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. © 2011 Elsevier Inc. All rights reserved.