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Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces

About Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces

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.- A. Complementation in the sense of de Branges.- B. Applications to reproducing kernel Pontryagin spaces.- 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).- A. Definition and basic properties.- B. Conformally invariant view.- C. Application to factorization of operator-valued functions.- D. A non-holomorphic kernel.- 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.- A. Rational unitary functions.- B. Symmetry in the state spaces.- C. Some consequences of Leech's theorem.- D. The scalar case: S(z)? ?(S) if and only if $$ \widetilde{S} $$(z) ? ?($$ \widetilde{S} $$).- 4: Structural Properties.- 4.1 Factorization and invariant subspaces.- A. Inclusion of spaces ?(S).- B. Inclusion of spaces D (S).- 4.2 Kre?n-Langer factorization.- A. Existence and properties.- B. Strongly regular representations.- 4.3 The Potapov-Ginzburg transform.- 4.4 Applications to the realization theory.- A. Kre?n space inner and outer spaces F andB.- B. Other base points.- C. Examples.- 4.5 Canonical models.- Epilogue: Open Questions and Directions for Further Work.- Appendix: Some Finite-Dimensional Spaces.- Notes.- References.- Notation Index.- Author Index

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  • Language:
  • English
  • ISBN:
  • 9783764357634
  • Binding:
  • Hardback
  • Pages:
  • 252
  • Published:
  • July 14, 1997
  • Dimensions:
  • 170x244x16 mm.
  • Weight:
  • 603 g.
Delivery: 2-3 weeks
Expected delivery: January 12, 2025
Extended return policy to January 30, 2025
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Description of Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces

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.- A. Complementation in the sense of de Branges.- B. Applications to reproducing kernel Pontryagin spaces.- 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).- A. Definition and basic properties.- B. Conformally invariant view.- C. Application to factorization of operator-valued functions.- D. A non-holomorphic kernel.- 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.- A. Rational unitary functions.- B. Symmetry in the state spaces.- C. Some consequences of Leech's theorem.- D. The scalar case: S(z)? ?(S) if and only if $$ \widetilde{S} $$(z) ? ?($$ \widetilde{S} $$).- 4: Structural Properties.- 4.1 Factorization and invariant subspaces.- A. Inclusion of spaces ?(S).- B. Inclusion of spaces D (S).- 4.2 Kre?n-Langer factorization.- A. Existence and properties.- B. Strongly regular representations.- 4.3 The Potapov-Ginzburg transform.- 4.4 Applications to the realization theory.- A. Kre?n space inner and outer spaces F andB.- B. Other base points.- C. Examples.- 4.5 Canonical models.- Epilogue: Open Questions and Directions for Further Work.- Appendix: Some Finite-Dimensional Spaces.- Notes.- References.- Notation Index.- Author Index

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