Linear preservers on upper triangular operator matrix algebras

Abstract

In this paper, we obtain several characterizations of rank preserving linear maps and completely rank nonincreasing linear maps on upper triangular Hilbert space operator matrix algebras and apply them to get some algebraic results. We show that every automorphism of an upper triangular operator matrix algebra is inner and every weakly continuous surjective local automorphism is in fact an automorphism. A weakly continuous linear bijection on an upper triangular operator matrix algebra is idempotent preserving if and only if it is a Jordan homomorphism, and in turn, if and only if it is an automorphism or an anti-automorphism. As an application, we also obtain a result concerning the asymptotic joint-similarity of matrix tuples.