Additive Maps Preserving Similarity of Operators on Banach Spaces

Abstract

Let X be an infinite–-dimensional complex Banach space and denote by ℬ(X) the algebra of all bounded linear operators acting on X. It is shown that a surjective additive map Φ from ℬ(X) onto itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible linear or conjugate linear operator A and a 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 where X has infinite multiplicity, in particular, when X is an infinite–dimensional Hilbert space, the above similarity invariant additive functional φ is always zero.