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E-raamat: Handbook of Analytic Operator Theory

Edited by (State University of New York, Albany, NY, U.S.A)
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Handbook of Analytic Operator Theory thoroughly covers the subject of holomorphic function spaces and operators acting on them. The spaces covered include Bergman spaces, Hardy spaces, Fock spaces and the Drury-Averson space. Operators discussed in the book include Toeplitz operators, Hankel operators, composition operators, and Cowen-Douglas class operators.

The volume consists of eleven articles in the general area of analytic function spaces and operators on them. Each contributor focuses on one particular topic, for example, operator theory on the Drury-Aversson space, and presents the material in the form of a survey paper which contains all the major results in the area and includes all relevant references.

The overalp between this volume and existing books in the area is minimal. The material on two-variable weighted shifts by Curto, the Drury-Averson space by Fang and Xia, the Cowen-Douglas class by Misra, and operator theory on the bi-disk by Yang has never appeared in book form before.

Features:

The editor of the handbook is a widely known and published researcher on this topic

The handbook's contributors are a who's=who of top researchers in the area

The first contributed volume on these diverse topics
Preface ix
1 Fock Space, the Heisenberg Group, Heat Flow, and Toeplitz Operators
1(16)
Lewis A. Coburn
1.1 Introduction
1(1)
1.2 Toeplitz operators and the Heisenberg group
2(2)
1.3 Toeplitz operators and the Bargmann transform
4(3)
1.4 Limit behavior of ||Tf(t)||t and ||Hf(t)||t, as t→0
7(4)
1.5 Examples
11(6)
References
14(3)
2 Two-Variable Weighted Shifts in Multivariable Operator Theory
17(48)
Raul E. Curto
2.1 Introduction
18(2)
2.2 2-variable weighted shifts
20(1)
2.3 The lifting problem for commuting subnormals
21(2)
2.4 Hyponormality, 2-hyponormality and subnormality for 2-variable weighted shifts
23(1)
2.5 Existence of nonsubnormal hyponormal 2-variable weighted shifts
24(8)
2.6 Propagation in the 2-variable hyponormal case
32(2)
2.7 A measure-theoretic necessary (but not sufficient!) condition for the existence of a lifting
34(3)
2.8 Reconstruction of the Berger measure for 2-variable weighted shifts whose core is of tensor form
37(2)
2.9 The subnormal completion problem for 2-variable weighted shifts
39(4)
2.10 Spectral picture of hyponormal 2-variable weighted shifts
43(4)
2.11 A bridge between 2-variable weighted shifts and shifts on directed trees
47(2)
2.12 The spherical Aluthge transform
49(16)
References
58(7)
3 Commutants, Reducing Subspaces, and von Neumann Algebras
65(22)
Kunyu Guo
Hansong Huang
3.1 Introduction
65(3)
3.2 Commutants and reducing subspaces for multiplication operators on the Hardy space H2(D)
68(3)
3.3 The case of the Bergman space L2a(D)
71(3)
3.4 The case of Bergman space over a polygon
74(2)
3.5 The case of the Bergman space over high dimensional domains
76(4)
3.6 Further questions
80(7)
References
81(6)
4 Operators in the Cowen-Douglas Class and Related Topics
87(52)
Gadadhar Misra
4.1 Introduction
88(16)
4.2 Some future directions and further thoughts
104(35)
References
133(6)
5 Toeplitz Operators and Toeplitz C*-Algebras
139(32)
Harald Upmeier
5.1 Introduction
139(1)
5.2 Toeplitz operators on Hilbert spaces of multi-variable holomorphic functions
140(3)
5.3 Strongly pseudoconvex domains
143(3)
5.4 Symmetric domains and Jordan triples
146(6)
5.5 Holomorphic function spaces on symmetric domains
152(4)
5.6 Toeplitz C*-algebras on symmetric domains
156(4)
5.7 Hilbert quotient modules and Kepler varieties
160(4)
5.8 Toeplitz operators on Reinhardt domains
164(7)
References
167(4)
6 Mobius Invariant 2p and 2K Spaces
171(32)
Hasi Wulan
6.1 Introduction
172(1)
6.2 Background
173(1)
6.3 Basic properties of 2P spaces
174(2)
6.4 Carleson measures
176(3)
6.5 The boundary value characterizations
179(2)
6.6 spaces
181(3)
6.7 2K-Carleson measures
184(7)
6.8 Boundary spaces
191(2)
6.9 Composition operators on 2p and 2K spaces
193(10)
References
195(8)
7 Analytical Aspects of the Drury-Arveson Space
203(20)
Quanlei Fang
Jingbo Xia
7.1 Introduction
203(1)
7.2 Von Neumann inequality for row contractions
204(2)
7.3 The multipliers
206(4)
7.4 A family of reproducing-kernel Hilbert spaces
210(2)
7.5 Essential normality
212(2)
7.6 Expanding on Drury's idea
214(3)
7.7 Closure of the polynomials
217(6)
References
218(5)
8 A Brief Survey of Operator Theory in H2 (D2)
223(36)
Rongwei Yang
8.1 Introduction
224(1)
8.2 Background
224(4)
8.3 Nagy-Foias theory in H2(D2)
228(5)
8.4 Commutators
233(3)
8.5 Two-variable Jordan block
236(3)
8.6 Fredholmness of the pairs (R1, R2) and (S1, S2)
239(2)
8.7 Essential normality of quotient module
241(2)
8.8 Two single companion operators
243(5)
8.9 Congruent submodules and their invariants
248(3)
8.10 Concluding remarks
251(8)
References
251(8)
9 Weighted Composition Operators on Some Analytic Function Spaces
259(28)
Ruhan Zhao
9.1 Introduction
259(1)
9.2 Preliminaries
260(3)
9.3 Carleson measures
263(2)
9.4 Weighted composition operators between weighted Bergman spaces
265(6)
9.5 Weighted composition operators between Hardy spaces
271(3)
9.6 Weighted composition operators between weighted spaces of analytic functions
274(3)
9.7 Weighted composition operators between Bloch type spaces...
277(10)
References
283(4)
10 Toeplitz Operators and the Berezin Transform
287(32)
Xianfeng Zhao
Dechao Zheng
10.1 Introduction
287(2)
10.2 Basic properties of Toeplitz operators and the Berezin transform
289(4)
10.3 Positivity of Toeplitz operators via the Berezin transform
293(10)
10.4 Invertibility of Toeplitz operators via the Berezin transform
303(16)
References
316(3)
11 Towards a Dictionary for the Bargmann Transform
319(32)
Kehe Zhu
11.1 Introduction
319(2)
11.2 Hermite polynomials
321(1)
11.3 The Fourier transform
322(3)
11.4 Dilation, translation, and modulation operators
325(3)
11.5 Gabor frames
328(6)
11.6 The canonical commutation relation
334(2)
11.7 Uncertainty principles
336(2)
11.8 The Hilbert transform
338(5)
11.9 Pseudo-differential operators
343(2)
11.10 Further results and remarks
345(6)
References
347(4)
Index 351
Kehe Zhu is professor of mathematics at the State University of New York at Albany. His research areas are complex analysis, functional analysis, and operator theory. He has published over 110 papers in leading research journals in mathematics. He has also published several books including Theory of Bergman Spaces, Spaces of Holomorphic Functions in the Unit Ball, Analysis on Fock Spaces, Mobius Invariant Qk Spaces, Operator Theory in Function Spaces, and An Introduction to Operator Algebras (CRC Press).
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