A spectral mapping theorem for polynomial operator matrices
Klaus-J. Engel · Differential and Integral Equations · 1989
Systems of linear evolution equations can be written as a single equation u(t) = Au(t), (*) where u is a function with values in a product space En and A = (A;j )nxn is an operator matrix.Often the entries A; 1 are polynomials Pii (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 a(A) of A. One expects a kind of spectral mapping theorem based on the spectrum a(A) of A and the structure of the matrix (p; 1 ).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.1. How to define an operator matrix?We study n x n matrices A whose entries are polynomials p; 1 (A) in a fixed-possibly unbounded-operator A on some Banach space E.For bounded A it is obvious that the operator matrix A defines a bounded operator on the product space £ := En.The situation for unbounded A is more complicated.In fact, the matrix A only induces a formal map but leaves open a wide choice of possible domains.If one wants "nice" properties of A, such as closedness for example, then a more careful analysis is necessary.