Schur functions, operator colligations, and reproducing kernel Pontryagin spaces

Daniel Alpay, Aad Dijksma, James Rovnyak, hendrik de Snoo

Research output: Book/ReportBookpeer-review


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.
Original languageEnglish
Place of PublicationBasel, Switzerland
Number of pages229
ISBN (Electronic)3034889089, 3764357630
ISBN (Print)3764357630
StatePublished - 1997

Publication series

NameOperator theory, advances and applications


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


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