Zolotarev V.A. Analytic Methods of Spectral Representations of Non-Selfadjoint (Non-Unitary) Operators

Cham: Springer, 2025. — 550 p.Preface
Local Colligations and Model Representations of Linear Non-Self-Adjoint Bounded Operators
Local Colligations. Open Systems. Characteristic Functions
Dilations of Semigroups. The Scattering Operator
Translational and Functional Models
Diagonal of an Open System. Existence Theorem
Triangular Models
Universal and Functional Models of Dissipative Colligations
Universal Models of Linear Operators with Restrictions on the Resolvent Growth
De Branges Spaces of Entire Functions. Functional Models of Non-dissipative Operators
Potapov-Ginzburg Transforms and Functional Models of Non-dissipative Operators
Triangular Model of an Operator with Infinite-Dimensional Component
Jacobi Operators. Matrix-Valued Polynomials. De Branges Vector Spaces
Abstract Scattering Problem
Unitary Metric Colligations and Their Model Representations
Metric Operator Colligations, Open Systems, and Their Characteristic Functions
Existence Theorem
Functional Model
Triangular Models
The Cayley Transform of a Unitary Metric and Local Colligations. Diagonal of a Discrete Open System
Scattering Scheme for J-Unitary Dilations That Correspond to a Unitary Metric Colligation
Colligations Corresponding to Unbounded Operators and Their Model Representations
Dilations of Strongly Continuous Contraction Semigroups. Local Colligation and Open Systems for Unbounded Operators
Main Properties of the Characteristic Function of an Unbounded Operator
Scattering Scheme. Translational and Functional Models of Strongly Continuous Contractive Semigroups
Triangular Models
Elements of V.P. Potapov´s J-Theory and the Factorization Problem of J-Contractive Matrix Functions
Block Matrix Lemma. Schwarz-Pick Inequalities
Infinite Multiplicative Products and Stieltjes Multiplicative Integrals
The Modulus Theorem
Detachment of Elementary Factors
Multiplicative Representation of J-Non-contractive Matrix Functions
Comments and Remarks
Appendix
Second-Order Open Systems
Functional Model of a Self-Adjoint Operator
Model Representations of Quadratic Pencils
Characteristic Functions
References
Index