Additive Maps Preserving Nilpotent Perturbation of Scalars

Abstract

Let X be a Banach space over $\mathbb{F} (=\mathbb{R} \rm{or} \mathbb{C})$ with dimension greater than 2. Let $\mathcal{N}(X)$ be the set of all nilpotent operators and $\mathcal{B}_0(X)$ the set spanned by $\mathcal{N}(X)$. We give a structure result to the additive maps on $\mathbb{F}I+\mathcal{B}_0(X)$ that preserve rank-1 perturbation of scalars in both directions. Based on it, a characterization of surjective additive maps on $\mathbb{F}I+\mathcal{B}_0(X)$ 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 $T\in\mathbb{F}I+\mathcal{B}_0(X)$ or Φ(T) = cAT* A⁻¹ + ϕ(T)I for all $T\in\mathbb{F}I+\mathcal{B}_0(X)$, 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.