The Hardy Space of a Slit Domain

ISBN-13: 9783034600972 / Angielski / Miękka / 2009 / 144 str.

Alexandru Aleman; Nathan S. Feldman

Opis

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 .

Recenzje

From the reviews:

"This memoir is concerned with the description of the shift-invariant subspaces of a Hardy space on a slit domain ... . this brief monograph represents an interesting and valuable contribution to the literature on the subject of shift-invariant subspaces. It should be helpful for researchers and advanced graduate students specializing in the field." (Dragan Vukotic, Mathematical Reviews, Issue 2011 m)

Spis treści

Preface; Notation; List of Symbols; Preamble; 1 Introduction; 2 Preliminaries; 3 Nearly invariant subspaces; 4 Nearly invariant and the backward shift; 5 Nearly invariant and de Branges spaces; 6 Invariant subspaces of the slit disk; 7 Cyclic invariant subspaces; 8 The essential spectrum; 9 Other applications; 10 Domains with several slits; 11 Final thoughts; 12 Appendix