The Hardy Space of a Slit Domain

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Springer Science & Business Media, Jan 8, 2010 - Mathematics - 144 pages
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If H is a Hilbert space and T : H ? H is a continous linear operator, a natural question to ask is: What are the closed subspaces M of H for which T M ? M? Of course the famous invariant subspace problem asks whether or not T has any non-trivial invariant subspaces. This monograph is part of a long line of study of the invariant subspaces of the operator T = M (multiplication by the independent variable z, i. e. , M f = zf )on a z z Hilbert space of analytic functions on a bounded domain G in C. The characterization of these M -invariant subspaces is particularly interesting since it entails both the properties z of the functions inside the domain G, their zero sets for example, as well as the behavior of the functions near the boundary of G. The operator M is not only interesting in its z own right but often serves as a model operator for certain classes of linear operators. By this we mean that given an operator T on H with certain properties (certain subnormal operators or two-isometric operators with the right spectral properties, etc. ), there is a Hilbert space of analytic functions on a domain G for which T is unitarity equivalent to M .
 

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Contents

Introduction
1
12 Invariant subspaces of the slit disk
2
13 Nearly invariant subspaces
5
14 Cyclic invariant subspaces
6
15 Essential spectrum
7
Preliminaries
9
22 Harmonic measure
12
23 Slit domains
15
52 de Branges spaces and nearly invariant subspaces
60
Invariant subspaces of the slit disk
64
62 Second description of the invariant subspaces
68
Cyclic invariant subspaces
79
72 Cyclic subspaces
80
73 Polynomial approximation
82
The essential spectrum
84
Other applications
93

24 More about the Hardy space
20
Nearly invariant subspaces
24
32 Normalized reproducing kernels
26
33 The operator J
34
34 The Wold decomposition
37
35 Proof of the main theorem
42
36 Uniqueness of the parameters
46
Nearly invariant and the backward shift
47
42 A new description of nearly invariant subspaces
48
Nearly invariant and de Branges spaces
59
92 The parameters
95
Domains with several slits
97
102 Some technical lemmas
99
103 A localization of Yakubovich
101
104 Finally the proof
111
Final thoughts
113
Appendix
115
Bibliography
117
Index
122
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