Linear Maps Preserving Operators of Local Spectral Radius Zero

Abstract

Let X be a complex Banach space and let ${\mathcal{B}(X)}$ be the space of all bounded linear operators on X. For ${x \in X}$ and ${T \in \mathcal{B}(X)}$, let ${r_{T}(x) =\limsup_{n \rightarrow \infty} \| T^{n}x\| ^{1/n}}$ denote the local spectral radius of T at x. We prove that if ${\varphi : \mathcal{B}(X) \rightarrow \mathcal{B}(X)}$ is linear and surjective such that for every ${x \in X}$ we have r _T (x) = 0 if and only if ${r_{\varphi(T)}(x) = 0}$, there exists then a nonzero complex number c such that ${\varphi(T) = cT}$ for all ${T \in \mathcal{B}(X) }$. We also prove that if Y is a complex Banach space and ${\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)}$ is linear and invertible for which there exists ${B \in \mathcal{B}(Y, X)}$ such that for ${y \in Y}$ we have r _T (By) = 0 if and only if ${ r_{\varphi ( T) }(y)=0}$, then B is invertible and there exists a nonzero complex number c such that ${\varphi(T) =cB^{-1}TB}$ for all ${T \in \mathcal{B}(X)}$.