An Introduction to Operator Polynomials

This book provides an introduction to the modern theory of polynomials whose coefficients are linear bounded operators in a Banach space - operator polynomials. This theory has its roots and applications in partial differential equations, mechanics and linear systems, as well as in modern operator theory and linear algebra. Over the last decade, new advances have been made in the theory of operator polynomials based on the spectral approach. The author, along with other mathematicians, participated in this development, and many of the recent results are reflected in this monograph. It is a pleasure to acknowledge help given to me by many mathematicians. First I would like to thank my teacher and colleague, I. Gohberg, whose guidance has been invaluable. Throughout many years, I have worked wtih several mathematicians on the subject of operator polynomials, and, consequently, their ideas have influenced my view of the subject; these are I. Gohberg, M. A. Kaashoek, L. Lerer, C. V. M. van der Mee, P. Lancaster, K. Clancey, M. Tismenetsky, D. A. Herrero, and A. C. M. Ran. The following mathematicians gave me advice concerning various aspects of the book: I. Gohberg, M. A. Kaashoek, A. C. M. Ran, K. Clancey, J. Rovnyak, H. Langer, P.

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.