This book exposes the internal structure of non-self-adjoint operators acting on complex separable infinite dimensional Hilbert space, by analyzing and studying the commutant of operators. A unique presentation of the theorem of Cowen-Douglas operators is given. The authors take the strongly irreducible operator as a basic model, and find complete similarity invariants of Cowen-Douglas operators by using K-theory, complex geometry and operator algebra tools.
Author(s): Chunlan Jiang, Zongyao Wang
Publisher: World Scientific
Year: 2006
Language: English
Pages: 260
City: Hackensack, NJ
Contents ......Page 10
Preface ......Page 6
1.1 Banach Algebra ......Page 12
1.2 K-Theory of Banach Algebra ......Page 14
1.3 The Basic of Complex Geometry ......Page 15
1.4 Some Results on Cowen-Douglas Operators ......Page 16
1.5 Strongly Irreducible Operators ......Page 18
1.7 Similarity Orbit Theorem ......Page 20
1.8 Toeplitz Operator and Sobolev Space ......Page 21
2.1 Generalized Eigenspace and Minimal Idempotents ......Page 24
2.2 Similarity Invariant of Matrix ......Page 25
2.3 Remark ......Page 29
3.1 Sum of Strongly Irreducible Operators ......Page 30
3.2 Approximate Jordan Decomposition Theorem ......Page 40
3.4 Remark ......Page 53
4. Unitary Invariant and Similarity Invariant of Operators ......Page 54
4.1 Unitary Invariants of Operators ......Page 55
4.2 Strongly Irreducible Decomposition of Operators and Similarity Invariant: Ko-Group ......Page 68
4.3 (SI) Decompositions of Some Classes of Operators ......Page 80
4.4 The Commutant of Cowen-Douglas Operators ......Page 91
4.5 The Sobolev Disk Algebra ......Page 105
4.6 The Operator Weighted Shift ......Page 137
4.8 Remark ......Page 158
5.1 The Cowen-Douglas Operators with Index 1 ......Page 160
5.2 Cowen-Douglas Operators with Index n ......Page 165
5.3 The Commutant of Cowen-Douglas Operators ......Page 168
5.4 The Commutant of a Classes of Operators ......Page 180
5.5 The (SI) Representation Theorem of Cowen-Douglas Operators ......Page 187
5.6 Maximal Ideals of The Commutant of Cowen-Douglas Operators ......Page 200
5.7 Some Approximation Theorem ......Page 203
5.9 Open Problem ......Page 212
6.1 K0-Group of Some Banach Algebra ......Page 214
6.2 Similarity and Quasisimilarity ......Page 217
6.3 Application of Operator Structure Theorem ......Page 248
6.5 Open Problems ......Page 250
Bibliography ......Page 252
Index ......Page 258
