Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces

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

   PDF

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.

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

• 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

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