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Book review

Author(s) Rodman, Leiba
Title An Introduction to Operator Polynomials
Publisher Birkhäuser
Year of publication 1989
Reviewed by Michal Zajac

This book is an introduction to the modern theory of polynomials (in complex variable) whose coefficients are bounded linear operators in a Banach space. A part of exposition is parallel to the finite-dimensional case presented in the book I. Gohberg, P. Lancaster, L. Rodman: Matrix Polynomials, Academic Press, New York 1982. The main topics treated in the book are linearizations and factorizations of a polynomial into the product of polynomials of lower degrees. The nice correspondence between the factorizations of an operator polynomial and the invariant subspaces of its linearization is treated as well.

The book consists of 10 chapters: Linearizations, Representations and divisors of monic polynomials, Vandermonde operators and common multiples, Stable factorizations of monic polynomials, Self-adjoint operator polynomials, Spectral triples and divisibility of non-monic polynomials, Polynomials with given spectral pairs and exactly controllable systems, Common divisors and common multiples, Resultant and Bezoutian operators, Wiener-Hopf factorization.

The book is accessible for graduate students after a basic course in operator theory. To understand the exposition the background in the following topics is needed: Fredholm operators, compact operators in Banach spaces, spectral theorem for bounded self-adjoint operators, the closed graph theorem and its applications, basics in Banach algebras.

The author is one of the best expert in the field. The book is probably as much self-contained as it can be. Besides a clear exposition of the theory of operator polynomials it contains also applications in differential equations, interpolation and in system theory. There are many exercises in the book, too. This makes the text suitable for a graduate course. There is a sufficiently detailed table of contents in the book, on the other hand the index contains only a few notions.

Surely every good mathematical library should have this book. It is also interesting for control and system engineers. Probably many specialists in operator theory and differential equations should buy it for their personal library.