An introduction to operator polynomials

Content: 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.