Similarity of Operator on Analytic Function Spaces, livre ebook

This book summarizes the study of similarity and reducing subspaces for a class of multiplication operators on the Bergman space, Dirichlet space, and Sobolev disc algebra.

It also investigates the compactness of certain radial and Hankel operators on the weighted Bergman space. The research methods combine operator theory techniques with infinite matrix manipulations.

The book consists of five chapters. In the first chapter, we characterize the similarity and commutant of certain analytic Toeplitz operators. The second chapter focuses on the similarity of multiplication operator induced by finite Blaschke product and characterize the compactness of a class of Hankel operators. The third chapter discusses the quasi-similarity and reducing subspaces of a class of multiplication operator with monomial symbol in the Fock space. The fourth chapter explores the similarity and reducing subspaces of a class of multiplication operator induced by monomial symbol in the Sobolev disc algebra. The fifth chapter addresses the algebra matrix and similarity classification of certain Cowen-Douglas operators.

The theme of the book is straightforward. Its contents can be comfortably understood by readers with a background in single-variable complex analysis and basic knowledge of operator theory.

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III

CHAPTER 1

Operator on the Bergman Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Similarity Invariant of Analytic Toeplitz Operators . . . . . . . . . . . . . . . 1

1.2 Commutant of Analytic Toeplitz Operators. . . . . . . . . . . . . . . . . . . . . 17

1.3 Compactness of a Class of Radial Operators . . . . . . . . . . . . . . . . . . . . 31

1.4 Similarity of a Class of Multiplication Operators . . . . . . . . . . . . . . . . . 42

1.5 n-Berezin Transform and Radial Operator . . . . . . . . . . . . . . . . . . . . . . 57

1.6 A Class of Hilbert–Schmidt Operators on the Harmonic Bergman Space . . .. . . .. 71

1.7 The Operator Mzn1zn2 on Subspaces of Bergman Spaces over the Biannulus .. . . . . . . . . 83

1.8 Local Quasi-Similarity and Reducing Subspaces of Multiplication Operator . . ... . . . . . 97

1.9 Quasi-Affinity and Reducing Subspaces of Multiplication Operator . . . 106

1.10 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

CHAPTER 2

Operator on the Dirichlet Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

2.1 Similarity and Commutant of a Class of Multiplication Operators . . . . 117

2.2 The Properties of Canonical Solution Operator to @ . . . . . . . . . . . . . . 131

2.3 Compactness of Hankel Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

2.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

CHAPTER 3

Operator on the Fock Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

3.1 Quasi-Similarity and Reducing Subspaces of Multiplication Operator . 161

3.2 The Norm of Hankel Operator Restricted to the Fock Space . . . . . . . . 168

3.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

CHAPTER 4

Operator on the Sobolev Disk Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

4.1 Similarity and Reducing Subspaces of Multiplication Operator . . . . . . 179

4.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

CHAPTER 5

Operator on Banach Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

5.1 Algebra Matrix and Similarity Classification of Operators . . . . . . . . . . 187

5.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

List of Symbols and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207