A spectral mapping theorem for polynomial operator matrices

Abstract

Systems of linear evolution equations can be written as a single equation (t) = Au(t), where u is a function with values in a product space En and A = (Aij)nXn is an operator matrix. Often the entries Aij are polynomials Pij (A) with respect to a single (unbounded) operator A onE (see, e.g., [1], [2], [3], [6], [11]). In order to solve (*) one has to determine the properties of the operator matrix A. In particular one has to find an appropriate domain D(A) such that A is closed. This will be discussed in the first part of this paper. Then it is important to compute the spectrum σ(A) of A. One expects a kind of spectral mapping theorem based on the spectrum σ(A) of A and the structure of the matrix (pij). We show in Part 2 in which sense such a spectral mapping theorem holds. An application to stability theory, i.e., the computation of an estimate for the spectral bound s(A) concludes this paper. In a subsequent paper we discuss which operator matrices (Pij (A)) generate strongly continuous semigroups on En and give applications to systems of differential equations. © 1989, Khayyam Publishing. All rights reserved.

Publication Title

Differential and Integral Equations

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