Author | Rodman, Leiba. author |
---|---|
Title | An Introduction to Operator Polynomials [electronic resource] / by Leiba Rodman |
Imprint | Basel : Birkhรคuser Basel, 1989 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-9152-3 |
Descript | XII, 391 p. online resource |
1. Linearizations -- 1.1 Definitions and examples -- 1.2 Uniqueness of linearization -- 1.3 Existence of linearizations -- 1.4 Operator polynomials that are multiples of identity modulo compacts -- 1.5 Inverse linearization of operator polynomials. -- 1.6 Exercises -- 1.7 Notes -- 2. Representations and Divisors of Monic Operator Polynomials -- 2.1 Spectral pairs -- 2.2 Representations in terms of spectral pairs -- 2.3 Linearizations -- 2.4 Generalizations of canonical forms -- 2.5 Spectral triples -- 2.6 Multiplication and division theorems -- 2.7 Characterization of divisors in terms of subspaces -- 2.8 Factorable indexless polynomials -- 2.9 Description of the left quotients -- 2.10 Spectral divisors -- 2.11 Differential and difference equations -- 2.12 Exercises -- 2.13 Notes -- 3. Vandermonde Operators and Common Multiples -- 3.1 Definition and basic properties of the Vandermonde operator -- 3.2 Existence of common multiples -- 3.3 Common multiples of minimal degree -- 3.4 Fredholm Vandermonde operators -- 3.5 Vandermonde operators of divisors -- 3.6 Divisors with disjoint spectra -- Appendix: Hulls of operators -- 3.7 Application to differential equations -- 3.8 Interpolation problem -- 3.9 Exercises -- 3.10 Notes -- 4. Stable Factorizations of Monic Operator Polynomials -- 4.1 The metric space of subspaces in a Banach space -- 4.2 Spherical gap and direct sums -- 4.3 Stable invariant subspaces -- 4.4 Proof of Theorems 4.3.3 and 4.3.4 -- 4.5 Lipschitz stable invariant subspaces and one-sided resolvents -- 4.6 Lipschitz continuous dependence of supporting subspaces and factorizations -- 4.7 Stability of factorizations of monic operator polynomials -- 4.8 Stable sets of invariant subspaces -- 4.9 Exercises -- 4.10 Notes -- 5. Self-Adjoint Operator Polynomials -- 5.1 Indefinite scalar products and subspaces. -- 5.2 J-self-adjoint and J-positizable operators -- 5.3 Factorizations and invariant semidefinite subspaces -- 5.4 Classes of polynomials with special factorizations -- 5.5 Positive semidefinite operator polynomials -- 5.6 Strongly hyperbolic operator polynomials -- 5.7 Proof of Theorem 5.6.4 -- 5.8 Invariant subspaces for unitary and self-adjoint operators in indefinite scalar products -- 5.9 Self-adjoint operator polynomials of second degree -- 5.10 Exercises -- 5.11 Notes -- 6. Spectral Triples and Divisibility of Non-Monic Operator Polynomials -- 6.1 Spectral triples: definition and uniqueness -- 6.2 Calculus of spectral triples -- 6.3 Construction of spectral triples -- 6.4 Spectral triples and linearization -- 6.5 Spectral triples and divisibility -- 6.6 Characterization of spectral pairs -- 6.7 Reduction to monic polynomials -- 6.8 Exercises -- 6.9 Notes -- 7. Polynomials with Given Spectral Pairs and Exactly Controllable Systems -- 7.1 Exactly controllable systems -- 7.2 Spectrum assignment theorems -- 7.3 Analytic dependence of the feedback -- 7.4 Polynomials with given spectral pairs -- 7.5 Invariant subspaces and divisors -- 7.6 Exercises -- 7.7 Notes -- 8. Common Divisors and Common Multiples -- 8.1 Common divisors -- 8.2 Common multiples -- 8.3 Coprimeness and Bezout equation -- 8.4 Analytic behavior of common multiples -- 8.5 Notes -- 9. Resultant and Bezoutian Operators -- 9.1 Resultant operators and their kernel -- 9.2 Proof of Theorem 9.1.4 -- 9.3 Bezoutian operator -- 9.4 The kernel of a Bezoutian operator -- 9.5 Inertia theorems -- 9.6 Spectrum separation -- 9.7 Spectrum separation problem: deductions and special cases -- 9.8 Applications to difference equations -- 9.9 Notes -- 10. Wiener-Hopf Factorization -- 10.1 Definition and the main result -- 10.2 Pairs of finite type and proof of Theorem 10.1.1 -- 10.3 Finite-dimensional perturbations -- 10.4 Notes -- References -- Notation