Spectral preservers and approximate spectral preservers on operator algebras

Let A and beta be unital primitive Banach algebras with minimal idempotents. We prove that every surjective spectral isometry from A onto beta is of the form lambda Psi where lambda is an element of C with vertical bar lambda vertical bar = 1 and Psi is either an isomorphism or an anti-isomorphism from A onto beta. As an application we show that, for all Banach spaces X and Y, the spectral nearisometries and the approximate spectrum-preserving maps from L(X) onto L(Y) are perturbations of actual spectral isometries and spectrum preserving maps, respectively. (C) 2016 Elsevier Inc. All rights reserved.