Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces

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.

1: Pontryagin Spaces and Operator Colligations.- 1.1 Reproducing kernel Pontryagin spaces.- 1.2 Operator colligations.- 1.3 Julia operators and contractions.- 1.4 Extension of densely defined linear relations.- 1.5 Complementation and reproducing kernels.- 2: Schur Functions and their Canonical Realizations.- 2.1 Pontryagin spaces ?(S), ?($$ \widetilde{S} $$ ), and D(S).- 2.2 Canonical coisometric and isometric realizations.- 2.3 Canonical unitary realization.- 2.4 Unitary dilations of coisometric and isometric colligations.- 2.5 Classes SK(F,B).- 3: The State Spaces.- 3.1 Invariance under difference quotients.- 3.2 Spaces ?(S).- 3.3 Spaces ?$$ \widetilde{S} $$.- 3.4 Spaces D(S).- 3.5 Examples and miscellaneous results.- 4: Structural Properties.- 4.1 Factorization and invariant subspaces.- 4.2 Kre?n-Langer factorization.- 4.3 The Potapov-Ginzburg transform.- 4.4 Applications to the realization theory.- 4.5 Canonical models.- Epilogue: Open Questions and Directions for Further Work.- Appendix: Some Finite-Dimensional Spaces.- Notes.- References.- Notation Index.- Author Index.