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
