Additive Maps Preserving Nilpotent Perturbation of Scalars

Let X be a Banach space over F (=R or C) with dimension greater than 2. Let N (X) be the set of all nilpotent operators and B₀(X) the set spanned by N (X). We give a structure result to the additive maps on FI + B₀(X) that preserve rank-1 perturbation of scalars in both directions. Based on it, a characterization of surjective additive maps on FI + B₀(X) that preserve nilpotent perturbation of scalars in both directions are obtained. Such a map Φ has the form either Φ(T)=cATA^-1 +φ(T)I for all T ∈ FI + B₀(X) or Φ(T)=cAT^* A^-1 + φ(T)I for all T ∈ FI + B₀(X), where c is a nonzero scalar, A is a τ-linear bijective transformation for some automorphism τ of F and φ is an additive functional. In addition, if dim X=∞, then A is in fact a linear or conjugate linear invertible bounded operator.