Resultant operators and the Bezout equation for analytic matrix functions, I

Abstract

This paper is a continuation of [4]. It contains our results on resultant operators for a family of matrix functions that are analytic in a finitely connected domain. The proofs are based on the results about the Bezout equation obtained in [4]. It is assumed that the reader is familiar with the concepts and results from [4]. Let f denote a Cauchy contour in the complex plane consisting of several non intersecting simple smooth closed contours which form the positively oriented boundary of a finitely connected bounded domain A + Throughout this chapter we assume for convenience that 0 E A + The general case can be obtained by a shift in the complex plane. Let A, ,..., A, be regular n x n matrix functions which are analytic in A + and continuous in A + u r, and assume that C(A,) n r= $3. The paper consists of two sections. In Section 1 we show that the role of the resultant operator of the matrix functions A,,..., A, is played by the operator R,: L;;(T) + L;(/‘) (1 <p<co) defined by