ) = A(r) ∗ Λ(r), where Λ is the Toeplitz matrix r associated to the function λ(r) = (1−r) 2. Since 2π dt r |λ(reit )| M1 (λ, r) := = 2π 1 − r2 0 and 2

Mp (B, r ) :=

2π

0

2 it

||B(r e

we obtain that 1 (1−r2 )pk+p ||rA (r)||pCp 2dr = 0

0

1

= 0

1

0

4 pk+p

4r (1 − r ) 2π

=C 0

3

)||pCp

1

dt 2π

2π

0

Mpp (B, r 2 )dr

1/p ≤ M1 (λ, r)Mp (A, r),

1 (1−r2 )pk+p ||reit A (reit )||pCp dtdr π

≤C

1 0

1 (1 − r2 )pk ||A(reit )||pCp dtrdr = C π

r(1 − r 2 )pk Mpp (A, r)dr

1 0

(1 − r 2 )pk ||A(r)||pCp 2rdr.

Conversely, we take A such that (1 − r2 )k+1 Mp (A ,r) ∈ 1 L ((0, 1), dr). We may, without loss of generality, assume that 0 (1 − r)(k+1)p Mpp (A , r)dr = 1 and also that A0 = 0. r Since Mp (A, r) ≤ 0 Mp (A , s)ds we have that 1 1 (1 − r2 )kp ||A(r)||pCp 2rdr = (1 − r2 )kp Mpp (A, r)2rdr ≤ p

0

0

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1 0

vers*11*oct*2013

2r(1 − r2 )kp (

0

r

Mp (A , s)ds)p dr ≤ C

1 0

(1 − r)kp (

For p = 1 we get that 1 1 2 k k (1 − r ) ||A(r)||C1 2rdr ≤ C (1 − r) ( 0

0

0

r

r 0

Mp (A , s)ds)p dr.

M1 (A , s)ds)dr

1 C = (1 − s)k+1 M1 (A , s)ds = C . k+1 0 For p > 1 we write, for each t ∈ (0, 1), t r (1 − r)kp ( Mp (A , s)ds)p dr. It = 0 0 r 1 (1 − r)pk+1 and v(r) = ( 0 Mp (A , s)ds)p . Let u(r) = − pk+1 Since u(t)v(t) < 0 and v(0) = 0, we have that t t u (r)v(r)dr ≤ − u(r)v (r)dr. It = 0

Thus, It ≤ =

p pk + 1

0

t r p (1 − r)pk+1 Mp (A , r)( Mp (A , s)ds)p−1 dr pk + 1 0 0 r t (1 − r)k+1 Mp (A , r)(1 − r)(p−1)k Mp (A , s)ds)p−1 dr. 0

0

1/p

Then the assumption and H¨older’s inequality shows that It ≤ CIt . Hence, It ≤ C for all t and the proof is complete. Remark. We observe that for A ∈ L1a (D, 2 ) and n ∈ N we have that 1 2π 1 A(reit )rn e−int rdtdr = π 0 0 1 2π 1 An . (1 − r2 )A (reit )rn−1 e−i(n−1)t rdtdr = π n +1 0 0 See [14]-Proposition 2.6. Using the above remark it is easy to prove the following result (see [14]-Proposition 2.7): Proposition 6.8. If A ∈ L1a (D, 2 ), then 1 2π A(seiu )s 1 duds = A(reit ) = it se−iu )2 π (1 − re 0 0 1 π (1 − s2 )A(seiu )s 1 duds for all 0 ≤ r < 1. 2 it −iu )3 π 0 0 (1 − re se

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Lemma 6.9. Let A ∈ L2a (D, 2 ), 0 ≤ r < 1 and B ∈ C2 . Then the linear functional Fr,B (A) = tr A(r)B ∗ is continuous on L2a (D, 2 ). Proof. If A is an upper triangular matrix of ﬁnite order and A(r) = C(r)∗ A, then we consider the function fA (r, θ) on D given in the introduction. It is clear that the function above is an holomorphic C2 -valued function on D, k ikθ and, consequently, the function z → || ∞ ||C2 is subharmonic. k=0 Ak r e Thus, for 0 < r ≤ 1 − r we have that 1 ≤ 2 πr

||A(r)||2C2 ≤

1 r 2

∞ r

0

r 2π 0

0

||

∞

Ak |r + seiθ |k eikθ ||2C2 sdθds

k=0

||Ak ||2C2 (r + s)2k 2sds ≤ (since r ≤ 1 − r)

k=0

1 1 1 ||A(s)||2C2 2sds = 2 ||A||2L2a (D,2 ) . ≤ 2 r r 0 By taking above r = 1 − r we obtain that ||B||2C2 · ||A||2L2a (D,2 ) , |Fr,B (A)|2 = |tr A(r)B ∗ |2 ≤ ||A(r)||2C2 · ||B||2C2 ≤ (1 − r)2 which implies the continuity of Fr,B . In view of Lemma 6.9 and Riesz Theorem it follows that there is an unique matrix Kr,B ∈ L2a (D, 2 ) such that 1 2 Fr,B (A) = < A, Kr,B >La (D,2 ) = 2 tr A(s)[Kr,B (s)]∗ sds 0

for all B ∈ C2 , 0 ≤ r < 1 and A ∈ L2a (D, 2 ). Let i, j ∈ N and let B be the matrix whose entries b(k, l) are b(k, l) := δki δlj . Then the above formula yields that: 1 1 ∗ tr A(s)Kr,i,j (s)∗ sds = 2 tr [A(s)(Kr,i,j ∗ P (s))]sds A(r)(i, j) = 2 0

0

for all i, j ∈ N, where by Kr,i,j we denote the matrix Kr,B for the above matrix B, P (s) is the Toeplitz matrix associated to the Poisson kernel, that is ⎞ ⎛ 1 s s 2 s3 . . . ⎟ ⎜ ⎜ s 1 s s2 . . . ⎟ ⎟ ⎜ P (s) = ⎜ ⎟. ⎜ s2 s 1 s . . . ⎟ ⎠ ⎝ .. . . . . . . . . . . . . .

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Since A(r) is an analytic matrix, we have that 1 tr(P (s) ∗ A)[P (s) ∗ Kr,i,j ](2s)ds (P (r) ∗ A)(i, j) = 0

for all j ≥ i and all 0 ≤ r < 1. It is easy to see that , ∞ (j − i + 1)rj−i δi,m δj,l l,m=1 i ≤ j, Kr,i,j = ∞ i > j. (0)l,m=1 ˜ 2a (D, 2 ) Deﬁnition 6.10. Let r → A(r) be an element of L2 (D, 2 ). Since L 2 is a closed subspace in the Hilbert space L (D, 2 ), there is an unique or˜ 2 (D, 2 ), called Bergman projection. We denote thogonal projection P˜ on L a

by P the corresponding operator from L2 (D, 2 ) onto L2a (D, 2 ).

Proposition 6.11. For all functions r → A(r) from L2 (D, 2 ) and for all i, j ∈ N we have that , 1 2(j − i + 1)r j−i 0 aij (s)sj−i+1 ds if i ≤ j, {[P (A(·))](r)}(i, j) = 0 if i > j. We have that

Proof.

[P (A(·))](r)(i, j) = Fr,i,j (P (A(·))) = = (since P is a selfadjoint projection) = < A, P (Kr,i,j ) > = (since Kr,i,j is an analytic matrix) = < A, Kr,i,j >L2a (D,2 ) =2 0

1

∗ tr[A(s)Kr,i,j (s)]sds

= 2(j − i + 1)r

1

j−i 0

aij (s) · sj−i+1 ds,

if j ≥ i and < A, B > means trAB ∗ . If j < i, then it is easy to see that ([P (A(·))]) (i, j) = 0.

Unfortunately the Bergman projection is unbounded on L1 (D, 2 ), but instead we can consider a version of it. Let α > −1. Then we put ⎧ ∞ ⎨ Γ(j−i+2+α) rj−i δ δ i ≤ j, i,l j,m (j−i)!Γ(2+α) l,m=1 Kr,i,j,α = ⎩0 i>j

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and we have that, for an analytic matrix A(s) = P (s) ∗ A, 1 ∞ j−i = (α + 1)2 < Ak , [Kr,i,j,α ]k > s2k+1 (1 − s2 )α ds aij r 0 k=0

where A = (aij )∞ i,j=1 . Then , [Pα A(·)](r) =

1 (α+1)Γ(j−i+2+α) j−i r (2 0 (j−i)!Γ(α+2)

0

∀ i, j,

aij (s)sj−i+1 (1 − s2 )α ds) if j ≥ i if j < i.

Theorem 6.12. P1 is a continuous operator (precisely a continuous projection) from L1 (D, 2 ) onto L1a (D, 2 ). Proof. By Theorem 1.2, the topological dual of L1 (D, 2 ) is L∞ (D, 2 ) with respect to the duality pair 1 < A(·), B(·) > := tr (A(s)[B(s)]∗ )2sds, 0

where A(·) ∈ L∞ (D, 2 ), B(·) ∈ L1 (D, 2 ). Now we are looking for the adjoint P1∗ of P1: < P1∗ A(·), B(·) > = 2 ∞ ∞

=

1 0

i=1 j=1

∞ ∞ 1 0

(P1∗ A(·))(r)(i, j)bij (r)rdr

i=1 j=1

(P1∗ A(·))(r)(i, j)bij (r)(2r)dr.

On the other hand <

P1∗ A(·), B(·)

=

> = < A(·), P1 B(·) > =

∞ ∞ 0

i=1 j=1

=

A(r)(i, j)(P1 B)(r)(i, j)(2rdr)

(j − i)!Γ(2)

1 0

tr A(r)(P1 B)∗ (r)(2rdr)

1

∞ ∞ Γ(j − i + 3) i=1 j=i

0

1

1

[A(s)](i, j)s

j−i

0

bij (s)sj−i (1 − s2 )(2sds) .

(2sds) ×

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We take B(s)(i, j) = χIk (s)/(μ(Ik )) and B(s)(l, k) = 0, (l, k) = (i, j), ∀(i, j) ∈ N×N, where Ik r is a sequence of intervals such that lim μ(Ik ) = k→∞

0, and dμ = 2sds. By Lebesgue’s diﬀerentiation theorem we have that , 1 Γ(j−i+3) j−i r (1 − r2 ) 0 A(s)(i, j)sj−i (2sds) if j > i, ∗ (j−i)!Γ(2) (P1 A(·)) (r)(i, j) = 0 if j ≤ i, a.e. for all r ∈ [0, 1). We show that P1∗ : L∞ (D, 2 ) → L∞ (D, 2 ) is a bounded operator. In order to prove this we have to remark that ||A(r)||2L∞ (D,2 ) = ess sup ||A(r)||2B(2 ) = 0≤r<1

ess sup

0≤r<1

∞

sup

j=1

∞ ∞ | aij (r)hj |2 .

|hj |2 ≤1 i=1

j=1

Consequently, since L1 [0, 1] has cotype 2, there is a constant K > 0 such that ||P1∗ A(·)||2L∞ (D,2 ) = ess sup

sup

∞ ∞ |

0≤r<1 ||h||2 ≤1 i=1

j=i

1 0

hj rj−i

Γ(j − i + 3) (1 − r 2 )aij (s)sj−i (2sds)|2 = (j − i)!Γ(2)

∞ 1 ∞ Γ(j − i + 3) | ( aij (s)((rs)j−i ess sup sup (1−r ) )hj )(2sds)|2 (j − i)! 0≤r<1 ||h||l2 ≤1 0 i=1 j=i 2 2

≤ Kess sup (1 − r2 )2 0≤r<1

1

(

[ 0

sup ×

||h||l2 ≤1

∞ ∞ | aij (s)[(rs)j−i (j − i + 2)(j − i + 1)]hj |2 )1/2 (2sds)]2 . i=1

j=i

Since the Toeplitz matrix C(rs) given by the sequence of functions (cij )(rs)∞ i,j=1 , where

(rs)j−i (j − i + 2)(j − i + 1) if j ≥ i cij (rs) := cj−i (rs) = 0 if j < i,

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∞ k ikθ is a Schur multiplier (we remark that = k=0 u (k + 2)(k + 1)e 2 ), then, by Bennett’s Theorem, the multiplier norm of the maiθ 3 (1−ue ) 2 trix C(rs) is exactly the L1 (T)-norm of (1−rse iθ )3 , that is it is equal to ∞ Γ(n+3/2)2 2n 2 n=0 (n!)2 Γ(3/2)2 (rs) . Thus, ⎛ ⎞1/2 ∞ ∞ ⎝ | aij (s)(rs)j−i (j − i + 2)(j − i + 1)hj |2 ⎠ sup ∞ j=1

|hj |2 ≤1

i=1

j=i

= ||A(s) ∗ C(rs)||B(2 ) ≤ ||A(s)||B(2 ) ·

∞ Γ(n + 3/2)2 (rs)2n . 2 Γ(3/2)2 (n!) n=0

Consequently, ||P1∗ A(·)||2L∞ (D,2 ) ≤ 2 2

Kess sup(1 − r ) r<1

1 0

∞ Γ(n + 3/2)2 2n 2n+1 ||A(s)||B(2 ) · r s (2ds) (n!)2 Γ(3/2)2 n=0

2 2

Kess sup(1 − r ) r<1

||A(·)||2L∞ (D,2 )

∞

Γ(n + 3/2)2 r2n (n!)2 (n + 1)Γ(3/2)2 n=0

(by Stirling’s formula) ∼ ess sup(1 − r2 )2 r<1

2 ≤

2 ∼

1 ||A(·)||2L∞ (D,2 ) (1 − r2 )2

∼ ||A(·)||2L∞ (D,2 ) , which shows in turn that P1∗ : L∞ (D, 2 ) → L∞ (D, 2 ) is bounded. The proof is complete. We have the following duality theorem: Theorem 6.13. Let 1 = 0

Proof. We use the boundedness of the projection P : Lp (D, 2 ) → Lpa (D, 2 ). Since the proof is similar to that of Theorem 7.11 we leave out the details. Cf, also [14] Theorem 3.6.

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Some inequalities in Bergman-Schatten classes

The following Hardy-Littlewood inequalities are useful in the study of classical Hardy spaces H p , with 1 ≤ p ≤ ∞. See [23].

Hardy-Littlewood Theorem. (i) If 0 < p ≤ 2, then f ∈ H p implies np−2 |an |p < ∞, and ∞ 1/p p−2 p (n + 1) |an | ≤ Kp ||f ||H p , n=0

n where Kp denotes a constant depending only on p and f (z) = ∞ n=0 an z , for all |z| < 1. p−2 (ii) If 2 ≤ p ≤ ∞, then n |an |p < ∞ implies f ∈ H p , and ∞ 1/p p−2 p (n + 1) |an | . ||f ||H p ≤ Kp n=0

In the book [28] it is given a more complete form of Hardy-Littlewood theorem for classical Bergman spaces. In what follows we give the matrix version of these last inequalities, more precisely we state and prove the matrix version of Theorem 3-p.83 [28]. See also [14]-Proposition 4.1. unc We denote by Lp, (D, 2 ) the space a ⎛ ⎞1/p 1 1 || k (t)Ak rk ||pCp rdrdt⎠ < ∞}, {A| ||A||p,unc := ⎝ 0

0

k≥0

where k means the kth Rademacher function. ∞ Theorem 6.14. 1. If p = 1 and A ∈ L1a (D, 2 ), then n=1 n−2 ||An ||C1 < ∞. (D, 2 ), then we have that 2. If 1 < p ≤ 2 and A ∈ Lp,unc a ∞ p p−3 ||An ||Cp < ∞. n=1 n ∞ 3. If 2 ≤ p < ∞ and n=1 np−3 ||An ||pCp < ∞, then A ∈ Lp,unc (D, 2 ). a ∞ p −1 4. If 2 ≤ p < ∞ and A ∈ La (D, 2 ), then n=1 n ||An ||Cp < ∞. 1 ||An ||pCp < ∞, then A ∈ Lpa (D, 2 ). 5. If 1 < p ≤ 2 and n≥0 n+1 6. If 1 ≤ p < ∞ and A ∈ Lpa (D, 2 ), then ||An ||Cp = o(n1/p ). Proof. 1. Let A ∈ L1a (D, 2 ). We use the results of Shields [84] applied to the function z → A(rz), with a ﬁxed 0 < r < 1. Then, we ﬁnd that ∞ (n + 1)−1 ||An r n ||C1 ≤ C{M1 (A, r)}. n=0

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Multiplying the above inequality by r and integrating over (0, 1) we have that 1 ∞ (n + 1)−1 ||An ||C1 rn+1 dr ≤ ||A||L1a , 0

n=0

or ∞

(n + 1)−2 ||An ||C1 ≤ C||A||L1a .

n=0

2. Let A ∈ Ap,unc and 1 < p ≤ 2. By Theorem 4.5 applied for Ar (z) = A(rz), 0 < r < 1, with Ar ∈ Tp (2R ) + Tp (2C ), we ﬁnd that 1 ∞ ∞ p−2 n p (n + 1) ||An r ||Cp ≤ C(p) || k (t)Ak r k ||pCp dt. 0

n=0

k=0

Proceeding as in the case 1 we get that 1 ∞ (n + 1)p−2 ||An ||pCp r np+1 dr ≤ C(p)||A||pp,unc , 0

n=0

or ∞

(n + 1)p−3 ||An ||pCp ≤ C(p)||A||pp,unc .

n=0

3. By Theorem 4.7 applied to Ar (z) = A(rz), for 0 < r < 1, we have that ∞ ||Ar ||pp,unc ≤ C(p) (n + 1)p−2 ||An rn ||pCp , n=0

and, consequently, 1 ∞ ∞ || k (t)Ak r k ||pCp dt ≤ C(p) (n + 1)p−2 ||An rn ||pCp . 0

0

n=0

k=0

Hence, denoting C(p) = C,

∞ ∞ 1 1 k p || k (t)Ak r ||Cp rdr dt ≤ C (n + 1)p−2 ||An ||pCp 0

n=0

k=0

1

r np+1 dr

0

and ﬁnally ||A||pp,unc ≤ C(p)

∞

(n + 1)p−3 ||An ||pCp .

n=0

4. In the same way as in Corollary 7.6 we get that Lpa (D, 2 ) coincides with the interpolation space [L2a (D, 2 ), B(D, 2 )]θ for 2/p = 1−θ. Moreover,

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let T : B(D, `2 ) → `∞ (`∞ ); be given by T (A) = (An )n≥0 . We note that T maps L2a (D, `2 ) continuously into `2 (`2 , w), where w(n) = 1/(n + 1), for all n ≥ 0, and (`2 , w) is the space of all sequences x = (xn )n≥0 such that P 2 n≥0 |xn | w(n) < ∞, with the natural norm. Indeed, similarly to Theorem at page 84-[28] we get that X 1 ||T (A)||2 = ||An ||2∞ ≤ C(2)||A||2L2a (D,`2 ) . n+1 n≥0

Hence, T : Lpa (D, `2 ) → [`2 (`2 , w), `∞ (`∞ )]θ = `p (`p , w), where θ = 1 − 2/p, is a bounded operator. (See [93] for the last equation.) Hence,  1/p X 1  ||An ||pCp  ≤ C(p)||A||Lpa (D,`2 ) . n+1 n≥0

5. By Theorem 6.13 we find that [Lpa (D, `2 )]∗ = Lqa (D, `2 ), 1/p+1/q = 1. On the other hand, [`p (`p , w)]∗ = `q (`q , w), where 1/p + 1/q = 1. Therefore, the conclusion follows from 4. 6. For each n and r ∈ (0, 1) we have that Z 2π 1 A(reit )e−int dt. An rn = 2π 0 This implies that, for any n ∈ N and 0 < r < 1, it yields that ||An ||Cp rn ≤ M1 (A, r).

(6.1)

Since Mp (A, ·) is increasing in (0, 1), from (6.1) it follows that Z 1 p np p (1 − r)||An ||Cp r ≤ (1 − r)Mp (A, r) ≤ Mpp (A, s)ds r

for each r ∈ (0, 1). Hence, for any n, by taking r = 1 − n1 , we see that Z 1 1 1 np 1 p n ||An ||Cp ≈ ||An ||Cp (1 − ) ≤ Mpp (A, s)ds. n n n 1−1/n This shows that

||An ||Cp n1/p

→ 0.

Remark. In connection with the above results we mention that O. Blasco showed in [14]-Theorem 4.4 that, for 1 ≤ p < ∞, there exist K1 , K2 > 0 such that !1/p ∞ X p K1 sup ||An ||Cp k=0 2

k−1 ≤n<2k

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⎛ ≤ ||A||Lpa (D,2 ) ≤ K2 ⎝

∞

⎞1/p

(

||An ||Cp )p ⎠

.

k=0 2k−1 ≤n<2k

We are now ready to present the Hausdorﬀ-Young Theorem for Bergman-Schatten classes extending Theorem 2-[28] at pages 81-82. p Theorem 6.15. Let 1 ≤ p ≤ ∞ and let q = p−1 be its conjugate exponent. p (i) If 1 ≤ p ≤ 2, then A ∈ La (D, 2 ) implies that ⎞1/q ⎛ ⎝ (n + 1)1−q ||An ||qTp ⎠ ≤ ||A||Lpa (D,2 ) . n≥0

(ii) If 2 ≤ p ≤ ∞, then

+ 1)1−q ||An ||qTp < ∞ implies that 1/q ∞ q 1−q (n + 1) ||A || . n Tp n=0

n≥0 (n

A ∈ Lpa (D, 2 ) and ||A||Lpa (D,2 ) ≤

Proof. In case (ii), let μ be the discrete measure on the set N which assigns the mass μ(n) + 1 to the integer n = 0, 1, 2, . . . Consider the linear op 1 An } to the formal series n≥0 An z n . erator T that maps the sequence { n+1 We want to show that T is bounded as an operator from Lq (N, dμ; Tp ) to Lp (D, dσ; Tp ), with norm ||T || ≤ 1, where Lp (D, dσ; Tp ) is the space of all p-Lebesgue Bochner integrable Tp -valued functions deﬁned on the unit disk D with respect to the area measure dσ on D. In the case p = 2 this follows from ||T ({An /(n + 1)})||2L2 (D,dσ;T2 ) = ||{An /(n + 1)}||2L2 (N,dμ;T2 ) . For p = ∞ it is the clear that sup|z|≤1 || n An z n ||T∞ ≤ n ||An ||T∞ . The result follows by using these estimates and complex interpolation of vector-valued Lp spaces. p the linear operator T (A) = Case (i). Let A(z) ∈ L (D, dσ; Tp ). Deﬁne p n {bn }, where bn = D A(z)z dσ. If A ∈ La (D, 2 ), then bn = An /(n + 1). With the measure μ deﬁned as before, we want to show that T is a bounded operator from Lp (D, dσ; Tp ) to Lq (N, dμ; Tp ), with norm ||T || ≤ 1. For p = 1 this is the trivial inequality ||bn ||T1 ≤ ||A||L1 (D,dσ;T1 ) for all n ∈ N. For p = 2 it follows from the relation ∞ ∞ 2 (n + 1)||bn ||T2 = (n + 1)|| A(z)z n dσ||2L2a (D,2 ) n=0

=

∞ ∞ k=0 l=1

n=0

1 k+1

D

D

A(z)lk A(ζ)lk

D

∞

(n + 1)(zζ)n dσ(z)dσ(ζ)

n=0

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=

∞ ∞ k=0 l=1

1 k+1

D

D

A(z)lk A(ζ)lk dσ(z)dσ(ζ) = , (1 − zζ)2

where P denotes the Bergman projection. Since | | ≤ ||P A(·)||L2a (D,2 ) ||A(·)||L2a (D,2 ) ≤ ||A(·)||2L2a (D,2 ) , this gives the desired inequality for p = 2. By using these estimates and complex interpolation as before we obtain that ∞ 1/q 1/p q p (n + 1)||bn ||Tp ≤ ||A(z)||Tp dσ , A(·) ∈ Lp (D, dσ; Tp ). D

n=0

Specializing to A ∈ Lpa (D, 2 ) and recalling that bn = An /(n + 1), we arrive at the desired result. 6.3

A characterization of the Bergman-Schatten space

We give a characterization of the space L1a (D, 2 ) completely similar to those obtained by Mateljevic and Pavlovic in [61]. First we prove some necessary Lemmas, which are also of independent interest. Lemma 6.16. Let A = nk=m Ak , where m ≤ n. Then ||A||C1 r n ≤ ||A(r)||C1 ≤ ||A||C1 rm , for all 0 < r < 1. Proof.

We put B[r] =

n k=m

A−k r n−k . Then

||A(r)||C1 = ||A(r)∗ ||C1 = ||

n k=m

1 A−k r k ||C1 = ||B[ ]||C1 rn . r

1 r

Since > 1 and the function seit → ||B[seit ]||C1 , for 0 < s < 1 and t ∈ [0, 2π], is a subharmonic function we get that 2π 2π 1 it dt dt 1 ||B[ ](e )||C1 ||B[1](eit )||C1 ≥ = ||B[1]||C1 ||B[ ]||C1 = r r 2π 2π 0 0 = ||A∗ ||C1 = ||A||C1 , so that the left hand side inequality holds. The other inequality can be proved similarly. The proof is complete.

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n k Let σn (A) = k=0 1 − n+1 Ak be the Cesaro mean of the order n of the upper triangular matrix A, and ||σn (A)||1 := sup0
∞

||σn (A)||1 (n + 1)rn .

n=0

n |k| )eikt ||L1 (T) ≤ 1 for all n ∈ N, it follows Proof. Since || k=−n (1 − n+1 that ||A(r)||C1 ≥ ||σn (A)(r)||C1 for all 0 < r < 1 and for all n ∈ N. Using this inequality and Lemma 6.16 we get the left hand side inequality of Lemma 6.17. For the right hand side of the inequality of the Lemma we use the ∞ elementary formula: A(r) = (1 − r)2 n=0 σn (A)(n + 1)rn . The proof is complete. Lemma 6.18. Let A be an upper triangular matrix. Then, for 0 ≤ k < n, we have that (n − k + 1)||σk (A)||1 ≤ (n + 1)||σn (A)||1 . Since by [17] we have that ||σk (B)||C1 ≤ ||B||C1 it follows that r ||σn (A)||1 ≥ ||σk σn (A)||1 = ||σk (A) − σ (A)||1 ≥ ||σk (A)||1 n+1 k

Proof.

−

1 ||σ (A)||1 . n+1 k

Now it is clear that Bernstein’s inequality ||

n

kAk ||1 ≤ n||

k=0

n

Ak ||1

k=0

holds. Therefore, ||σn (A)||1 ≥ ||σk (A)||1 − The proof is complete.

k ||σk (A)||1 . n+1

We aim to state and prove an analogue of Theorem 2.2 in [61]. In order to do that we recall some results about functions, which were presented in Section 4 in [61].

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Let φ be a non-negative increasing function deﬁned on (0, 1] for which φ(tr) ≤ Ctα φ(r),

0 < t < 1,

(6.2)

and φ(tr) ≥ C −1 tβ φ(r),

0 < t < 1,

(6.3)

where C and β are positive real numbers. Note that β ≥ α > 0. Lemma 6.19. Let ψ(r) = φ(r)q r − , where q < ∞, qα − > −1, and let α satisfy the condition (6.2). Then 1 ψ(1 − r)rx−1 dr ≤ Cx−1 ψ(1/x), x ≥ 1. C −1 x−1 ψ(1/x) ≤ 0

Proof.

We have that 1 x−1 −1 ψ(1 − r)r dr = x I(x) := 0

x

0

ψ(t/x)(1 − t/x)x−1 dt.

Since φ satisﬁes the conditions (6.2) and (6.3) it yields that φ(t/x) ≤ C(tα + tβ )φ(1/x), and, consequently, I(x) ≤ Cx−1 ψ(1/x)

x 0

0
(β ≥ α),

(tα + tβ )q t− (1 − t/x)x−1 dt

≤ Cx−1 ψ(1/x). This proves the right-hand side inequality. The left-hand side inequality is easy and does not depend on the conditions (6.2) and (6.3). The proof is complete. Lemma 6.20. Let ψ(r) = φ(r)r− , ≤ α and let α satisfy (6.2). Then C −1 ψ(1/x) ≤ sup ψ(1 − r)rx−1 ≤ Cψ(1/x), r

Proof.

x ≥ 1.

The proof is similar to that of Lemma 6.19.

Besides these lemmas we shall use the well-known estimate ∞

n

2nβ r2 ≤ Cr(1 − r)−β ,

β > 0.

(6.4)

n=0

The proof of Theorem 6.22, which follows, is based on Lq -behaviour of the functions n

F1 (r) = (1 − r)−1/q φ(1 − r) sup{λn r 2 : n ≥ 0}

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and F2 (r) = (1 − r)−1/q φ(1 − r)

∞

n

λn r 2 ,

n=0

where (λn ) is a sequence of non-negative real numbers. Proposition 6.21. Let F = F1 or F = F2 . Then C −1 ||F ||Lq ≤ ||(φ(2−n )λn )||q ≤ ||F ||q . Proof. We consider only the case q < ∞. In the case q = ∞ the proof is similar and is based on Lemma 6.20. Let q < ∞. Then k

F (r)q ≥ F1 (r)q ≥ (1 − r)−1 φ(1 − r)q λqk r 2

q

for all k

and, by (6.4), F (r)q ≥ C −1 φ(1 − r)q

∞

n

k

2n r 2 λk r 2 q .

n=0

Hence, F (r) ≥ C q

−1

φ(1 − r)

q

∞

2n λqn r 2

n

(1+q)

.

(6.5)

n=0

On the other hand, from Lemma 3.4 and hypothesis (6.3) it follows that 1 n φ(1 − r)q r2 (q+1) dr ≥ C −1 2−n φq (2−n ). 0

Combining this with (6.5), we obtain the right-hand side inequality in Proposition 3.6. To prove the left-hand side inequality, let ηn = 2nδ r 2

n−1

,

θn = 2−nδ λn r2

n−1

,

where δ = α/2 and α satisﬁes (6.2). Then ∞ q ∞ q ∞ q ∞ n λn r 2 = ηn θn ≤ ηn θnq n=0

n=0

≤ C(1 − r)−qδ

n=0 ∞

n−1

2−nqδ λqn r 2

n=0 q

,

n=0

where we have used (6.4). Hence, F (r)q ≤ F2 (r)q ≤ Cψ(1 − r)

∞

2−nqδ λqn r 2

n−1

q

,

n=0

where ψ(r) = φ(r)q r−qδ−1 . Now the desired result follows from Lemma 6.19.

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Our main result in this section reads: Theorem 6.22. Let A be an upper triangular matrix. Then A ∈ L1a (D, 2 ) if and only if ∞ ||σn (A)||1 < ∞. (n + 1)2 n=1 Proof.

First we remark that A ∈ L1a (D, 2 ) if and only if 1 ||A(r)||C1 dr < ∞. 0

Let A ∈ 6.17,

L1a (D, 2 ).

Then, by the left hand side inequality of Lemma

||A(r)||C1 = (1 − r)

∞

||A(r)||C1 rn ≥

n=0

(1 − r)

∞

||σn (A)||1 r 2n .

n=0

Consequently, an integration with respect to r yields 1 1 ∞ ∞> ||A(r)||C1 dr ≥ (1 − r) ||σn (A)||1 r 2n dr = 0

0

n=0

∞

∞ 1 1 1 ||σn (A)||1 ≥ ||σn (A)||1 . (2n + 1)(2n + 2) 8 (n + 1)2 n=0 n=0

Conversely, suppose that ∞

1 ||σn (A)||1 < ∞. (n + 1)2 n=0

Let xn =

n

(k + 1)(n − k + 1)||σk (A)||1 .

k=0

Then, summing by parts as in (4.4) in [61], we get that ∞ n=0

∞

||σn (A)||1 (n + 1)r n = (1 − r)2

n=0

On the other hand, using Lemma 6.18, we see that xn ≤ C(n + 1)3 ||σn (A)||1

xn r n .

(6.6)

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and, therefore, ∞

∞ 1 1 x ≤ C ||σn (A)||1 < ∞. n 5 (n + 1) (n + 1)2 n=0 n=0

By Lemma 4.8 in [61], with q = 1, φ(r) = r, r ∈ (0, 1], we get that 1 ∞ 4 (1 − r) xn rn dr < ∞. 0

n=0

Using (6.6) we arrive at 1 ∞ (1 − r)2 ||σn (A)||1 (n + 1)rn dr < ∞ 0

n=0

and, by the right hand side inequality in Lemma 6.17, ﬁnally we obtain that 1 ||A(r)||C1 dr < ∞ 0

i.e. A ∈ 6.4

L1a (D, 2 ).

The proof is complete.

Usual multipliers in Bergman-Schatten spaces

In this section we characterize the multipliers with respect to usual product of matrices for the Bergman-Schatten spaces of index 2, L2a (2 ). More speciﬁc, let A be an inﬁnite upper triangular matrix. A is called an usual multiplier for L2a (2 ) and one writes A ∈ M (L2a ), if for all B ∈ L2a (2 ), it follows that AB ∈ L2a (2 ), or equivalently (by theorem of closed graph) there is the smallest constant M (A) < ∞ such that ||A · B||L2a (2 ) ≤ M (A)||B||L2a (2 ) .

(6.7)

Now let b ∈ N. Let us denote by n2 (w) = {x = (x1 , x2 , . . . , xn ) with the 1/2 def n |xj |2 }. norm ||x||2,w = j=1 n−j+1 Then, A being as above we denote by ||A||n = ||A||B(n2 (w),22 (w)) , n ∈ N. We have the following result: Theorem 6.23. A is an usual multiplier for L2a (2 ) if and only if sup ||A||n < ∞ and M (A) = sup ||A||n . n

n

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Proof.

First we denote by P (r) the Toeplitz matrix ⎞ ⎛ 1 r r2 r3 . . . ⎜ r 1 r r2 . . . ⎟ ⎟ ⎜ ⎜ r2 r 1 r . . . ⎟ , 0 ≤ r < 1 ⎠ ⎝ .. . . . . . . . . . . . . .

and we denote by A(r) = A ∗ P (r), where, as usual in this book ∗ means the Schur product of matrices. Let us remark that A(r) · B(r) = (A · B)(r) for all B upper triangular matrices and for all r ∈ [0, 1). Then, for an usual multiplier A for L2a (2 ), we have 1 1 ∞ ∞ || (AB)k r k ||2C2 (2rdr) ≤ M (A)2 || Bk r k ||2C2 2rdr, (6.8) 0

0

k=0

k=0

(n) for all upper triangular matrices B. Take now B = B , the nth column j of B. Then, substituting in (6.8) we have, for A = ak and B (n) = k≥0,j≥1 ⎛ ⎞ 0 0 . . . 0 b1n−1 0 . . . ⎜ 0 0 . . . 0 b2n−1 0 . . . ⎟ ⎜ ⎟ ⎜ .. .. .. .. .. .. .. ⎟ , ⎜. . . . . . . ⎟ ⎜ ⎟ ⎝ 0 0 . . . 0 bn 0 . . . ⎠ 0 0 0 ... 0 0 0 ...

⎛

a10 ⎜0 ⎜ ⎜ ⎜0 =⎜ ⎜ .. ⎜. ⎜ ⎝0 0 ⎛

a11 a20 0 .. .

a12 a21 a30 .. .

a13 a22 a31 .. .

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

0 ... ... ... 0 ... ... ...

0 0 ... ⎜0 0 ... ⎜ ⎜ = ⎜ ... ... ... ⎜ ⎝0 0 ... 0 0 ...

AB (n) ⎞ ⎛ a1n−1 . . . 0 0 ... ⎜ ⎟ 2 an−2 . . . ⎟ ⎜ 0 0 . . . ⎟ ⎜ a3n−3 . . . ⎟ ⎜ 0 0 . . . ⎜ .. .. ⎟ ⎟ · ⎜. . . . . ⎟ ⎜ .. .. .. ⎟ ⎜ an0 . . . ⎠ ⎝ 0 0 . . . 0 ... 0 0 ...

0 b1n−1 0 b2n−2 0 b3n−3 .. .. . . 0 bn0 00

0 a10 b1n−1 + a11 b2n−2 + · · · + a1n−1 bn0 0 a20 b2n−2 + a21 b3n−3 + · · · + a2n−2 bn0 .. .. . . 0 an0 bn0 00

Hence ||AB (n) ||2L2a (2 ) =

⎞ ... ...⎟ ⎟ ⎟ ...⎟ .. ⎟ ⎟ . ⎟ ⎟ 0 ...⎠ 0 ... 0 0 0 .. .

⎞ 0 ... 0 ...⎟ ⎟ .. .. ⎟ . . . ⎟ ⎟ 0 ...⎠ 0 ...

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1 0

⎛ ⎝|

n−1

2 2n−2 a1j bj+1 n−1−j | r

+|

n−2

j=0

=|

⎞ 2 2n−4 a2j bj+2 n−2−j | r

+ · · · + |an0 bn0 |2 ⎠ (2rdr)

j=0

n−1

2 a1j bj+1 n−1−j |

j=0

n−2 1 1 2 · +| a2j bj+2 + · · · + |an0 bn0 |2 , n−2−j | · n n − 1 j=0

and ||B (n) ||2L2a (2 ) = |b1n−1 |2 ·

1 1 + |b2n−2 |2 · + · · · + |bn0 |2 . n n−1

Hence ⎛ ⎞1/2 n−1 n−2 1 1 2 2 ⎝| a1j bj+1 a2j bj+2 ≤ +| + · · · + |an0 bn0 |2 ⎠ n−1−j | · n−2−j | · n n − 1 j=0 j=0 ⎛ M (A) ⎝

n−1

⎞1/2 |bn−j |2 · j

j=0

1 ⎠ n−j

n = || bjn−j

j=1

||n2 (w)

or ||A||n ≤ M (A) for all n. Consequently sup ||A||n ≤ M (A). n

Conversely, if supn ||A||n = M < ∞, if k is arbitrary but ﬁxed and (k) n−1 the sequence B = k=0 B (k) ∈ L2a (2 ), we have, denoting by AB (k) placed on the kth column beginning from the top, n n (k) ||AB (k) ||2L2a (2 ) = || AB (k) ||2k (w) ||AB||2L2a (2 ) = k=1

≤M

2

k=1

n n || B (k) ||2k (w) ≤ M ||B (k) ||2L2a (2 ) = M ||B||2L2a (2 ) k=1

2

k=1

for all k. Hence AB ∈ L2a (2 ) ∀ B ∈ L2a (2 ) and M (A) = sup ||A||n . n

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Now we can characterize the matrices A such that supn ||A||n < ∞. Theorem 6.24. If A is an upper triangular matrix, then it follows that supn ||A||n < ∞ if and only if A ∈ B(2 ). Proof. Let A = alk l≥1,k≥0 such that supn ||A||n = 1 < ∞. Then, for a column matrix B such that ⎛ 1 ⎞ bn−1 ⎜ b2n−2 ⎟ ⎟ ⎜ (B) = ⎜ . ⎟ ∈ n2 (w) . ⎠ ⎝. bn0 with ||(B)||22,w = we have ||(AB)||22,w |

n−1

1 k=0 (ak

√

=

|

n−1 k=0

n − k)

|b2 |2 |b1n−1 |2 + n−2 + · · · + |bn0 |2 = 1 n n−1

2 | a1k bk+1 n−k−1 | + n

bk+1 n−k−1 2 √ | n−k

n

+

|

n−2

2 a2k bk+2 n−k−1 | + · · · + |an0 bn0 |2 = n−1

k=0

n−2

2 k=0 (ak

√

bk+2

n − k − 1) √n−k−2 |2 n−k−1 n−1

bk+1

+· · ·+|an0 bn0 |2 .

n−k−1 , where k = 0, 1, . . . , n − 1 we have If we denote by yk = √ n−k 2 2 2 ||(B)||2,w = |y0 | + · · · + |yn−1 | = ||(yk )||22 . Hence 0 0 n−1 n−2 n−k 2 2 n−k−1 2 1 yk | +| yk+1 |2 +· · ·+|an0 yn−1 |2 ak ak ||(AB)||2,w = | n n−1

k=0

k=0

= ||An y||22 , where

⎛

a10 a11

⎜ ⎜ ⎜0 ⎜ An = ⎜ ⎜0 ⎜ ⎜ .. ⎝. 0

>

n−1 n

a12

a20

a12

0 .. . 0

a30 .. . 0

> >

n−2 n

...

n−2 n−1

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

a1n−1 √ n a2n−2 √ n−1 a3 √n−3 n−2

.. . an0

0 ...

⎞

⎟ ⎟ 0 ...⎟ ⎟ ⎟ 0 ...⎟. ⎟ ⎟ .. . ...⎠ 0 ...

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Hence ||An ||B(2 ) ≤ ||A||n , which in turn implies that supn ||An ||B(2 ) ≤ supn ||A||n = 1, in other words the sequence (An )∞ n=1 belongs to the unit ball B(B(2 )) of B(2 ). It is easy to see that An (k, l) → A(k, l) for all k, l ∈ N. Since the unit ball B(B(2 )) is σ(B(2 ), C1 )-compact it follows that A ∈ B(B(2 )), that is ||A||B(2 ) ≤ 1 = supn ||A||n . The converse implication is easy, it is enough to verify that such a matrix is an usual multiplier on L2a (2 ) and apply the previous theorem. Each matrix Φ ∈ L2a (2 ) is said to generate a subspace [Φ], the closure of the set of multiples of Φ by matrices of ﬁnite band type. When Φ ∈ H ∞ (2 ), it is important to observe that [Φ] does not necessarily coincide with ΦL2a (2 ) = {ΦF : F ∈ L2a (2 )}, since the latter set need not be closed. The crucial requirement is that the operator of multiplication by Φ be bounded below: in other words, that there exist a constant c > 0 such that ||ΦF ||L2a (2 ) ≥ c||F ||L2a (2 ) for all F ∈ L2a (2 ). Theorem 6.25. Let Φ be an invertible matrix from H ∞ (2 ). Then the set ΦL2a (2 ) is closed in L2a (2 ) if and only if the operator of multiplication MΦ is bounded below on L2a (2 ). Proof. Suppose ﬁrst that MΦ is bounded below. Then if An ∈ L2a (2 ) and {ΦAn } converges to some G ∈ L2a (2 ), it follows that An → A for some A ∈ L2a (2 ). Hence ΦAn → ΦA because Φ ∈ H ∞(2 ) is an usual multiplier on L2a (2 ). Hence G = ΦA ∈ ΦL2a (2 ), which is consequently a closed subspace. Consequently, we deﬁne T (G) = Φ−1 G with the domain D(T ) = {G ∈ [Φ] : Φ−1 G ∈ L2a (2 )}. It is easy to remark that T is a closed operator. Indeed if Gn → G ∈ L2a (2 ), where Gn ∈ D(T ) and T (g n ) → F, that is Φ−1 Gn → F ∈ L2a (2 ), then it follows that Gn → MΦ (F ) = ΦF in L2a (2 ). Consequently G = ΦF and Φ−1 G = F ∈ L2a (2 ), that is G ∈ D(T ) and T is a closed operator. Observe ﬁrst that ΦL2a (2 ) is closed if and only if ΦL2a (2 ) = [Φ], because the matrices of ﬁnite band type are dense in L2a (2 ). Since by hypothesis ΦL2a (2 ) is a closed subspace and D(T ) = {G = ΦA, A ∈ L2a (2 ), Φ−1 G ∈ L2a (2 )} = ΦL2a (2 ), by closed graph theorem it follows that T is a bounded operator. Hence ||A||L2a (2 ) ≤ C||ΦA||L2a (2 ) for some constant C > 0 and for all A ∈ L2a (2 ). Hence MΦ is a bounded below operator.

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Now we give some interesting examples of matrices Φ such that MΦ be bounded below operator on L2a (2 ). Let α ∈ C with |α| < 1, α = 0. Let 0 ≤ r0 < |α| and Φ = Φα ∗ C(r0 ), where Φα is the Toeplitz matrix associated to the Moebius transform α−z φα (z) = 1−αz , |z| < 1. Then MΦ is bounded below linear and bounded operator on L2a (2 ). Indeed we have ⎛ ⎞ α |α|2 − 1 (|α|2 − 1)α (|α|2 − 1)α2 . . . ⎜0 α |α|2 − 1 (|α|2 − 1)α . . . ⎟ ⎜ ⎟ ⎜ .. ⎟ 2 ⎜0 0 ⎟ . α |α| − 1 Φα = ⎜ ⎟. ⎜ ⎟ . . ⎜0 0 .⎟ 0 α ⎝ ⎠ .. .. .. .. .. . . . . . Taking 0 ≤ r < 1 and B ∈ L2a (2 ) we have ⎛ α (|α|2 − 1)r0 r (|α|2 − 1)αr02 r2 ⎜0 α (|α|2 − 1)r0 r ⎜ ⎜ ⎜ α [MΦ (B)](r) = ⎜ 0 0 ⎜ ⎜0 0 0 ⎝ .. .. .. . . . ⎛

b10 ⎜0 ⎜ ⎜ ⎜0 ⎜ ⎜ ⎜0 ⎝ .. .

⎞ (|α|2 − 1)α2 r03 r 3 . . . (|α|2 − 1)αr02 r2 . . . ⎟ ⎟ .. ⎟ 2 ⎟ . (|α| − 1)r0 r ⎟· .. ⎟ .⎟ α ⎠ .. .. . .

⎞ b11 r b12 r 2 b13 r 3 . . . b20 b21 r b22 r 2 . . . ⎟ ⎟ . ⎟ 0 b30 b31 r . . ⎟ ⎟= .. ⎟ 4 .⎟ 0 0 b0 ⎠ .. .. .. .. . . . .

αB(r)−(1−|α|2 )r0 τ1 [(B −B 1 )(r)]−(1−|α|2 )αr02 τ2 [(B −B 1 −B 2 )(αr0 r)]− (1 − |α|2 )α2 r03 τ3 [(B − B 1 − B 2 − B 3 )(r)] − . . . Here B k is the kth row of B and τk is the upper translation with k rows of the corresponding matrix. Then it follows that 1 1/2 ) ||MΦ (B)(r)||2C2 (2rdr) ≤ |α| + (1 − |α|2 )r0 + ||MΦ (B)||L2a (2 ) = 0

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* (1 − |α|2 )r02 |α| + (1 − |α|2 )r03 |α|2 + . . . ||B||L2a (2 ) = "

|α| + (1 − |α|2 )r0

# 1 ||B||L2a (2 ) ≤ [|α| + (1 + |α|)r0 ] ||B||L2a (2 ) = 1 − r0 |α| C(α)||B||L2a (2 ) ,

hence the multiplication operator MΦ is bounded on L2a (2 ) (even if r0 is not less than |α|). Now we will show that MΦ , if 0 ≤ r0 < |α| is bounded below on L2a (2 ). Indeed ||Φ−1 α ∗ C(r0 )||B(2 ) ≤ K(α). More speciﬁc

⎛ ⎜ ⎜ ⎜ = Φ−1 α ⎜ ⎝

Let 0 ≤ r0 < |α|. Then

Φ−1 α

⎛ ⎜ ⎜ ∗ C(r0 ) = ⎜ ⎜ ⎝

Hence ||Φ−1 α ∗ C(r0 )||B(2 )

2 2 1 1−|α| 1−|α| α α2 α3 1−|α|2 0 α1 α2 1 0 0 α

.. .

.. .

.. .

...

⎞

⎟ ...⎟ ⎟. ...⎟ ⎠ .. .

2 2 1 1−|α| r0 1−|α| r0 2 (α) α (α) α α 1−|α|2 r0 1 0 (α) α α 1 0 0 α

.. .

.. .

$⎛ $ 0 1 − |α|2 $ 1 $⎜ 0 + ≤ $⎝ |α| |α| $ . $ ..

.. .

r0 α

0 .. .

...

⎞

⎟ ...⎟ ⎟. ...⎟ ⎠ .. .

⎞$ ( rα0 )2 . . . $ $ r0 $ ...⎟ α ⎠$ .. . . $ . $ .

= B(2 )

1 1 − |α|2 r0 1 1 − |α|2 r0 1 1 + sup + r0 it ≤ r0 = |α| |α| |α| t∈R |1 − |α| e | |α| |α| |α| 1 − |α| 1 1 − |α|r0 = K(α, r0 ). = |α| − r0 φ|α| (r0 ) Since * ) −1 Φα ∗ C(r0 ) [Φα ∗ C(r0 )] = Φ−1 α Φα ∗ C(r0 ) = I ∗ C(r0 ) = I,

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it follows that −1 Φ−1 . α ∗ C(r0 ) = [Φα ∗ C(r0 )]

But

n o −1 B ∗ C(r) = [Φα ∗ C(r0 )] ∗ C(r) {Φα ∗ C(r0 ) ∗ C(r)} {B ∗ C(r)} =

Φ−1 α ∗ C(r0 ) ΦB ∗ C(r).

Hence ||B||2L2a (`2 ) Z 0

1

||

Z = 0

1

||B ∗ C(r)||2C2 2rdr =

−1 Φα ∗ C(r0 ) ΦB ∗ C(r)||2C2 2rdr =

2 || Φ−1 α ∗ C(r0 ) ΦB||L2a (`2 ) ≤ (by Theorems 4.2 and 4.3) ≤ 2 2 2 2 ||Φ−1 α ∗ C(r0 )||B(`2 ) ||ΦB||L2a (`2 ) ≤ K(α, r0 ) ||ΦB||L2a (`2 ) .

Notes In Section 6.1 we introduce a version of matrix valued Bergman spaces studied previously independently by O. Blasco [14]. We call these spaces Bergman-Schatten spaces. They are appropriate spaces in order to develop a theory similar to classical Harmonic Analysis. For instance see Theorem 6.7 which is a perfect analogue of Theorem 4.2.9 [94]. Moreover, a matrix version of Bergman Projection is introduced and a duality theorem between Bergman-Schatten spaces (Theorem 6.13) is given. In Section 6.2 we prove some inequalities similar with those from the monograph [28]. In particular, Theorem 6.15 is the Hausdorff-Young Theorem for Bergman Schatten classes. In Section 6.3 we derive a characterization of the Bergman-Schatten space (see Theorem 6.22) which is completely similar with that obtained by Mateljevic and Pavlovic in [61]. In Section 6.4 we characterize the usual multipliers on the BergmanSchatten space of index 2.

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Chapter 7

A matrix version of Bloch spaces

7.1

Elementary properties of Bloch matrices

The Bloch functions and the Bloch space have a long history behind them. They were introduced by the French mathematician Andr´e Bloch in the beginning of the last century. Many mathematicians paid attention to these spaces e.g. the following: L. Ahlfors, J. M. Anderson, J. Clunie, Ch. Pommerenke, P. L. Duren, B. W. Romberg and A. L. Shields. Correspondingly, there are a lot of interesting results in this area (see for example [27], [3], and the following recent monographs [94] and [28]). Our aim is to introduce the concept of Bloch matrix, which extends the notion of Bloch function and to prove some results generalizing those of the earlier cited paper [3]. The basic idea behind our considerations is to consider an inﬁnite matrix A as the analogue of the formal Fourier series associated to a 2π-periodic distribution, the diagonals Ak , k ∈ Z, being the analogues of the Fourier coeﬃcients associated to the above distribution. In this manner we get a one-to-one correspondence between inﬁnite Toeplitz matrices and formal Fourier series associated to periodic distributions. Hence, an inﬁnite matrix appears in a natural way as a more general concept than those of a periodic distribution on the torus. Deﬁnition 7.1. The matriceal Bloch space 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 ) := sup (1 − r2 )||A (r)||B(2 ) + ||A0 ||B(2 ) < ∞, 0≤r<1

where B(2 ) is the usual operator norm of the matrix A on the sequence ∞ space 2 , and A (r) = k=0 Ak kr k−1 . 149

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A matrix A ∈ B(D, 2 ) is called a Bloch matrix. It is clear that the Toeplitz matrices, which belong to the Bloch space of analytic matrices B(D, 2 ) coincide with Bloch functions. Hence, B(D, 2 ) appears as an extension of the classical Bloch space of functions B. An important class of Bloch matrices consists of the space L∞ a (D, 2 ) as the following proposition shows: Proposition 7.2. The Banach space L∞ a (D, 2 ) is a subspace of B(D, 2 ), ||A||B(D,2 ) ≤ 6||A||L∞ a (D,2 ) and L∞ a (D, 2 )) B(D, 2 ). More precisely, the analytic Toeplitz matrix A given by the sequence 1 ∞ }∞ { k+1 k=0 is not in La (D, 2 ), but A ∈ B(D, 2 ). We note that (1 − r 2 )A (r) = C1 (r) ∗ A1 (r), where

(1 − r 2 )(j − k)r (j−k−1)/2 if j ≥ k + 1, C1 (r)(k, j) = 0 if j < k + 1, ∞ 1 (k−1)/2 . and A (r) = A0 + k=1 Ak r But ||C1 (r)||M (2 ) ≤ 2. Thus, Proof.

||(1 − r2 )A (r)||B(2 ) = ||C1 (r) ∗ A1 (r)||B(2 ) ≤ 2||A1 (r)||B(2 )

∀ 0 ≤ r < 1,

implying that ||A||B(D,2 ) ≤ 2 sup ||A1 (r)||B(2 ) + ||A0 ||B(2 ) ≤ 3 sup ||A1 (r)||B(2 ) 0≤r<1

0≤r<1

≤ 9||A||L∞ . a (D,2 ) L∞ a (D, 2 ),

then clearly A ∈ B(2 ). Moreover, if A ∈ But, taking h1 = h2 = · · · = hn = √1n and hn+1 = hn+2 = · · · = 0, it ∞ follows that i=1 |hi |2 = 1 and ||A||2B(2 ) ≥

1 1 2 1 1 1 2 1 (1 + 1 + + · · · + ) + (1 + 1 + + · · · + ) n 2 n−1 n 2 n−2

1 2 (1) ≥ C ln(n − 1) → ∞. n On the other hand A ∈ B(D, 2 ). Indeed, it is easy to see that +··· +

sup (1 − r2 )||A (r)||B(2 ) = sup r 2 = 1.

0≤r<1

The proof is complete.

0≤r<1

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The matrix version of Bloch space can be considered as the limit case of Lpa (D, 2 ) as p → ∞. First we note that if A ∈ L∞ (D, 2 ), then r → A(r) is a w∗ -measurable function and, consequently, each function aij (r) is a Lebesgue measurable function on [0, 1) for all i and j and we may introduce P A(·) as in Proposition 6.11. ∞ (D, ) → = Theorem 7.3. P : L∞ (D, 2 ) → B(D, 2 ) and P |L ∞ (D, ) : L 2 2 B(D, 2 ) are bounded surjection operators.

Proof. Clearly it is enough to prove only the ﬁrst assertion. Let A(·) ∈ L∞ (D, 2 ) and B = P A(·). We show that B ∈ B(D, 2 ). It yields that ||B (r)||2B(2 ) ≤ ⎡ ⎢ sup ⎣

||h||2 ≤1

1 0

⎛ ⎤ 2 ⎞ ∞ ∞ ⎟ ⎜ ⎥ aij (s)rj−i−1 sj−i (j − i + 1)(j − i)hj ⎠ (2sds)⎦ ⎝ i=1 j=i+1 ≤

1

0

where

C(r, s)(i, j) =

2

A(s) ∗ C(r, s)B(2 ) (2sds), (j − i + 1)(j − i)(rs)j−i−1 s if j > i, 0 if j ≤ i.

Thus,

||B (r)||B(2 ) ≤ C||A(·)||L∞ (D,2 ) ·

1

0

π −π

s dθ iθ 3 |1 − rse | π

∼ (by Lemma 1.3) ∼ C||A(·)||L∞ (D,2 ) ·

1/2

2 sds

1 . 1 − r2

Consequently, ||B||B(D,2 ) ≤ C||A(·)||L∞ (D,2 ) , that is, P : L∞ (D, 2 ) → B(D, 2 ) is a bounded operator. In order to show that P is onto we take B ∈ B(D, 2 ) and put B 1 = B − B0 − B1 . Easy calculations show us that P [(B 1 ) (r)(1 − r2 )] = B 1 ∗ T, 4|k|(|k|+1) . where T = (tj−i )i,j∈Z , with tk = (2|k|−1)(2|k|+1) Clearly T 1 = (t1j−i )i,j∈Z , where t1k = plier.

(2|k|−1)(2|k|+1) , 4|k|(|k|+1)

is a Schur multi-

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Hence, it follows that B 2 (r) := T 1 ∗ (B 1 (r)) (1 − r2 ) ∈ L∞ (D, 2 ) and P [B 2 (·)] = B 1 . We only need to prove that B0 + B1 r ∈ L∞ (D, 2 ). Since P [B0 + B1 r] = B0 + B1 , it suﬃces to show that r → B1 r ∈ L∞ (D, 2 ), (since B0 ∈ B(2 ) by the hypothesis B ∈ B(D, 2 )). Then, since ||B1 ||B(D,2 ) = sup (1 − r2 )||B1 ||B(2 ) = ||B1 ||B(2 ) , 0≤r<1

it follows that B1 ∈ B(2 ). Thus r → B1 r ∈ L∞ (D, 2 ). The proof is complete.

Remark 7.4. Note that B(D, 2 ) endowed with the norm || · ||B(D,2 ) is a Banach space and, by the open mapping theorem, it follows that B(D, 2 ) is isomorphic to the quotient space L∞ (D, 2 )/Ker P, endowed with quotient norm. Theorem 7.5. The projection P1 is a bounded operator from L∞ (D, 2 ) (D, 2 )) onto B(D, 2 ). (respectively from L∞ Proof. The proof is an easy adaptation of the proof of Theorem 6.12 and thus we leave out the details. By using Theorem 7.5 and Theorem 6.12 we easily get the following corollary: Corollary 7.6. Lpa (D, 2 ) = [L1a (D, 2 ), B(D, 2 )]θ with equivalent norms, for 1 < p < ∞ and 1 − θ = 1/p. Indeed we use the known result about the interchangeability of the interpolation functor and a bounded projection. (See [93].) Now we give some properties of these Bloch matrices, which extend the corresponding properties for Bloch functions. It is known that in [3] a characterization of Taylor coeﬃcients of Bloch functions in terms of a quadratic form is given. We want to extend this result to the case with inﬁnite matrices. We recall the deﬁnition of the space I from the paper [3]: 1 2π 1 I = {g : D → C | |g (z)|dθ dr + |g(0)| < ∞}, 2π 0 0 1 2π 1 |g (z)|dθ dr. Then the equipped with the norm gI := |g(0)| + 2π 0 0 following lemma holds:

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∞ ∞ Lemma 7.7. Let A ∈ B(D, 2 ); A = n=0 An z n and g(z) = n=0 bn z n ∈ ∞ I. Then h(z) = n=0 An bn z n : D → B(2 ) is a continuous function in |z| ≤ 1 and, moreover, h(z)B(2 ) ≤ 2AB(D,2 ) gI

(7.1)

for all z ≤ 1. In particular, it follows that there exists A, g = lim

∞

ρ→1−

An bn ρn = lim

ρ→1−

n=0

1 2π

2π

A(ρe−ıθ )g(ρeıθ )dθ,

0

for all f ∈ B(D, 2 ), g ∈ I. Let |ζ| < 1. We have that

Proof.

∞

A (z) = fA (z) =

nAn z n−1

n=1

and ∞ d (n + 1)bn z n . [z(g(z) − b0 )] = dz n=1

Then, we get easily that, for z = reıθ and ζ ∈ D, 1 π

1

0

2π 0

(1 − r2 )A (ζz)

∞ d [z(g(z) − b0 )] e−ıθ dθ dr = An bn ζ n−1 . dz n=1

By applying H¨older’s inequality and integrating term by term we ﬁnd that ∞ An bn ζ n B(2 ) n=1

≤ sup (1 − |z|2 )A (ζz)B(2 ) |z|<1

1 π

1

0

Furthermore, we have that 1 2π iθ |g(re ) − b0 |dθ dr ≤ 0

2π 0

0

1 0

1 t

2π

0

1

0

(|g(z) − b0 | + r|g (z)|)dθ dr.

2π

0

dr |g (teiθ )|dtdθ =

0

2π

r 0

1 0

|g (teiθ )|dtdθ dr =

(1 − t)|g (teiθ )|dtdθ.

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Since z → A (z)B(2 ) is a subharmonic function, we get that

∞

An bn ζ n B(2 ) ≤ A0 b0 B(2 )

n=0

+ sup (1 − |z|2 )A (z)B(2 ) z∈D

1 π

1

0

2π

0

|g (teiθ )|dθdt.

Hence, h(ζ)B(2 ) ≤ 2AB(D,2 ) gI , for |ζ| < 1. In order to show the continuity of h in |z| ≤ 1, we take ζ1 , ζ2 ∈ D and note that ∞ h(ζ1 ) − h(ζ2 )B(2 ) = An (bn ζ1n − bn ζ2n )B(2 ) ≤ n=0

2AB(D,2 ) g(ζ1 ) − g(ζ2 )I . But it is known that the last norm converges to 0 as |ζ1 − ζ2 | → 0. (See Theorem 2.2 [3].) Hence, h can be extended by continuity to D and we get (7.1). The proof is complete. Theorem 7.8. Let A = k Ak be a Bloch matrix. Then the following inequality holds: ||

∞ ∞ ∞ Aμ+ν+1 |wν |2 wμ wν ||B(2 ) ≤ K , μ+ν+1 2ν + 1 μ=0 ν=0 ν=0

(7.2)

where wν , ν = 0, 1, 2, . . . are complex numbers and K ≤ 2||A||B(D,2 ) . Conversely, (7.2) implies that A ∈ B(D, 2 ) and ||A||B(D,2 ) ≤ 2K. Proof. It is clear that the double series converges if the right hand series converges. Therefore we have that n ∞ ∞ ∞ Aμ+ν+1 An+1 wν wn−ν = A, g , (7.3) wμ wν = μ+ν+1 n + 1 ν=0 μ=0 ν=0 n=0 where ∞

1 g(z) = n + 1 n=0

n ν=0

wν wn−ν

z n+1 ,

z∈D

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and by A, g we mean that A, g = lim− ρ→1

∞

∞

An bn ρn = lim

ρ→1−

n=0

1 2π

2π

for A(ρe−ıθ ) = n=0 An ρn e−ınθ and g(ρeıθ ) = series are || · ||B(2 ) -convergent. Moreover, g (z) =

A(ρe−ıθ )g(ρeıθ )dθ,

0

∞

n=0 bn ρ

n ınθ

e

, where the

∞ n ∞ ( wν wn−ν )z n = ( wn z n )2 . n=0 ν=0

n=0

Hence, by the Parseval formula, we get that 1 2π 1 2π ∞ 1 1 |g (z)|dθ dr = | wn z n |2 dθ dr = gI = 2π 0 0 0 2π 0 n=0 1 ∞ 0 n=0

|wn |2 r2n dr =

∞ |wn |2 . 2n + 1 n=0

Therefore, by Lemma 6.7, we have that < A, g > B(2 ) ≤ 2AB(D,2 )

∞ |wn |2 , 2n + 1 n=0

that is (7.2) holds for A ∈ B(D, 2 ). b) Conversely, if (7.2) holds we take ζ ∈ D and ﬁnd wn such that g(z) ∞ ∞ n 1 z n = n=1 nζ n−1 z n = (1−ζz) 2 = n=0 n+1 ( ν=0 wν wn−ν )z . (See Theorem 3.5-[3].) Using the computations done in [3] at page 17 it follows that ∞ 2 |wn |2 = gI ≤ . 2n + 1 1 − |ζ|2 n=0

By (7.2) and (7.3) we get that

A (ζ)B(2 )

∞ z |wn |2 2K = < A(z), > ≤ K . ≤ B( ) 2 (1 − zζ)2 2n + 1 1 − |ζ|2 n=0

Therefore AB(D,2 ) ≤ 2K. The proof is complete.

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We can give an elementary class of Bloch matrices. (See also Proposition 1.5 in [14].) ∞ Theorem 7.9. Let A = k=0 A2k . Then A ∈ B(D, 2 ) if and only if supk ||Ak ||B(2 ) < ∞. Proof. By Theorem 7.8 it follows that there is a constant C > 0 such that C||A||B(D,2 ) ≥ supk ||Ak ||B(2 ) , for all inﬁnite matrices A. We consider a lacunary matrix A as in the statement of the theorem. Then $ $ ∞ ∞ $ $ ||zfA (z)||B(2 ) n $ k 2k $ = |z| A2k 2 z $ ≤ sup ||A2k ||B(2 ) · $ $ $ 1 − |z| k n=0

⎛

∞

⎝

n=1

⎞

2k ⎠ |z|n ≤ 2sup ||A2k ||B(2 ) k

2k ≤n

B(2 )

k=0

∞

n|z|n =

n=1

2|z| sup ||A2k ||B(2 ) . (1 − |z|)2 k

Consequently, (1 − r2 )||A (r)||B(2 ) ≤ 4sup||A2k ||B(2 ) , k

which, obviously, implies that ||A||B(D,2 ) ≤ 4sup||Ak ||B(2 ) .

k

It was remarked in [3] that the classical Bloch space of functions B is a Banach algebra with respect to convolution, or, equivalently, to Hadamard ikθ (Schur) composition of functions, that is, for f = ∞ ∈ B and g = k=0 ak e ∞ ∞ ikθ ikθ b e ∈ B, f ∗ g = a b e ∈ B. (See [3].) k k k k=0 k=0 Now we extend this remark in the framework of matrices with respect to the Schur product. This result was kindly communicated to us by V. Lie. Theorem 7.10. The space B(D, 2 ) is a commutative Banach algebra with respect to Schur product of matrices. Proof.

Let

⎛

a01 ⎜0 ⎜ A = ⎜0 ⎝ .. .

a11 a02 0 .. .

... a12 a03 .. .

⎞ ... ...⎟ ⎟ , ...⎟ ⎠ .. .

fj (reit ) =

∞ k=0

akj r k eikt ,

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where 0 ≤ r < 1, t ∈ [0, 2π) and ||A||B(D,2 ) = sup(1 − r 2 )||A (r)||B(2 ) . r<1

Then, denoting by fj the partial derivative of fj with respect to r, ||A||B(D,2 ) is given by: ||A||B(D,2 )

∞ = sup(1 − r ){ sup ( |

2π

2

||h||2 ≤1 j=1

r<1

0

fj (reit )eijt h(e−it )

dt 2 1/2 | ) }. 2π

(See [18].) Hence, (||A ∗ B||B(D,2 ) )2 = sup{ sup (1 − r2 )2 r<1 ||h||2 ≤1

∞ | j=1

2π 0

(fj ∗ gj ) (reit )eijt h(e−it )

dt 2 | }, 2π

where (fj )j corresponds to A as above and (gj )j corresponds to B. Then, we have that √r 2π dθ it r(fj ∗ gj ) (re ) = 2 fj (sei(θ+t) )gj (se−iθ )s ds, 2π 0 0 for all j. By the Cauchy-Schwarz inequality it follows that 2 ∞ 2π it ijt −it dt (fj ∗ gj ) (re )e h(e ) = 2π 0 j=1

2 √ 2π r 2π dθ s s 1 dt gj ( iθ ) ijθ ( fj (sei(θ+t) )eij(t+θ) h( it ) ) ds 0 e e e 2π 2π 0 0

∞ 1 4 r2 j=1

≤

∞

4r

−2

√ r

0

2π

0

r

0

j=1

√

s

2π

0

2π 0

dθ |gj (se−iθ )|2

2π

sds ×

2 dθ ds := I. 2π 2π

dt fj (sei(θ+t) )eij(t+θ) h(e−it )

Moreover, 2 2

sup sup(1 − s ) j≥1 s<1

2π 0

−it 2 dθ gj (se ) 2π

≤ (||B||B(D,2 ) )2

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and for ||h||2 = 1 we also have that 2 ∞ 2π it ijt −it dt f (se )e h(e ) ≤ (||A||B(D,2 ) )2 . (1 − s2 )2 j 2π j=1

0

Consequently, I ≤ 4r

−2

√ r 0

s(||B||B(D,2 ) )2 (1 − s2 )2

√

ds 0

r

s(||A||B(D,2 ) )2 (1 − s2 )2

ds =

(1 − r)−2 (||A||B(D,2 ) )2 (||B||B(D,2 ) )2 , that is ||A ∗ B||B(D,2 ) ≤ C||A||B(D,2 ) ||B||B(D,2 ) .

The proof is complete.

Theorem 7.11. The Banach space L1a (D, 2 )∗ (dual of L1a (D, 2 )) may be identiﬁed with B(D, 2 ). Namely, let A ∈ L1a (D, 2 ) and B ∈ B(D, 2 ). Then we have that 1 tr [A(r)B ∗ (r)](2rdr)| ≤ C||A||L1a (D,2 ) · ||B||B(D,2 ) , | < A, B > | = | 0

where C > 0 is a constant. Proof. Since C1 , the Schatten trace class of operators, is a separable Banach space with C1∗ = B(2 ), by < A, B > = tr(AB ∗ ), we have, in view Theorem 1.2, that L1 (D, 2 )∗ = L∞ (D, 2 ), using the duality map 1 < A(r), B(r) > = 0 tr [A(r)B ∗ (r)](2rdr). ˜ 1 (D, 2 )∗ = L∞ (D, 2 )/(L ˜ 1 (D, 2 ))⊥ . Then we have that L a a 1 ˜ 1 (D, 2 ), we Using the fact that La (D, 2 ) is canonically isomorphic to L a have to show that ˜ 1 (D, 2 ))⊥ Ker P = Ker P˜ = (L a

in L∞ (D, 2 ).

˜ 1 (D, 2 ))⊥ , since for A(r) ∈ L∞ (D, 2 ) such that P˜ A(·) But Ker P˜ ⊂ (L a = 0, we have, at least for ﬁnitely order matrices A(·), B(·), that < P˜ A(·), B(·) > = < A(·), P˜ B(·) >, ˜ 1 (D, 2 )), then and if B ∈ L a < A(·), B(·) > = < A(·) − P˜ A(·), B(·) > = < A(·), B(·) − P˜ B(·) > = 0, ˜ 1a (D, 2 ))⊥ . and, consequently, A(·) ∈ (L

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Conversely, let A(·) ∈ (L1a (D, 2 ))⊥ , that is < A(·), B(·) > = 0 ∀B ∈ Taking B(r)(i, j) = rj−i for j > i, with ﬁxed j, i and 1 B(r)(i, j) = 0 otherwise, we get that 0 aij (r)(2rdr) = 0 for all j > i. Thus (P˜ A)(r)(i, j) = 0 for all i, j, that is A(·) ∈ Ker P˜ . L1a (D, 2 )).

For B(r) ∈ L∞ (D, 2 ) and A ∈ L1a (D, 2 ), we easily get that 1 1 ∗ | tr [A(r)B (r)](2rdr)| ≤ |tr [A(r)B ∗ (r)]|(2rdr) ≤ 0

1 0

0

||A(r)||C1 · ||B(r)||B(2 ) 2rdr ≤ ||A||L1a (D,2 ) · ||B(·)||L∞ (D,2 ) ,

so using Remark 7.4 we get the required inequality, since for A ∈ L1a (D, 2 ), B ∈ B(D, 2 ) we have obviously that | < A, B > | = | < A(r), B(r) > | ≤ ||A||L1a (D,2 ) ||B(·)||L∞ (D,2 ) ∀ B(·) ∈ L∞ (D, 2 ) such that P B(·) = B. The proof is complete. Lemma 7.12. Let A be a matrix of ﬁnite band-type, that is A = such that Ak ∈ C1 for k = 1, 2, . . . and let B ∈ B(D, 2 ). Then < A, B > =

∞

n k=1

Ak ,

1 tr (Ak ∗ B k ). k+1

k=0

1 Proof. We recall that < A, B > = 0 tr [A(r)B ∗ (r)]2rdr. We denote the entries of the diagonal matrix Ak , where k ∈ Z, by (alk )∞ l=1 . Then it is easy to see that ∞ n alk blk r 2k tr A(r)B ∗ (r) = l=1

k=0

and, consequently,

n 1

< A, B > = 0

=

l=1

∞ k=0

∞

2rdr =

n l=1

alk blk

∞ 1

0 k=0

k=0

1 k+1

The proof is complete.

alk blk r 2k

=

∞ k=0

2r

2k+1

n

alk blk

dr

l=1

1 tr (Ak ∗ B k ). k+1

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Next we present a result, which shows us the intrinsic connection between the matricial Bloch space and the Bergman metric. (See Thm. 5.1.6 [94] for the classical Toeplitz (function) case.) Theorem 7.13. If A ∈ B(D, 2 ), then sup (1 − r2 )||A (r)||B(2 ) =

0≤r<1

sup

z,w∈D z =w

||fA (z) − fA (w)||B(2 ) . β(z, w)

Proof. The proof is simply a translation of the proof of Theorem 5.1.6 [94], replacing f by fA . We omit the details. In particular, we conclude that, in fact, the Bloch matrices are those upper triangular matrices such that B ∗ C(z) are Lipschitz B(2 )-valued functions with respect to Bergman metric on D. We can use this result for proving that B(D, 2 ) is a Banach space. We note that by Theorem 7.11 this is clear, but the next proof is much simpler. Theorem 7.13 has as an immediate consequence the following inequality: ||fA (z)||B(2 ) ≤ ||A||B(D,2 ) log

1 + |z| 1 − |z|

for all |z| ≥ 12 , which implies that the point evaluation of a matrix A, that is the linear operator Δz : B(D, 2 ) → B(2 ), given by Δz (A) = fA (z), is bounded for all z ∈ D. This, in its turn, implies that if a sequence An of matrices converges in the Bloch norm, then Δz (An ) does so locally uniformly with respect to z. In particular, we can use the inequality above to prove the completeness of B(D, 2 ). Proposition 7.14. The space of Bloch matrices B(D, 2 ) is a Banach space. Proof. Denoting the dilations of fA by (fA )r (z) = fA (rz) = fA∗C(r) (z), we have that ||A ∗ C(r)||B(D,2 ) = sup (1 − |z|2 )||(fA )r (z)||B(2 ) + ||(fA )r (0)||B(2 ) = z∈D

r sup (1 − |z|2 )||(fA ) (rz)||B(2 ) + ||fA (0)||B(2 ) , z∈D

which increases to ||A||B(D,2 ) as r increases to 1. Using this fact, we can now show that B(D, 2 ) is complete.

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Let {An } be a Cauchy sequence in B(D, 2 ). By (∗), this implies that {fAn } is a uniform Cauchy sequence on each compact subset of D, and, hence, it converges locally uniformly to some vector-valued analytic function fA . It remains to show that ||An − A||B(D,2 ) → 0. Given > 0, choose N such that ||An − Am ||B(D,2 ) < when n, m ≥ N. Then, for r < 1, ||An ∗ C(r) − A ∗ C(r)||B(D,2 ) ≤ ||An ∗ C(r) − Am ∗ C(r)||B(D,2 ) + ||Am ∗ C(r) − A ∗ C(r)||B(D,2 ) ≤ ||An − Am ||B(D,2 ) + ||Am ∗ C(r) − A ∗ C(r)||B(D,2 ) < + ||Am ∗ C(r) − A ∗ C(r)||B(D,2 ) . Observe that the last term approaches 0 as m → ∞, since (Am ∗ C(r)) converges to (A ∗ C(r)) uniformly on D. Thus ||An ∗ C(r) − A ∗ C(r)||B(D,2 ) ≤ 2 for n ≥ N and all r < 1. Finally, we let r → 1 to arrive at the desired conclusion. The proof is complete. 7.2

Matrix version of little Bloch space

Now we introduce another space of matrices, the so-called little Bloch space of matrices. Deﬁnition 7.15. The space B0 (D, 2 ) is the space of all upper triangular inﬁnite 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 space is endowed with the norm of B(D, 2 ). Let A ∈ B(D, 2 ) and Ar (s) = A(rs) = A(r) ∗ P (s) for all 0 ≤ r < 1 and 0 ≤ s < 1, where P (s) is the Toeplitz matrix associated to the Poisson kernel, that is ⎞ ⎛ 1 s s2 s 3 . . . ⎜ . ⎟ ⎜ s 1 s s2 . . ⎟ ⎟ ⎜ ⎜ 2 .. ⎟ ⎟. . P (s) = ⎜ s 1 s s ⎟ ⎜ ⎟ ⎜ . ⎜ s3 s2 s 1 . . ⎟ ⎠ ⎝ .. . . . . . . . . . . . . .

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Then it follows that Ar is a matrix belonging to B0 (D, 2 ), for all 0 ≤ r < 1, since π (1 − s2 ) dθ 2 lim (1 − s )||Ar (s)||B(2 ) ≤ lim ||A||B(D,2 ) · r iθ s→1 s→1 1 − r 2 s2 −π |1 − se | ∼ ||A||B(D,2 ) ·

r 1 lim (1 − s2 ) log = 0. 1 − r 2 s→1 1 − s2

Theorem 7.16. Let A ∈ B(D, 2 ). Then A ∈ B0 (D, 2 ) if and only if lim− ||Ar − A||B(D,2 ) = 0.

r→1

Proof. By the remark above it follows that Ar ∈ B0 (D, 2 ) and we use the obvious fact that B0 (D, 2 ) is a closed subspace of B(D, 2 ) in order to conclude that the condition is suﬃcient. Conversely, let A ∈ B0 (D, 2 ). Then ∀ > 0 there is 0 < δ < 1 such that (1 − s2 )||A (s)||B(2 ) < ∀ δ 2 < s < 1. We note that ||Ar − A||B(D,2 ) = sup (1 − s2 )||Ar (s) − A (s)||B(2 ) 0≤s<1

≤ sup (1 − s2 )||Ar (s) − A (s)||B(2 ) δ<s<1

+ sup (1 − s2 )||Ar (s) − A (s)||B(2 ) . 0≤s≤δ

For δ < r < 1, the ﬁrst term is smaller than (1 − r2 s2 )||A (rs)||B(2 ) + (1 − s2 )||A (s)||B(2 ) < 2. The second term converges to 0 whenever r → 1− . Indeed, for 0 ≤ s ≤ δ < δ < 1, letting u = δs , we get that ||Ar (s) − A (s)||B(2 ) = ||A (s) ∗

∞

(rk − 1)Ek ||B(2 )

k=0

≤ ||A (u)||B(2 ) · ||

∞

(r k − 1)(δ )k−1 Ek ||M (2 )

k=0

= ||A (u)||B(2 ) · ||

∞

(rk − 1)δ k−1 eikθ ||M (T)

k=0

≤ ||A (u)||B(2 ) ·

π

−π

|1 − δ eiθ − 1 + rδ eiθ | dθ |1 − rδ eiθ | · |1 − δ eiθ | 2π

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≤ ||A (

s (1 − r)δ )||B(2 ) · . δ (1 − rδ )(1 − δ )

We recall that E is the Toeplitz matrix having all its entries equal to 1. Therefore, sup (1 − s2 )||A (rs) − A (s)||B(2 ) ≤

0≤s≤δ

sup

0≤s≤δ<δ

s 1 − s2 s (1 − ( )2 )||A ( )||B(2 ) · 2 δ δ 1 − δs2 ||A||B(D,2 ) ·

·

δ (1 − r) ≤ (1 − δ )(1 − rδ )

1 − δ2 δ (1 − r) , · 2 1 − δδ 2 (1 − δ )(1 − rδ )

for all t ≥ 0. Consequently lim sup(1 − s2 )||Ar (s) − A (s)||B(2 ) = 0.

r→1− s≤δ

The proof is complete.

Corollary 7.17. B0 (D, 2 ) is the closure of all matrices of ﬁnite band type in the Bloch norm. In particular, this implies that B0 (D, 2 ) is a separable space. n Proof. Let A ∈ B0 (D, 2 ) and An = k=0 Ak . . Then, by Theorem 7.16, we have that ∀ > 0 there is a r0 < 1 such that ||Ar − A||B(D,2 ) < /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 sn s0 ||A(sn ) − A(s0 )||B(2 ) = ||[C( ) − C( )] ∗ A(s )||B(2 ) s s sn s0 ) − C( )||M (2 ) , s s where sn → s0 < s < 1. Hence, by putting δ = ss0 < 1 and reasoning as in the proof of Theorem 7.16, we get that ≤ ||A(s )||B(2 ) · ||C(

||A(sn ) − A(s0 )||B(2 ) ≤ ||A(s )||B(2 ) · ||

∞

δ k ((

k=1

sn k ) − 1)eikθ ||M (T) → 0. s0

Moreover, for a ﬁxed r0 < 1 we have that sup ||Ar0 (s)||B(2 ) := M (r0 ) = ||A(r0 )||B(2 ) < ∞

0≤s≤1

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for all analytic matrices. Thus, for r0 < r < 1 and by using the notation n r r 0 0 Ck , Cn = r r k=0

we ﬁnd that ||Ar0 (·) − (Ar0 )n (·)||L∞ (D,2 ) = ess sup ||(A − An )(r0 s)||B(2 ) s<1

= ||(A − An )(r0 )||B(2 ) = ||[C(

r0 r0 r0 r0 ) − C n ( )] ∗ A(r )||B(2 ) ≤ ||C( ) − C n ( )||M (2 ) · ||A(r )||B(2 ) r r r r = ||

∞ r0 ( )k eikθ ||M (T) · ||A(r )||B(2 ) → 0. r

k+1

Consequently, ||Ar0 (·) − (Ar0 )n (·)||L∞ (D,2 ) → 0 and ||A(·) − (Ar0 )n (·)||B(D,2 ) ≤ , whenever r0 < 1 is ﬁxed as before and n is suﬃciently large. The proof is complete. The next theorem expresses a natural relation between the Bergman projection and the Bloch spaces. Theorem 7.18. Let A ∈ B(D, 2 ). The following assertions are equivalent: 1) A ∈ B0 (D, 2 ). 2) There is a continuous function r → B(r), deﬁned on [0, 1] and B(2 )valued, such that P (B(·))(r) = A(r). 3) There is a function r → B(r), which is a continuous B(2 )-valued function such that limr→1 B(r) = 0 and satisfying that P (B(·))(r) = A(r). Proof. To prove that 1) implies 3), we take A ∈ B0 (D, 2 ). We deﬁne ∞ ∞ A1 (r) = k=2 Ak r k , with r < 1. Thus A1 (r) = k=2 kAk r k−1 and A(r) = A0 + A1 r + A1 (r). We take now B2 (r) = (1 − r2 )T ∗ P (r) ∗ A1 (r), where P (r) is the Toeplitz matrix associated to the Poisson kernel and T = (tij )i,j is a Schur multiplier, which will be deﬁned later on.

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Thus, by the deﬁnition of Bergman projection P, we get that [P B2 (·)](r)(i, j) , =

tij aij (j − i + 1)(j − i)rj−i

1 3(j−i)+1 [ 32 (j−i+1)] 2

0

Consequently, by taking , tij =

3[3(j−i)+1] 4(j−i) 9 4

for j − i ≥ 2, if j − i < 2.

for all j = i, i, j ≥ 1 if j = i, i ≥ 1,

it follows that T is a Schur multiplier, and P [B2 (·)] = A1 (·). Let now B(r) = 2(1 − r2 )A0 + 3(1 − r 2 )rA1 + B2 (r). It is clear that [P B(·)](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, and limr→1 B(r) = 0. Thus 3) holds and we have proved that 1) implies 3). It is obvious that 3) implies 2). It remains to prove that 2) implies 1). Let 2) hold and choose B(r) ∈ B(2 ) such that [P B(·)](r) = A(r) for r ∈ [0, 1]. Assume that r → B(r) is a continuous B(2 )-function on [0, 1] and let M = sup0≤r≤1 ||B(r)||B(2 ) < ∞. Let 0 ≤ r0 < 1 be ﬁxed and consider Ar0 (r) given by the formula , 1 (j − i + 1)(rr0 )j−i (2 0 bij (s)sj−i+1 ds) if j − i ≥ 0, Ar0 (r)(i, j) = 0 otherwise. Consequently, according to Theorem 7.3, we ﬁnd that Ar0 = P [P (r0 ) ∗ B(·)] = P [Br0 (·)] ∈ B(D, 2 ), where P (r0 ) is the Toeplitz matrix associated to the Poisson kernel. Let C(D, 2 ) denote the space of all continuous B(D, 2 )-valued functions deﬁned on [0, 1]. Next we prove that the function s → P (P (r0 ) ∗ B(s)) belongs to C(D, 2 ) if B is a continuous B(2 )-valued function such that lim sup ||Ar (s) − A(s)||B(2 ) = 0.

r→1 s∈[0,1]

(7.4)

This, in its turn, implies that limr→1 Ar = A in B(D, 2 ). Therefore, by the Theorem 7.16, it follows that A ∈ B0 (D, 2 ).

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Let s, s0 ∈ [0, 1]. Then ||P (r0 )∗[B(s)−B(s0 )]||B(2 ) ≤ ||

|k|

r0 eikθ ||M (T) ·||B(s)−B(s0 )||B(2 ) → 0

k∈Z

for s → s0 and B(s) is a continuous function on [0, 1]. Here we have used Bennett’s Theorem and the fact that 1 − r2 r |k| eikθ ||M (T) = || ||M (T) ≤ 1. || |1 − reiθ |2 k∈Z

Thus, the function s → P [P (r0 ) ∗ B(s)] belongs to C(D, 2 ). Hence, it only remains to prove that (7.4) holds. In fact, |k| ||P (r0 ) ∗ B(s) − B(s)||B(2 ) ≤ || (r0 − 1)eikθ ||M (T) · ||B(s)||B(2 ) k∈Z

≤ M · ||

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

k∈Z

Denoting by μr (θ) the measure k∈Z (r|k| − 1)eikθ , then , for a trigonom metric polynomial φ(θ) = n=−m an einθ , we have that μr (φ) =

m

(r|n| − 1)an

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

n=−m

where φ(r) is the value of the Poisson extension of φ in the point r. Consequently μr is a measure with the norm less than 2. But limr→1 μr (φ) = 0 for all trigonometric polynomes φ. Thus, w∗ - limr→1 μr = 0 in M (T) and then it is clear that limr→1 ||μr || = 0 and, according to Theorem 7.3, the relation (7.4) is proved. Thus, also the implication 2) ⇒ 1) is proved and the proof is complete. Now, using the notation which preceeds Theorem 6.12, we get: Theorem 7.19. P2 is a continuous operator (precisely a continuous projection) from L1 (D, 2 ) onto L1a (D, 2 ). Proof. By Theorem 1.2 the topological dual of L1 (D, 2 )) is L∞ (D, 2 ) with respect to the duality pair: 1 tr (A(s)[B(s)]∗ )2sds, < A(·), B(·) > := 0

where A(·) ∈ L∞ (D, 2 ), B(·) ∈ L1 (D, 2 ). By using a duality argument it is suﬃcient to prove that P2∗ : L∞ (D, 2 ) → L∞ (D, 2 ) is bounded.

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We are looking for the adjoint P2∗ of P2 : 1 ∞ ∞ ∗ (P2∗ A(·))(r)(i, j)bij (r)rdr < P2 A(·), B(·) > = 2 0

=

∞ ∞

1 0

i=1 j=1

i=1 j=1

(P2∗ A(·))(r)(i, j)bij (r)(2r)dr.

On the other hand it yields that < P2∗ A(·), B(·) > = < A(·), P2 B(·) > =

=

∞ ∞ 0

i=1 j=1

=

0

1

tr A(r)(P2 B)∗ (r)(2rdr)

A(r)(i, j)(P2 B)(r)(i, j)(2rdr)

(j − i)!Γ(3)

0

1

1

∞ ∞ Γ(j − i + 4) i=1 j=i

1 0

[A(s)](i, j)sj−i (2sds) ×

bij (s)sj−i (1 − s2 )2 (2sds) .

Now we consider {Ik }, a sequence of intervals such that lim μ(Ik ) = 0, dμ = 2sds and r ∈ Ik .

k→∞

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 we have that , 1 Γ(j−i+4) j−i r (1 − r2 )2 0 A(s)(i, j)sj−i (2sds) if j ≥ i, ∗ (j−i)!Γ(3) (P2 A(·))(r)(i, j) = 0 if j < i, a.e. for all r ∈ [0, 1). We will now prove that P2∗ : L∞ (D, 2 ) → L∞ (D, 2 ) is a bounded operator. In order to prove that we ﬁrst note that ||A(r)||2L∞ (D,2 ) = ess sup ||A(r)||2B(2 ) 0≤r<1

= ess sup

0≤r<1

∞

sup

j=1

∞ ∞ | aij (r)hj |2 .

|hj |2 ≤1 i=1

j=1

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Consequently, because L1 [0, 1] has cotype 2, there is a constant K > 0, such that ∞ ∞ |

|| sup

||h||l2 ≤1 i=1

j=1 2 4

|| sup (1−r ) ||h||l2 ≤1

||P2∗ A(·)||2L∞ (D,2 ) = 1 0

∞

hj rj−i

|

i=1

1

( 0

Γ(j − i + 4) (1 − r2 )2 aij (s)sj−i (2sds)|2 ||L∞ = (j − i)!Γ(3)

∞

aij (s)((rs)j−i

j=i

Γ(j − i + 4) )hj )(2sds)|2 ||L∞ ≤ (j − i)!Γ(3)

∞ ∞ (j − i + 3)! 2 1/2 | aij (s)[(rs)j−i hj | ) (sds)]2 ||. (j − i)! ||h||l2 ≤1 0 i=1 j=i Since the Toeplitz matrix C(rs) = (cij )(rs)∞ i,j=1 , where

(rs)j−i (j − i + 3)(j − i + 2)(j − i + 1) if j ≥ i cij (rs) := cj−i (rs) = 0 if j < i, is a Schur multiplier with ∞ 6 ||L1 (T) = 6 (n + 1)2 (rs)2n , ||C(rs)||L1 (T) = || (1 − rseiθ )4 n=0

K||(1 − r2 )4

1

sup [

(

we get that

∞

sup

j=1

|hj |2 ≤1

⎛ ⎞1/2 ∞ ∞ ⎝ | aij (s)(rs)j−i (j − i + 3)(j − i + 2)(j − i + 1)hj |2 ⎠ i=1

j=i

= ||A(s) ∗ C(rs)||B(2 ) ≤ 6||A(s)||B(2 ) ·

∞

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

n=0

Hence,

≤ Kess sup(1 − r2 )4 r<1

||P2∗ A(·)||2L∞ (D,2 )

1 0

3||A(s)||B(2 ) ·

∞

2 (n + 1)2 r2n s2n (2sds)

n=0

2 4

≤ 9K · ess sup(1 − r ) r<1

||A(·)||2L∞ (D,2 ))

∞

2 (n + 1)r

2n

∼

n=0

1 ||A(·)||2L∞ (D,2 ) ∼ ||A(·)||2L∞ (D,2 ) , (1 − r2 )4 which shows in turn that P2∗ : L∞ (D, 2 ) → L∞ (D, 2 ) is bounded. The proof is complete. ess sup(1 − r2 )4 r<1

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We 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 norm of B(2 ). Lemma 7.20. Let V = (P2 )∗ , that is (P2 )∗ (A(r))(i, j) = ,

(j−i+3)(j−i+2)(j−i+1) j−i r (1 2

0

− r2 )2

1 0

aij (s)sj−i (2sds) if j − i ≥ 0, otherwise.

Then V is an isomorphic embedding of B0 (D, 2 ) in C0 (D, 2 ). Proof. According to Theorem 7.18, for B ∈ B0 (D, 2 ) we can ﬁnd some A(·) ∈ C0 (D, 2 ) such that [P A(·)](r) = B(r). Clearly, we have that P2∗ B = P2∗ P A = T 1 ∗ (P2∗ A1 ), where A1 (r) = T ∗ A(r), for T = (tj−i )i,j with , 2(j−i+1) for j − i = −2, j−i+2 tj−i = 0 otherwise. T is a Schur multiplier and the same is true for T 1 = (t1j−i )i,j , where , j−i+2 for j − i = −1, 1 tj−i = 2(j−i+1) 0 otherwise. Thus ||A1 (r)||B(2 ) ∼ ||A(r)||B(2 ) for all r ∈ [0, 1]. Hence, we obtain that ||P2∗ A1 (r)||2B(2 ) =

K

1 ∞ ∞ Γ(j − i + 4) (1 − r2 )2 | hj r j−i a1ij (s)sj−i (2sds)|2 ≤ (j − i)!Γ(3) ||h||2 ≤1 i=1 j=i 0 sup

2 4

sup (1 − r ) [

||h||2 ≤1

1

( 0

∞ ∞ Γ(j − i + 4) 2 1/2 hj | ) (2sds)]2 . | a1ij (s)(rs)j−i 2(j − i)! j=1

j=i

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

(rs)j−i (j − i + 3)(j − i + 1)2 if j ≥ i cj−i (r, s) = 0 otherwise, is a Schur multiplier, since ∞ (rs)k (k + 3)(k + 1)2 eikθ ∼ k=0

1 (1 − rseiθ )4

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and

π −π

∞ 1 dθ ∼ n2 (rs)2n . |1 − rseiθ |4 2π n=0

Therefore, we have that ||P2∗ A1 (r)||B(2 ) ≤ C(1 − r2 )2

1 0

||A(s)||B(2 ) · (

∞

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

n=0

Since lims→1 ||A(s)||B(2 ) = 0, for all > 0, there is δ > 0 such that ||A(s)|| < for all 1 ≥ s ≥ δ and, consequently, P2∗ A1 (r)B(2 ) ≤ C( + A(·)C(D,2 ) ·

(1 − r2 )2 δ 2 ). (1 − r2 δ 2 )2

It follows that limr→1 P2∗ A1 (r)B(2 ) = 0, and, since T 1 is a Schur multiplier, it follows that P2∗ B ∈ C0 (D, 2 ) and ||P2∗ B||B(2 ) ≤ C||A(·)||C(D,2 ) . Moreover, in view of the proof of Theorem 7.18, 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 arguments in the proof of Theorem 7.19 it follows that P2∗ : B0 (D, 2 ) → C0 (D, 2 ) is bounded. On the other hand, if A ∈ B0 (D, 2 ), then, since A(r) is an analytic matrix, it is obvious that A(r) = [P A(·)](r) = (P [P2∗ A(·)])(r) for all r ∈ [0, 1). Thus, by using Theorem 7.19, we conclude that there exists a constant C > 0 such that ||A(·)||B(D,2 ) ≤ C||P2∗ A(·)||C(D,2 ) , which implies that P2∗ : B0 (D, 2 ) → C0 (D, 2 )) is an isomorphic embedding. The proof is complete. 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 introduce B0, c (D, 2 ) as the closed Banach subspace of B0 (D, 2 ) consisting of all upper triangular matrices whose diagonals are compact operators. We are now ready to prove that this space B0,c (D, 2 ) is in fact the predual of the Bergman-Schatten space. More exactly, our last main result in this Section is the following duality result: Theorem 7.21. It yields that B0, c (D, 2 )∗ = L1a (D, 2 ) with respect to the usual duality, whenever B0 (D, 2 ) has the norm induced by B(D, 2 ).

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1 Proof. Let A ∈ L1a (D, 2 ). Then B → 0 tr[B(s)A∗ (s)](2sds) deﬁnes a linear and bounded functional on B0, c (D, 2 ) due to Theorem 7.11. Conversely, let us assume that F is a bounded linear functional on B0, c (D, 2 ). Then we shall show that there is a matrix C from L1a (D, 2 ) such that 1 tr[B(r)C ∗ (r)](2rdr), (7.5) F (B) = 0

for B from a dense subset of B0 (D, 2 ). In what follows we identify the matrices A ∈ B0 (D, 2 ) with the function r → A ∗ C(r). In particular, we identify nk=0 r k Ak with nk=0 Ak . By Lemma 7.20 it follows that P2∗ : B0 (D, 2 ) → C0 (D, 2 ), is an isomorphic embedding. Thus X = P2∗ (B0, c (D, 2 )) is a closed subspace in C0 (D, C∞ ) and F ◦ (P2∗ )−1 : X → C, is a bounded linear functional on X, where C0 (D, C∞ ) is the subset in C0 (D, 2 ) whose elements are C∞ -valued functions. By Hahn-Banach theorem F ◦ (P2∗ )−1 can be extended to a bounded linear functional on C0 (D, C∞ ). Let Φ : C0 (D, C∞ ) → C denote this functional. It follows that ˆ C∞ , Thus, Φ is a bilinear integral map, i.e., there C0 (D, C∞ ) = C0 [0, 1]⊗ 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) = f (r)tr(AB ∗ )dμ(r, B) ∀f ∈ C0 [0, 1] and A ∈ C∞ . [0,1]×UC1

n Thus, for the matrix k=0 Ak ∈ B0, c (D, 2 ), identiﬁed with the analytic n matrix k=0 Ak rk , we have that F(

n

Ak ) = F (

k=0

n

r k Ak ) = [F ◦ (P2∗ )−1 ][P2∗ (

k=0

n

r k Ak )]

k=0

n (k + 3)(k + 2) k r (1 − r2 )2 Ak ) = Φ( 2 k=0

=

n

tr[(

[0,1]×UC1 k=0

:= < μ(r, B), tr(

(k + 3)(k + 2) k r (1 − r2 )2 Ak )B ∗ ]dμ(r, B) 2

n (k + 3)(k + 2) k=0

2

rk Ak )B ∗ (1 − r2 )2 > .

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On the other hand, we wish to prove that 1 n n k ∗ Ak ) = tr( s Ak )(C(s) )(2sds) = F( 0

k=0

k=0 n

=

k=0

tr Ak (

1

tr( 0

n

s2k Ak Ck∗ )(2sds)

k=0

Ck∗ ). k+1

Now, letting A = ei,i+k , where ei,i+k is the matrix having 1 as the single nonzero entry on the ith-row and the (i + k)th-column, for i ≥ 1 and j ≥ 0, we ﬁnd that (k + 1)(k + 2)(k + 3) k Ck = < μ(r, B), r (1 − r2 )2 Bk >, k = 0, 1, 2, . . . . 2 Therefore, denoting [0, 1] × UC1 by V, we have 1 ||C(s)||C1 (2sds) 0

1

= 0

k=0

[

V

0

1

1

0

V

[

n (k + 3)! (sr)k (1 − r2 )2 Bk dμ(r, B)||C1 (2sds) ≤ || 2 k! V

||

n (k + 3)!

||

2 k!

k=0

n (k + 3)!

2 k!

k=0

[ ∼

0

V

= V

= V

0

1

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

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

1

0

2π

(1 − r2 )2 dθ (2sds)]d|μ|(r, B) |1 − rseiθ |4 2π

(1 − r2 )2

(1 − r2 )2

∞

(n + 1)2 (sr)2n (2sds)d|μ|(r, B)

k=0 ∞

(n + 1)r2n d|μ|(r, B) ≤ ||μ|| < ∞.

n=0

Consequently, C ∈ L1a (D, 2 ) and we get the relation (7.5), using the n fact that the set of all matrices k=0 Ak is dense in B0, c (D, 2 ). The proof is complete.

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Notes The idea behind our study of matrix versions of diﬀerent kinds of function spaces is to consider the diagonals of an upper triangular inﬁnite matrix as an analogue of Fourier coeﬃcients of an analytic function or distribution. Using this idea we consider a Bloch space of matrices (see [72]) and prove results similar to those which can be found in the well-known paper [3]. We mention the remarkable fact that this Bloch space is a commutative Banach algebra under the Schur product and that it is the topological dual of the Bergman-Schatten space. Section 7.2 is dedicated to introduce and study the little Bloch space of matrices. In particular, it is proved that the dual space of the subspace of the little Bloch space consisting of compact matrices is the BergmanSchatten space. (See Theorem 7.21.)

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Chapter 8

Schur multipliers on analytic matrix spaces

An interesting topic in matriceal harmonic analysis is the study of Schur multipliers on diﬀerent classes of Banach spaces of inﬁnite matrices. First we describe the Schur multipliers from B(2 ) into B(D, 2 ). Theorem 8.1. An upper triangular matrix A belongs to (B(2 ), B(D, 2 )) k−1 if and only if sup0≤r<1 (1 − r)|| ∞ ||M (2 ) < ∞, or, equivalently, k=1 kAk r ∞ if and only if the M (2 )-valued function fA (z) = k=0 Ak z k is a Lipschitz function with respect to the Bergman metric β(z, w). Proof. Let B ∈ B(2 ) and A ∈ (B(2 ), B(D, 2 )) with ||A||(B(2 ),B(D,2 )) = C. Then ∞ sup (1 − r)|| kAk ∗ Bk r k−1 ||B(2 ) ≤C · ||B||B(2 ) 0≤r<1

k=1

and it follows that sup (1 − r)||

0≤r<1

∞

kAk r k−1 ||M (2 ) ≤ C.

k=1

Conversely, if sup (1 − r)||

0≤r<1

∞

kAk r k−1 ||M (2 ) = C < ∞,

k=1

then, for an arbitrary B ∈ B(2 ), we get obviously that sup (1 − r)||

0≤r<1

∞

kAk ∗ Bk r k−1 ||B(2 ) ≤ C · ||B||B(2 ) ,

k=1

and, consequently, A ∈ (B(2 ), B(D, 2 )) and ||A||(B(2 ),B(D,2 )) ≤ C. 175

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On the other hand, it is clear that the proof of Theorem 5.1.6 in [94] holds also if we consider M (2 )-analytic functions instead of usual analytic ones. Thus, A ∈ (B(2 ), B(D, 2 )) if and only if the function fA (z) is a Lipschitz function with respect to the Bergman metric. The proof is complete. Remark. The previous theorem may be regarded as a matrix extension of Theorem 4.3 of Jevtic and Pavlovic [47], namely: Theorem of Jevtic and Pavlovic. (H ∞ , B) = H11,∞,1 . Next we consider the possibility to ﬁnd the space of all Schur multipliers from B(D, 2 ) into itself and expect that the following statement holds: Theorem 8.2. (B(D, 2 ), B(D, 2 )) = H11,∞,1 (2 ) = {A upper triangular matrices such that sup0
(1 − r)|g(reiθ )| < ∞.

Theorem of Shields and Williams: (H ∞,∞,1 , H ∞,∞,1 ) = H11,∞,1 . (See also Lemma 3.3 in [47].) The matrix version of this last statement is the following: Lemma 8.3. It yields that H11,∞,1 (2 ) ⊂ (H ∞,∞,1 (2 ), H ∞,∞,1 (2 )), where H ∞,∞,1 (2 ) = {A upper triangular matrices such that sup (1 − r)||A(r)||B(2 ) < ∞}.

0
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Schur multipliers on analytic matrix spaces

Let A be an upper triangular matrix such that

Proof.

sup(1 − r)||A (r)||M (2 ) < ∞ r<1

and B such that sup(1 − r)||B(r)||B(2 ) < ∞. r<1

We have to show that sup (1 − r)||(A ∗ B)(r)||B(2 ) < ∞.

0
We note that sup (1 − r)||(A ∗ B)(r)||B(2 ) ≤ C if ||(A ∗ B)(r)||B(2 ) ≤

0
C , (1 − r)2

for all 0 < r < 1. Indeed, using Minkowski’s inequality, we ﬁnd that ||(A ∗ B)(ρ)||B(2 ) = ||

∞ (A ∗ B)k

k

k=1

K 0

1

||

(A ∗ B)k r ρ ||B(2 ) dr ≤ K k k

k

0

1

ρk ||B(2 ) ≤

C 1 . dr = 2 (1 − rρ) 1−ρ

The proof is complete.

Proof of Theorem 8.2. It is clear that (H ∞,∞,1 (2 ), H ∞,∞,1 (2 )) ⊂ (B(D, 2 ), B(D, 2 )). Consequently, using Lemma 8.3 and Theorem 8.1, we get that (B(D, 2 ), B(D, 2 )) ⊂ (B(2 ), B(D, 2 )) = H 1,∞,1 (2 ) ⊂ (H ∞,∞,1 (2 ), H ∞,∞,1 (2 )) ⊂ (B(D, 2 ), B(D, 2 )).

Theorem 8.4. It yields that (L1a (D, 2 ), L1a (D, 2 )) = H11,∞,1 (2 ) := {A upper triangular matrices such that sup (1−r)||A (r)||M (2 ) < ∞}. 0
Proof. We use the obvious fact that the space (X, X) equals the space (X , X ), where X is a Banach space of inﬁnite matrices and X is the topological dual of X. Then the theorem follows from Theorem 8.2.

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Next we want to present an application of Bloch matrices to describing the Schur multipliers from a matriceal version of the Hardy space to a matrix version of the BM O space, extending a nice result of Mateljevic and Pavlovic [60]. First we give a formula for the norm in B(2 ) due to V. Lie [50] (see also Chapter 2): ∞ 1/2 1 B 2 sup | Lk (x)h(x)dx| . ||B||B(2 ) = ||h||L2 (0,1) ≤1

k=1

0

Moreover, we use this formula to prove an interesting inequality also due to V. Lie. Proposition 8.5. If A ∈ H 1 (2 ) and B ∈ B(D, 2 ), then we have that 1 1/2 2 ||(A ∗ B) (r)||B(2 ) (1 − r)dr ≤ C||A||H 1 (2 ) ||B||B(D,2 ) , 0

where C > 0 is a constant independent of the choice of A and B. Proof. From the above formula applied to (A ∗ B) we may ﬁnd a ﬁxed h ∈ L2 ([0, 1]) with ||h||2 = 1 such that ∞ 1 ∼ | ||(A ∗ B) (r)||2 LA ∗ LB (re2πit )e2πijt h(e2πit )dt|2 . B(2 )

j

0

j=1

j

We use now the following relation, which holds if f and g are analytic functions on the unit disk D and if z = qe2πiζ , where 1 ≥ |z| = q > 0 : √q 1 f (re2πi(θ+ζ) )g (re−2πiθ )re2πiζ dθdr, z(f ∗ g) (z) = 2 0

0

and we get, by duality, that there is a x = (xj )j≥1 ∈ 2 (N), with ||x||2 ≤ 1, such that 1 ∞ B (re2πit )e2πijt h(e2πit )dt| ∼ ||(A ∗ B) (r)||B(2 ) ≈ | xj LA j ∗ Lj 0

j=1

|

∞ j=1

√ 0

r

0

1

2πiθ (LA )xj ( j ) (se

≤

√ 0

r

1 0

⎛ ⎝

∞ j=1

1 0

−2πi(θ−t) 2πijt (LB )e h(e2πit )dt)sdsdθ| j ) (se

|xj |2 | LA j

⎞1/2 (se2πiθ )|2 ⎠

×

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Schur multipliers on analytic matrix spaces

⎛ ⎞1/2 ∞ 1 −2πi(θ−t) 2πijt ⎝ | (LB )e h(e2πit )dt|2 ⎠ sdsdθ ≤ j ) (se 0

j=1

C||B||B(D,2 ) ×

√

r

1 ∞ 0

j=1

2πiθ 2 |xj |2 |(LA )| j ) (se

1/2

dθ

1−s

0

ds.

From this relation we deduce that 1 ||(A ∗ B) (r)||2B(2 ) (1 − r)dr ≤ C||B||2B(D,2 ) × 0

1

sup ||x||2 ≤1

0

(1 − r)

√

r

1 ∞ 2πiθ 2 1/2 ( j=1 |xj |2 |(LA )| ) dθ j ) (se 0

0

1−s

2 ds

dr.

Using now Proposition 4.4 we ﬁnd that 1 ||(A ∗ B) (r)||2B(2 ) (1 − r)dr ≤ C||B||2B(D,2 ) ||A||2H 1 (2 ) . 0

The proof is complete.

Now we prove the extension of the result of Mateljevic and Pavlovic: Theorem 8.6. We have the following relation: B(D, 2 ) = H 1 (2 ), BM OA(2 ) with equivalence of norms. Proof.

First we prove that 1 H (2 ), BM OA(2 ) ⊃ B(D, 2 ).

It is enough to show that ||A ∗ B||BM OA(2 ) ≤ K||A||H 1 (2 ) ||B||B(D,2 ) , for all A ∈ H 1 (2 ), and B ∈ B(D, 2 ), where K > 0 is a constant not depending on A, that is, according to Corollary 5.4, we have to prove that: 1/2 1 ||(A ∗ B)||2B(2 ) (1 − r)dr ≤ C||A||H 1 (2 ) ||B||B(D,2 ) , 0

where C > 0 is a constant independent of the choice of A and B. But this is just Proposition 8.5.

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The opposite inclusion is easy. We have to prove that (H 1 (2 ), BM OA(2 )) ⊂ B(D, 2 ). In order to do this, we assume that the following relation holds: (H 1 (2 ), B(D, 2 )) ⊂ B(D, 2 ).

(8.1)

Then it is enough to prove that: BM OA(2 ) ⊂ B(D, 2 ). We remark that, for an upper triangular matrix A, we have that ||A||L2 (2 ) = sup ||LA k ||L2 (T) ,

(8.2)

k

where T is the unidimensional torus. With the same notations as above, by [18], we have that ∞ 1/2 1 2 ||A||B(2 ) = sup | LA ≤ sup ||LA k (s)h(s)ds| k ||L∞ (T) , ||h||2 ≤1

k=1

0

k

which, in its turn, by [33] and (8.2), implies that A ||A||B(D,2 ) ≤ sup ||LA k ||B(D) ≤ sup ||Lk ||BM OA ≤ ||A||BM OA(2 ) . k

k

Hence, we have to prove only (8.1). This is the assertion of the next Proposition so the proof is complete when this result is proved. Proposition 8.7. We have that (H 1 (2 ), B(D, 2 )) ⊂ B(D, 2 ). Proof. Let B be an upper triangular matrix such that A ∗ B ∈ B(D, 2 ), for all A ∈ H 1 (2 ). By the closed graph theorem there is a constant C > 0 such that ||A ∗ B||B(D,2 ) ≤ C||A||H 1 (2 ) , for all A ∈ H 1 (2 ). Now take the Toeplitz matrix A associated with function θ → Thus, 2π dθ 1 1 1 = , ||A||H (2 ) = 2π 0 |1 − reiθ |2 1 − r2

eiθ . (1−reiθ )2

which in turn implies that C . 1 − r2 ∞ On the other hand it is clear that A ∗ B = k=0 kBk r k−1 = B (r), and, C thus, ||B (r)||B(D,2 ) ≤ 1−r 2 for all 0 < r < 1. ||A ∗ B||B(D,2 ) ≤

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Moreover, using the deﬁnition of B(D, 2 ), we have that C ||B ∗ C (r) ∗ C (ρ)||B(2 ) ≤ , (1 − r2 )(1 − ρ2 ) for all 0 < r, ρ < 1, C > 0 depending only on B. Letting r = ρ we get that C , ∀ 0 < r < 1. (8.3) ||B ∗ C (r) ∗ C (r)||B(2 ) ≤ (1 − r2 )2 Moreover, r r ∞ ∞ k2 Bk r 2k−1 = [B ∗ C (s) ∗ C (s)]ds = k 2 Bk s2k−2 ds = 2k − 1 0 0 k=0

r

∞

k=0

Bk

∗

∞

lr2l−1

l=0

∗

k=0

∞

n 2n −1 n=1

= r[B ∗ C (r 2 ) ∗ T ],

k where T is a Toeplitz matrix with 2k−1 on the kth diagonal, ∀ k = 0, 1, 2, . . . Now let T1 be the Toeplitz matrix having on the diagonal (T1 )k , k = and on the main diagonal ±1, ±2, ±3, . . . , the constant sequence 2|k|∓1 |k| (T1 )0 the value 2. It follows that r 2 ||B ∗ C (s) ∗ C (s)||B(2 ) ds ≤ (by (8.3)) ≤ r||B ∗ C (r ) ∗ T ||B(2 ) ≤ 0

r C ds r 1 ≤ C2 ds ≤ C1 = C1 , 2 )2 2 )2 (1 − s (1 − s 1 − r 1 − r2 0 0 where C2 is an universal constant. Therefore,

r

||B ∗ C (r2 )||B(2 ) ≤ ||B ∗ C (r 2 ) ∗ T ||B(2 ) · ||T1 ||M (2 ) ≤ C2 ||T1 ||M (2 ) ≤ C4 ∀ 0 < r < 1, 1 − r2 that is, B ∈ B(D, 2 ). The proof is complete.

Remark 8.8. The result above complement the results of O. Blasco [12], [13]. In these papers Blasco introduced a space B(X) of X-valued Bloch functions, X a Banach space, which in the case X = B(2 ), is larger than our space B(D, 2 ), and working with another (more general) deﬁnition of multipliers space (X, Y ) he proved that (H 1 (X), BM OA(X)) = B(2 ) if and only if X is a Hilbert space. We also have to mention that if X is the scalar ﬁeld of complex numbers, then the result of Mateljevic and Pavlovic follows also from Blasco’s theorem.

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The following result extends Corollary 1 in [60]: Corollary 8.9. It yields that (H 1 (2 ), V M OA(2 )) = B(D, 2 ). Let A ∈ H 1 (2 ) and B ∈ B(D, 2 ). Then, by Theorem 8.6, (8.4) ||X ∗ B||BM OA(2 ) ≤ c||X||H 1 (2 ) for all X ∈ H 1 (2 ), where the constant c does not depend on X. If we substitute X = A(s) − A in (8.4), we get that ||A ∗ B(s) − A ∗ B||BM OA(2 ) ≤ c||A(s) − A||H 1 (2 ) . Since obviously the term on the right-hand side of the last inequality approaches 0 when s → 1, it follows that A ∗ B ∈ V M OA(2 ). The proof is complete. Proof.

Let n1 , n2 < . . . be a lacunary sequence of integers in the sense that nk+1 /nk ≥ q > 1. ∞ It is well-known that g(z) = k=1 z nk ∈ B, (see for instance the more general Theorem 7.9), so that we have by the previous corollary: Corollary 8.10. If A = An ∈ H 1 (2 ), then, for every lacunary sequence {nk }, ∞ A˜ := Ank ∈ V M OA(2 ). k=1

Remark. It follows by the deﬁnition of H 1 (2 ) that the space of all upper triangular Hilbert-Schmidt matrices T2 is contained in H 1 (2 ), so that if A ∈ T2 , then we have that ∞ k=1 A2k ∈ V M OA(2 ). Since clearly V M OA(2 ) ⊂ BM OA(2 ) ⊂ H 2 (2 ), we get the matrix version of Paley’s theorem: ∞ Theorem 8.11. If A = An ∈ H 1 (2 ), then P (A) := k=1 A2k ∈ H 2 (2 ). Another consequence of Corollary 8.10 is the following extension of a remark of [60]: Corollary 8.12. It yields that A = ∞ k=1 A2k ∈ V M OA(2 ) if and only if A ∈ H 2 (2 ), that is, if and only if ∞ sup |ak2j |2 < ∞, k

where A =

(ak2j )j,k≥1 .

j=1

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Proof. Since, clearly, H 2 (2 ) ⊂ H 1 (2 ) the assertion of Corollary 8.12 follows from Theorem 8.11 and the above discussion. We mention also the extension of Corollary 2 in [60], which follows by Theorems 7.16 and the proof (see relation (8.1)) of Theorem 8.6: Corollary 8.13. The space of all Schur multipliers (H 1 (2 ), B0 (D, 2 )) coincides with B(D, 2 ). Notes Various results about Schur multipliers on diﬀerent analytic matrix spaces are given in this chapter. For instance in Theorem 8.1 it is showed that ∞ kAk r k−1 ||M (2 ) < ∞. A ∈ (B(2 ), B(D, 2 )) if and only if sup (1 − r)|| 0≤r<1

k=1

Another important result is Theorem 8.2, which gives a characterization of the space (B(D, 2 ), B(D, 2 )). But the most important result of this section is an answer to the problem to ﬁnd a matrix version of the beautiful theorem of Matjelevic and Pavlovic [60] about the space of all Fourier multipliers from H 1 into BM OA. This is Theorem 8.6 mainly due to V. Lie and its proof uses results from Chapters 4 and 5. The statement is: B(D, 2 ) = (H 1 (2 ), BM OA(2 )). As one of its consequences we mention only a matrix version of Paley’s theorem: ∞ ∞ Theorem 8.11 If A = k=1 A2k ∈ H 1 (2 ), then P (A) = k=1 A2k ∈ H 2 (2 ).

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Bibliography

[1] J. Arazy, Some remarks on interpolation theorems and the boundness of the triangular projection in unitary matrix spaces, Int. Eq. Oper. Th. 1, (1978), 453-495. [2] J. Arazy, On the geometry of the unit ball of unitary matrix spaces, Int. Eq. Oper. Th., 4 (1981), 151-171. [3] J. M. Anderson, J. Clunie and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math.,270 (1974), 12-37. [4] J. M. Anderson and A. Shields, Coeﬃcient multipliers of Bloch functions, Trans. Amer. Math. Soc., 224 (1976), 255-265. [5] J. Arazy, S. D. Fisher and J. Peetre, M¨ obius invariant function spaces, J. Reine Angew. Math. 363(1985), 110-145. [6] A. B. Alexandrov, Essays on non locally convex Hardy classes, Lecture Notes in Math. 864 (1981), 1-89, Springer-Verlag, Berlin-Heidelberg-New York. [7] A. B. Alexandrov, On the boundary decay in the mean of harmonic functions, St. Petersburg Math. J. 5 (1996), 507-542. [8] A. B. Alexandrov and V. V. Peller, Hankel and Toeplitz-Schur multipliers, Math. Ann., 324 (2002), 277-327. [9] W. Arveson, Subalgebras of C ∗ − algebras, Acta Math. 123 (1969), 144-224. [10] W. Arveson, Interpolation problems in nest algebras, J. Funct. Analysis 20 (1975), 208-233. [11] G. Bennett, Schur multipliers, Duke Math. J., 44 (1977), 603-639. [12] O. Blasco, A characterization of Hilbert spaces in terms of multipliers between spaces of vector-valued analytic functions, Michigan Math. J., 42(1995), 537-543. [13] O. Blasco, Vector-valued analytic functions of bounded mean oscillation and geometry of Banach spaces, Illinois J. Math., 41 (1997), 532-558. [14] O. Blasco, Introduction to vector valued Bergman spaces, Function spaces and operator theory, 9–30, Univ. Joensuu Dept. Math. Rep. Ser., 8, Univ. Joensuu, Joensuu, 2005. [15] O. Blasco and A. Pelczynski, Theorems of Hardy and Paley for vector valued analytic functions and related classes of Banach spaces, Trans. Amer. Math. Soc. 323 (1991), 335-367.

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Index

(H 1 − 1 )-multiplier, 80 (X, Y ) the space of all Schur multipliers from X into Y , 1 (·, ·) the scalar product in a Hilbert space, 3 (p, q)-bounded, 80 A ∗ B the Schur product of the matrices A and B, 1 Ak the kth-diagonal matrix , 1 B(2 ) the space of all bounded linear operators on 2 , 2 BM OA the space of analytic functions of bounded mean oscillation, 4 BM OA(2 ), 109 BM OF (2 ), BM OAF (2 ), xi, 114 C1 the trace class, 3 C2 the Hilbert-Schmidt class, 3 Cp , 0 < p < ∞ Schatten class, 3 DA the defect operator, 92 Et the Toeplitz matrix with entries (ei(j−k)t )j,k≥1 , 71 F M , 84 H 2 (H) Hardy space of Hilbert space-valued functions, 112 H ∞,∞,1 , H ∞,∞,1 (2 ), 176 H ∞ (2 ), 114 H p (2 ) the matriceal Hardy space of index p, 67 H p , 0 < p ≤ ∞ Hardy spaces, 4 p vector-valued Hardy space, 79 HX H11,∞,1 (2 ), 176

HΦ matrix version of Hankel operator, 113 L1 (D, 2 ), L∞ (D, 2 ), 5 (D, 2 ), 132 Lp,unc a M (2 ) the space of all Schur multipliers on B(2 ), 2 M (T) the convolution algebra of Borel measures on T, 101 P, P Bergman projection, 128 PT the triangular projection, 69 Pα , 129 Tp (2R ), T p(2C ) spaces of upper triangular matrices, 70 Tp , 0 < p < ∞ the matrix analogue of Hardy space H p , 69 V M OF (2 ), 118 ΓΩ the block Hankel matrix, 96 β(z, w) the Bergman metric, 7 ∞ (∞ ), 2 (2 , w), 134 B Bloch space, 4 B(D, 2 ) the matriceal Bloch space, 150 B0,c (D, 2 ), 170 B0 the little Bloch space, 4 B0 (D, 2 ) little Bloch space, 161 C(D, 2 ), 165 C0 (D, C∞ ), 171 C0 (D, 2 ), 169 C∞ -tensor product of C0 ⊗ corresponding Banach spaces, 171 I, 152 A LA k , Lk , LA (x, t), 65 191

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Mp,2 , 119 T ,2 ρ(z, w) the pseudo-hyperbolic distance, 7 σn (A), 137 ikt ,2 fA (r, t) = ∞ k=−∞ Ak (r)e mX the operator induced by the multiplier m, 80 w∗ -measurable function, 5 V M OA(2 ), 109 Bergman-Schatten classes, 122 Analytic matrices, 2 Banach space of (H 1 − 1 )-Fourier type, 81 Bennett’s Theorem, 2 Bloch matrix, 150 crudely ﬁnitely representable Banach space, 86 Matriceal Hausdorﬀ-Young Theorem, 75 Matrix version of Nehari theorem, xi, 114 Pavlovi´c Theorem, 97 Schmidt Theorem, 3 Shields inequality, 70 singular values of an operator T , 3 strongly measurable function, 5 the Hankel matrix Ha , 90 The matrix spaces Lpr (2 ), resp. Lpc (2 ), 1 ≤ p ≤ 2, 66 The noncommutative factorization theorem, 87 the projective tensor product, 88 Toeplitz matrices, 2 Vectorial Nehari theorem, 113

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