Additive maps preserving operators of inner local spectral radius zero

Abstract

Let \({\mathcal {B}}(X)\) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space \(X\). We determine the form of surjective additive maps \(\varphi :{\mathcal {B}}(X)\rightarrow {{\mathcal {B}}(X)}\) which preserve operators of inner local spectral radius zero at points of \(X\).