Preface i
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 r^Berezin Transform and Radial Operator 57
1.6 A Class of Hilbert-Schmidt Operators on the Harmonic Bergman Space 71
1.7 The Operator Mznzn 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 d 131
2.3 Compactness of Hankel Operators 141
2.4 Remarks 158
CHAPTER 3
Operator on the Fock Space 161
Quasi-Similarity and Reducing Subspaces of Multiplication Operator 161
The Norm of Hankel Operator Restricted to the Fock Space 168
Remarks 176
CHAPTER 4
Operator on the Soblev 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
Sample Pages Preview
Chapter 1 Operator on the Bergman Space 1.1 Similarity Invariant of Analytic Toeplitz Operators Let B and T be the unit disk and the boundary of D respectively, and let L denote the Bergman space of analytic functions which belong to. It is well known that is a Hilbert space.then where dA is the normalized area measure on D, and For , bnzn, then the inner product of and g on the Bergman space is defined by In this sense has an orthonormal basis, where en . Letdenote the algebra of the bounded analytic functions on ID, for G H (D), Mf is an analytic Toeplitz operator (multiplication operator) on the Bergman space defined by, for any g a bounded linear operator on with where represents the collection of all bounded linear operators on An operator T is said to be strongly irreducible see,if there is no non-trivial idempotent operator in Af(T). Let be an analytic map, and Cb be the composition operator on i defined by We consider a finite Blaschke product B(z) with the following form In fact, if B(z) is a finite Blaschke product with n zeros, then so is the function Mb and Mbx have the same commutant and the same lattice of the reducing subspace, we may assume,without loss of generality, that for convenience. So we always denote the first factor of by z. When has zeros with multiplicity greater than one, we can consider a new Blaschke product, and Aut. We claim that except a finite subset S C D each has distinct zeros. In fact, let, is a finite subset of D, either B, has distinct zeros (see [67,111]). Lemma 1.1.1. Lemma 1.1.2. Definition 1.1.3. Lemma 1.1.4. Green-Stokes formula. The main result of the following lemma is given by [111], but we make a little correction of the coefficient of the basis. Lemma 1.1.5. (1.1.1) Lemma 1.1.6. (1) (2) Proof. (1) (1.1.2) (1.1.3) (1.1.4) (1.1.5) (1.1.6)
