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Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces

  • Book
  • © 1997

Overview

Authors:
  1. Daniel Alpay

    1. Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel

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  2. Aad Dijksma

    1. Department of Mathematics, University of Groningen, Groningen, The Netherlands

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  3. James Rovnyak

    1. Department of Mathematics, University of Virginia, Charlottesville, USA

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  4. Hendrik Snoo

    1. Department of Mathematics, University of Groningen, Groningen, The Netherlands

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Part of the book series: Operator Theory: Advances and Applications (OT, volume 96)

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Table of contents (4 chapters)

  1. Front Matter

    Pages i-xi
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  2. Pontryagin Spaces and Operator Colligations

    • Daniel Alpay, Aad Dijksma, James Rovnyak, Hendrik de Snoo
    Pages 1-40

  3. Schur Functions and Their Canonical Realizations

    • Daniel Alpay, Aad Dijksma, James Rovnyak, Hendrik de Snoo
    Pages 41-82

  4. The State Spaces

    • Daniel Alpay, Aad Dijksma, James Rovnyak, Hendrik de Snoo
    Pages 83-128

  5. Structural Properties

    • Daniel Alpay, Aad Dijksma, James Rovnyak, Hendrik de Snoo
    Pages 129-184

  6. Back Matter

    Pages 185-235
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Keywords

About this book

Generalized Schur functions are scalar- or operator-valued holomorphic functions such that certain associated kernels have a finite number of negative squares. This book develops the realization theory of such functions as characteristic functions of coisometric, isometric, and unitary colligations whose state spaces are reproducing kernel Pontryagin spaces. This provides a modern system theory setting for the relationship between invariant subspaces and factorization, operator models, Krein-Langer factorizations, and other topics. The book is intended for students and researchers in mathematics and engineering. An introductory chapter supplies background material, including reproducing kernel Pontryagin spaces, complementary spaces in the sense of de Branges, and a key result on defining operators as closures of linear relations. The presentation is self-contained and streamlined so that the indefinite case is handled completely parallel to the definite case.

Authors and Affiliations

  • Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel

    Daniel Alpay

  • Department of Mathematics, University of Groningen, Groningen, The Netherlands

    Aad Dijksma, Hendrik Snoo

  • Department of Mathematics, University of Virginia, Charlottesville, USA

    James Rovnyak

Bibliographic Information

  • Book Title: Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces

  • Authors: Daniel Alpay, Aad Dijksma, James Rovnyak, Hendrik Snoo

  • Series Title: Operator Theory: Advances and Applications

  • DOI: https://doi.org/10.1007/978-3-0348-8908-7

  • Publisher: Birkhäuser Basel

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Basel AG 1997

  • Hardcover ISBN: 978-3-7643-5763-4Published: 15 July 1997

  • Softcover ISBN: 978-3-0348-9823-2Published: 16 October 2012

  • eBook ISBN: 978-3-0348-8908-7Published: 06 December 2012

  • Series ISSN: 0255-0156

  • Series E-ISSN: 2296-4878

  • Edition Number: 1

  • Number of Pages: XI, 232

  • Topics: Analysis, Engineering, general

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