Norm estimates for function Lyapunov equations and applications

摘要

We consider the function Lyapunov equation f*(A)X + Xf(A) = C, where A and C are given matrices, f(z) is a function holomorphic on a neighborhood of the spectrum sigma(A) of A. For a solution X of that equation, norm estimates are established. By these estimates we investigate perturbations of a matrix A whose spectrum satisfies the condition inf R sigma(f (A)) 0. In the case f (z) = z(v) with a positive integer v we obtain conditions that provide localization of the spectrum of a perturbed matrix in a given angle. (C) 2018 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.