Norm estimates for function Lyapunov equations and applications

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 σ(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 infℜσ(f(A))>0. In the case f(z)=zν with a positive integer ν we obtain conditions that provide localization of the spectrum of a perturbed matrix in a given angle.