Aleman A., Feldman N.S., Ross W.T. The Hardy Space of a Slit Domain

Basel: Birkhauser, 2009. — 143 p.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.Preface
Notation
List of Symbols
Preamble
Some history
Invariant subspaces of the slit disk
Nearly invariant subspaces
Cyclic invariant subspaces
Essential spectrum
Hardy space of a general domain
Harmonic measure
Slit domains
More about the Hardy space
Statement of the main result
Normalized reproducing kernels
The operator J
The Wold decomposition
Proof of the main theorem
Uniqueness of the parameters
The backward shift and pseudocontinuations
A new description of nearly invariant subspaces
de Branges spaces
de Branges spaces and nearly invariant subspaces
First description of the invariant subspaces
Second description of the invariant subspaces
Two-cyclic subspaces
Cyclic subspaces
Polynomial approximation
Essential spectrum
Compressions
The parameters
Statement of the result
Some technical lemmas
A localization of Yakubovich
Finally the proof
Final thoughts
Appendix
Bibliography
Index