Local Spectral Radius Preservers

Abstract

We characterize surjective maps on \({\fancyscript{B}(x)}\) , the space of all bounded operators on an infinite-dimensional complex Banach space X, which satisfy “rT-S(x) = 0 if and only if \({{{\rm r}_{\varphi(T)-\varphi(S)}(x) = 0}}\) for every \({x \in X}\) and \({{S, T \in \fancyscript{B}(x)}}\) ”. We do not assume \({\varphi}\) to be linear, or even additive, and thus this characterization is a step forward in generalizing some preceding results.