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） = cAT A-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.