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Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces

  • Daniel Alpay
  • Aad Dijksma
  • James Rovnyak
  • Hendrik de Snoo

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

Table of contents

  1. Front Matter
    Pages i-xi
  2. Daniel Alpay, Aad Dijksma, James Rovnyak, Hendrik de Snoo
    Pages 1-40
  3. Daniel Alpay, Aad Dijksma, James Rovnyak, Hendrik de Snoo
    Pages 41-82
  4. Daniel Alpay, Aad Dijksma, James Rovnyak, Hendrik de Snoo
    Pages 83-128
  5. Daniel Alpay, Aad Dijksma, James Rovnyak, Hendrik de Snoo
    Pages 129-184
  6. Back Matter
    Pages 185-235

About this book

Introduction

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.

Keywords

Operator theory differential equation function theory linear system theory mathematics system theory

Authors and affiliations

  • Daniel Alpay
    • 1
  • Aad Dijksma
    • 2
  • James Rovnyak
    • 3
  • Hendrik de Snoo
    • 2
  1. 1.Department of MathematicsBen-Gurion University of the NegevBeer-ShevaIsrael
  2. 2.Department of MathematicsUniversity of GroningenGroningenThe Netherlands
  3. 3.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

Bibliographic information

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