On linear maps approximately preserving the approximate point spectrum or the surjectivity spectrum

Abstract

Let $X$ and $Y$ be superreflexive complex Banachspaces and let $\mathcal{L}\mathcal{(}\mathcal{X}\mathcal{)}$ and $\mathcal{L}\mathcal{(}\mathcal{Y}\mathcal{)}$ be the Banachalgebras of all bounded linear operators on $X$ and $Y$,respectively. We describe a linear map $\varphi :\mathcal{L}\mathcal{(}\mathcal{X}\mathcal{)}\to \mathcal{L}\mathcal{(}\mathcal{Y}\mathcal{)}$ that almost preserves the approximate pointspectrum or the surjectivity spectrum. Furthermore, in the casewhere $X=Y$ is a separable complex Hilbert space, we show thatsuch a map is a small perturbation of an automorphism or ananti-automorphism.