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.
Author(s): Alexandru Aleman, Nathan S. Feldman, William T. Ross
Series: Frontiers in Mathematics
Edition: 1
Publisher: Birkhäuser Basel
Year: 2009
Language: English
Pages: 143
Preface......Page 4
Notation......Page 8
List of Symbols......Page 10
Preamble......Page 12
1.1 Some history......Page 20
1.2 Invariant subspaces of the slit disk......Page 21
1.3 Nearly invariant subspaces......Page 24
1.4 Cyclic invariant subspaces......Page 25
1.5 Essential spectrum......Page 26
2.1 Hardy space of a general domain......Page 28
2.2 Harmonic measure......Page 31
2.3 Slit domains......Page 33
2.4 More about the Hardy space......Page 39
3.1 Statement of the main result......Page 44
3.2 Normalized reproducing kernels......Page 45
3.3 The operator J......Page 53
3.4 The Wold decomposition......Page 56
3.5 Proof of the main theorem......Page 61
3.6 Uniqueness of the parameters......Page 65
4.1 The backward shift and pseudocontinuations......Page 66
4.2 A new description of nearly invariant subspaces......Page 67
5.1 de Branges spaces......Page 78
5.2 de Branges spaces and nearly invariant subspaces......Page 79
6.1 First description of the invariant subspaces......Page 84
6.2 Second description of the invariant subspaces......Page 87
7.1 Two-cyclic subspaces......Page 98
7.2 Cyclic subspaces......Page 99
7.3 Polynomial approximation......Page 101
8.2 Essential spectrum......Page 104
9.1 Compressions......Page 112
9.2 The parameters......Page 114
10.1 Statement of the result......Page 116
10.2 Some technical lemmas......Page 118
10.3 A localization of Yakubovich......Page 120
10.4 Finally the proof......Page 130
11 Final thoughts......Page 132
12 Appendix......Page 134
Bibliography......Page 136
Index......Page 142
