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=(Aij)n×n is an operator matrix. Often the entries Aij are polynomials pij(A) with respect to a single (unbounded) operator A on E. 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 is 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.
A spectral mapping theorem for polynomial operator matrices
ENGEL, KLAUS JOCHEN OTTO
1989-01-01
Abstract
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=(Aij)n×n is an operator matrix. Often the entries Aij are polynomials pij(A) with respect to a single (unbounded) operator A on E. 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 is 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.Pubblicazioni consigliate
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