The Hardy Space of a Slit Domain

by Aleman, Alexandru; Feldman, Nathan S.; Ross, William T.

• ISBN13: 9783034600972
• ISBN10: 3034600976
• Format: Paperback
• Copyright: 2009-11-01
• Publisher: Birkhauser

Summary

The book begins with an exposition of Hardy spaces of slit domains and then proceeds to several descriptions of the invariant subspaces of the operator multiplication by z. Along the way, we discuss and characterize the nearly invariant subspaces of these Hardy spaces and examine conditions for z-invariant subspaces to be cyclic. This work also makes important connections to model spaces for the standard backward shift operator as well as the de Branges spaces of entire functions. The book is written for a graduate student or professional with a reasonable knowledge of Hardy spaces of the disk and basic complex and functional analysis.

Table of Contents

Preface	p. v
Notation	p. ix
List of Symbols	p. xi
Preamble	p. xiii
Introduction	p. 1
Some history	p. 1
Invariant subspaces of the slit disk	p. 2
Nearly invariant subspaces	p. 5
Cyclic invariant subspaces	p. 6
Essential spectrum	p. 7
Preliminaries	p. 9
Hardy space of a general domain	p. 9
Harmonic measure	p. 12
Slit domains	p. 15
More about the Hardy space	p. 20
Nearly invariant subspaces	p. 25
Statement of the main result	p. 25
Normalized reproducing kernels	p. 26
The operator J	p. 34
The World decomposition	p. 37
Proof of the main theorem	p. 42
Uniqueness of the parameters	p. 46
Nearly invariant and the backward shift	p. 47
The backward shift and pseudocontinuations	p. 47
A new description of nearly invariant subspaces	p. 48
Nearly invariant and de Branges spaces	p. 59
de Branges spaces	p. 59
de Branges spaces and nearly invariant subspaces	p. 60
Invariant subspaces of the slit disk	p. 65
First description of the invariant subspaces	p. 65
Second description of the invariant subspaces	p. 68
Cyclic invariant subspaces	p. 79
Two-cyclic subspaces	p. 79
Cyclic subspaces	p. 80
Polynomial approximation	p. 82
The essential spectrum	p. 85
Fredholm theory	p. 85
Essential spectrum	p. 85
Other applications	p. 93
Compressions	p. 93
The parameters	p. 95
Domains with several slits	p. 97
Statement of the result	p. 97
Some technical lemmas	p. 99
A localization of Yakubovich	p. 101
Finally the proof	p. 111
Final thoughts	p. 113
Appendix	p. 115
Bibliography	p. 117
Index	p. 123
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