Linear operators preserving unitary similarity invariant norms

A norm N(·) on Hn the space of all n × n hermitian matrices, is unitary similarity invariant if N(UAU* ) = N(A) for all hermitian matrices A and all unitary matrices U. We characterize those linear isometries on Hn with respect to a unitary similarity invariant norm. A related result concerning the characterization of those linear operators on Hn which preserve the set of matrices whose vectors of eigenvalues belong to a certain compact subset of is also obtained.