Functional calculus for diagonalizable matrices

Abstract

For an arbitrary function f:\Omega \rightarrow C (where \Omega is a subset of the field C) and a positive integer k let f act on all diagonalizable complex matrices whose all eigenvalues lie in Omega in the following way: f[P Diag(z1,...,zk) P-1] = P Diag(f(z1),...,f(zk)) P-1 for arbitrary numbers z1,...,zk in \Omega and an invertible k \times k matrix P. The aim of the paper is to fully answer the question of when the function fop defined above is continuous for fixed k. In particular, it is shown that if \Omega is open in C, then fop is continuous for fixed k > 2 iff f is holomorphic; and if \Omega is an interval in R and k > 2, then fop is continuous iff f is of class Ck-2(\Omega) and f(k-2) is locally Lipschitz in \Omega. Also a full characterization is given when the domain of f is arbitrary as well as when fop acts on infinite-dimensional (diagonalizable) matrices.