Additive maps preserving nilpotent operators or spectral radius

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