Nonlinear maps preserving the local spectral subspace

Abstract

Let X be a complex Banach space, and let be the space of all bounded operators on X. For and , let denote the local spectral subspace of T associated with . In this paper, we prove that if is a surjective weakly continuous map which satisfies for all and , there exists a nonzero scalar c such that for all . We do not assume to be linear, or even additive.