Resultant and Bezoutian Operators

Abstract

We introduce and study operator generalizations of the classical resultant and Bezoutian matrices. It turns out that the theory of spectral pairs developed in Chapter 6 provides a very convenient tool in analysis of the operator resultant and Bezoutian. Many well-known properties of the classical Bezoutian and resultant can be extended to the operator theory framework. Subsequently, the Bezoutian operator is used to prove certain spectrum separation theorems for operator polynomials. In addition to the Bezoutian operator, we need also inertia theorems for Hilbert space operators. These are exposed in Section 9.5.