The Hardy Space of a Slit Domain

1. Front Matter

   Pages i-xix

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2. Introduction

   • Alexandru Aleman, William T. Ross, Nathan S. Feldman

   Pages 1-7

3. Preliminaries

   • Alexandru Aleman, William T. Ross, Nathan S. Feldman

   Pages 9-23

4. Nearly invariant subspaces

   • Alexandru Aleman, William T. Ross, Nathan S. Feldman

   Pages 25-46

5. Nearly invariant and the backward shift

   • Alexandru Aleman, William T. Ross, Nathan S. Feldman

   Pages 47-57

6. Nearly invariant and de Branges spaces

   • Alexandru Aleman, William T. Ross, Nathan S. Feldman

   Pages 59-63

7. Invariant subspaces of the slit disk

   • Alexandru Aleman, William T. Ross, Nathan S. Feldman

   Pages 65-77

8. Cyclic invariant subspaces

   • Alexandru Aleman, William T. Ross, Nathan S. Feldman

   Pages 79-83

9. The essential spectrum

   • Alexandru Aleman, William T. Ross, Nathan S. Feldman

   Pages 85-91

10. Other applications

    • Alexandru Aleman, William T. Ross, Nathan S. Feldman

    Pages 93-96

11. Domains with several slits

    • Alexandru Aleman, William T. Ross, Nathan S. Feldman

    Pages 97-112

12. Final thoughts

    • Alexandru Aleman, William T. Ross, Nathan S. Feldman

    Pages 113-114

13. Back Matter

    Pages 115-124

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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 .

• Hardy Space
• Invariant
• Invariant subspace
• Multiplication
• Slit plane
• character
• essential spectrum
• function
• functional
• functional analysis
• functions
• knowledge
• model
• operator

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 Vukotić, Mathematical Reviews, Issue 2011 m)

• Centre for Mathematical Sciences, Lund University, Lund, Sweden

  Alexandru Aleman

• Department of Mathematics and Computer Science, University of Richmond, Richmond, USA

  William T. Ross

• Department of Mathematics, Washington & Lee University, Lexington, USA

  Nathan S. Feldman

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