Matrix Spaces And Schur Multipliers: Matriceal Harmonic Analysis

Matriceal Harmonic Analysis

Lars-Erik Persson, Nicolae Popa

• Publisher: WSPC
• Year: 2013
• ISBN:
  9789814546799
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

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.

Contents:

• Introduction
• Integral Operators in Infinite Matrix Theory
• Matrix Versions of Spaces of Periodical Functions
• Matrix Versions of Hardy Spaces
• The Matrix Versions of BMOA
• Matrix Versions of Bergman Spaces
• A Matrix Version of Bloch Spaces
• Schur Multipliers on Analytic Functions

Readership: Graduates and mathematicians interested in this new area of matriceal harmonic analysis but also researchers in areas using harmonic analysis, wavelets etc., e.g. signal processing and image analysis.

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 = (a_ij), and an integer k we denote by A_k the matrix whose entries a′_i,j are given by

Then A_k 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 = (a_ij)_i,j and B = (b_ij)_i,j be two infinite matrices. Then the Schur product C = (c_ij)_i,j of A and B, denoted by A _* B, has the entries c_ij = a_ijb_ij for all i, j ∈

.

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

is denoted by (X, Y).

In the case X = Y = B(ℓ₂), where B(ℓ₂) is the space of all linear and bounded operators on ℓ₂, the space (X, Y) is denoted by M(ℓ₂) (an explanation of this notation is given later in this section) and a matrix A ∈ M(ℓ₂) 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 correspondence

where A_k(r) is the kth-diagonal of the matrix A(r), the preceding sum is a formal one and t belongs to the torus

.

We may consider f_A(r, t), or f_A(z), with z = re^it, 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 A_k(r) = A_kr^k, for all k ∈

.

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 = (a_ij)_i,j≥1 be an infinite matrix. If there is a sequence of complex numbers

such that a_ij = a_j–i for all i, j ∈

, then A is called a Toeplitz matrix.

...