This book gives a unified approach to the theory concerning a new matrix version of classical harmonic analysis. Most results in the book have their analogues as classical or newer results in harmonic analysis. It can be used as a source for further research in many areas related to infinite matrices. In particular, it could be a perfect starting point for students looking for new directions to write their PhD thesis as well as for experienced researchers in analysis looking for new problems with great potential to be very useful both in pure and applied mathematics where classical analysis has been used, for example, in signal processing and image analysis.

Preface vii

1. Introduction 1

1.1 Preliminary notions and notations 1

      1.1.1 Infinite matrices 1

      1.1.2 Analytic functions on disk 4

      1.1.3 Miscellaneous 5

      1.1.4 The Bergman metric 7

2. Integral operators in infinite matrix theory 9

2.1 Periodical integral operators 9

2.2 Nonperiodical integral operators 17

2.3 Some applications of integral operators in the classical theory of infinite matrices 18

      2.3.1 The characterization of Toeplitz matrices 18

      2.3.2 The characterization of Hankel matrices 24

      2.3.3 The main triangle projection 27

      2.3.4 B(l2) is a Banach algebra under the Schur product 30

3. Matrix versions of spaces of periodical functions 33

3.1 Preliminaries 34

3.2 Some properties of the space C(l2) 34

3.3 Another characterization of the space C(l2) and related results 36

3.4 A matrix version for functions of bounded variation 41

3.5 Approximation of infinite matrices by matriceal Haar polynomials 44

      3.5.1 Introduction 45

      3.5.2 About the space ms 50

      3.5.3 Extension of Haar's theorem 56

3.6 Lipschitz spaces of matrices; a characterization 61

4. Matrix versions of Hardy spaces 65

4.1 First properties of matriceal Hardy space 65

4.2 Hardy-Schatten spaces 69

4.3 An analogue of the Hardy inequality in T1 75

4.4 The Hardy inequality for matrix-valued analytic functions 79

      4.4.1 Vector-valued Hardy spaces HPX 79

      4.4.2 (HP-lq)-multipliers and induced operators for vector-valued functions 80

4.5 A characterization of the space T1 97

4.6 An extension of Shields's inequality 101

5. The matrix version of BMOA 109

5.1 First properties of BMOA (l2) space 109

5.2 Another matrix version of BMO and matriceal Hankel oprators 111

5.3 Nuclear Hankel operators and the space M1,2 119

6. Matrix version of Bergman spaces 121

6.1 Schatten class version of Bergman spaces 121

6.2 Some inequalities in Bergman-Schatten classes 132

6.3 A characterization of the Bergman-Schatten space 136

6.4 Usual multipliers in Bergman-Schatten spaces 141

7. A matrix version of Bloch spaces 149

7.1 Elementary properties of Bloch matrices 149

7.2 Matrix version of little Bloch space 161

8. Schur multipliers on analytic matrix spaces 175

Bibliography 185

Index 191

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