Additive Maps Preserving Nilpotent Operators or Spectral Radius

Let X be a (real or complex) Banach space with dimension greater than 2 and let .0(X) be the subspace of .(X) spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps . on .0(X) which preserve nilpotent operators in both directions. In particular, if X is infinite.dimensional, we prove that . has the form either .(T) = cATA .1 or .(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 .(X) preserving spectral radius has a similar form to the above with |c| = 1.