On a Spectral Version of Cartan's Theorem

For a domain Omega in the complex plane, we consider the domain S-n(Omega) consisting of those n x n complex matrices whose spectrum is contained in Omega. Given a holomorphic self-map Psi of S-n(Omega) such that Psi(A) = A and the derivative of Psi at A is identity for some A is an element of S-n(Omega), we investigate when the map Psi would be spectrum-preserving. We prove that if the matrix A is either diagonalizable or non-derogatory then for most domains Omega, Psi is spectrum-preserving on S-n(Omega). Further, when A is arbitrary, we prove that Psi is spectrum-preserving on a certain analytic subset of S-n(Omega).