Rank-1 preserving linear maps on nest algebras

In this paper, we characterize rank-1 preserving linear maps between nest algebras acting on real or complex Banach spaces. As applications, we show that every weakly continuous and surjective local automorphism (or, anti-automorphism) on a nest algebra with an additional property is either an automorphism or an anti-automorphism; furthermore, every weakly continuous and surjective local inner automorphism on such nest algebras is in fact an inner automorphism.