Additive Maps Preserving Nilpotent Operators or Spectral Radius

Abstract

Let X be a (real or complex) Banach space with dimension greater than 2 and let ℬ0(X) be the subspace of ℬ(X) spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Φ on ℬ0(X) which preserve nilpotent operators in both directions. In particular, if X is infinite–dimensional, we prove that Φ has the form either Φ(T) = cATA−1 or Φ(T) = cAT'A−1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T' denotes the adjoint of T. As an application of these results, we show that every additive surjective map on ℬ(X) preserving spectral radius has a similar form to the above with |c| = 1.