On a Spectral Version of Cartan’s Theorem

Abstract

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