Non-linear spectrum-preserving maps

Let M-n denote the space of complex n x n matrices, and let Omega(n) denote the spectral unit ball of M-n, namely the set of matrices in M-n whose eig envalues lie within the open unit disc. As a step towards the eventual clas sification of the holomorphic automorphisms of Omega(n), we prove that ever y such automorphism F satisfying F(0) = 0 and F'(0) = I has the property th at F(x) is conjugate to x for each x is an element of Omega(n). This result is obtained by combining earlier work of Ransford and White with a general theorem about spectrum-preserving maps proved in this paper.