Norm estimates for function Lyapunov equations and applications

We consider the function Lyapunov equation $f^{*}\left(A\right)X+Xf\left(A\right)=C,$ 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 $\inf ?\sigma \left(f\right(A\left)\right)>0$. In the case $f\left(z\right)=z^{\nu }$ with a positive integer ν we obtain conditions that provide localization of the spectrum of a perturbed matrix in a given angle.