Asymptotic similarity preserving additive maps on B(X) ✩

Let X be an infinite dimensional complex Banach space and denote B(X) the algebra of all bounded linear operators acting on X. We show that an additive surjective map on B(X) preserves asymptotic similarity in both directions if and only if there exist a nonzero scalar c, an invertible bounded linear or conjugate linear operator A and an asymptotic similarity invariant additive functional φ on B(X) such that either Φ(T ) = cATA−1 + φ(T )I for all T or Φ(T ) = cAT ∗A−1 +φ(T )I for all T . In the case that X has infinite multiplicity, especially if X is an infinite dimensional Hilbert space, above asymptotic similarity invariant additive functional φ is always zero.  2003 Elsevier Inc. All rights reserved.