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

摘要

Let X and Y be superre.exive complex Banach spaces and let L(X) and L(Y ) be the Banach algebras of all bounded linear operators on X and Y , respectively. We describe a linear map φ : L(X) → L(Y ) that almost preserves the approximate point spectrum or the surjectivity spectrum. Furthermore, in the case where X = Y is a separable complex Hilbert space, we show that such a map is a small perturbation of an automorphism or an anti-automorphism.