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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.
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.
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