Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces
Published on 16. October 2012
Book
Paperback/Softback
XI, 232 pages
978-3-0348-9823-2 (ISBN)
Description
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.
More details
Series
Edition
Softcover reprint of the original 1st ed. 1997
Language
English
Place of publication
Basel
Switzerland
Publishing group
Springer Basel
Target group
Professional and scholarly
Research
Illustrations
XI, 232 p.
Dimensions
Height: 244 mm
Width: 170 mm
Thickness: 14 mm
Weight
442 gr
ISBN-13
978-3-0348-9823-2 (9783034898232)
DOI
10.1007/978-3-0348-8908-7
Schweitzer Classification
Other editions
Additional editions
Daniel Alpay | Aad Dijksma | James Rovnyak
Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces
Book
06/1997
1st Edition
Springer
€85.55
Article exhausted; check different version
Persons
Prof. Daniel Alpay is a faculty member of the department of mathematics at Ben-Gurion University, Beer-Sheva, Israel. He is the incumbent of the Earl Katz Family chair in algebraic system theory. He has a double formation of electrical engineer (Telecom Paris, graduated 1978) and mathematician (PhD, Weizmann Institute, 1986). His research includes operator theory, stochastic analysis, and the theory of linear systems. Daniel Alpay is one of the initiators and responsible of the dual track electrical-engineering mathematics at Ben-Gurion University. He is the author of "An Advanced Complex Analysis Problem Book" (Birkhäuser, 2015). Together with co-authors, he has written seven books and close to 240 research papers, and edited fifteen books of research papers, and in particular the Springer Reference Work on Operator Theory.
Content
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.