Spectral preservers and approximate spectral preservers on operator algebras

Let AA and BB be unital primitive Banach algebras with minimal idempotents. We prove that every surjective spectral isometry from AA onto BB is of the form λ  Ψ where λ∈Cλ∈C with |λ|=1|λ|=1 and Ψ is either an isomorphism or an anti-isomorphism from AA onto BB. 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)L(X) onto L(Y)L(Y) are perturbations of actual spectral isometries and spectrum-preserving maps, respectively.