Approximately spectrum-preserving maps

Let X and Y be superreflexive complex Banach spaces and let B(X) and B(Y) be the Banach algebras of all bounded linear operators on X and Y, respectively. If a bijective linear map Φ:B(X)→B(Y) almost preserves the spectra, then it is almost multiplicative or anti-multiplicative. Furthermore, in the case where X=Y is a separable complex Hilbert space, such a map is a small perturbation of an automorphism or an anti-automorphism.