Linear Maps Preserving Operators of Inner Local Spectral Radius Zero at Some Fixed Vector

Abstract

Let X be a complex Banach space and let $$x_{0}\in X$$ be a fixed nonzero vector. Denote by $$\mathcal {L}\left( X\right) $$ the algebra of all linear and bounded operators on X, and for $$T \in \mathcal {L}\left( X\right) $$ denote by $$\sigma _{T}\left( x_{0}\right) $$ the local spectrum of T at $$x_{0}$$ . We characterize linear and surjective maps $$\varphi :\mathcal {L} \left( X\right) \rightarrow \mathcal {L}\left( X\right) $$ such that $$\varphi \left( I\right) \in \mathcal {L}\left( X\right) $$ is invertible and $$\begin{aligned} 0\in \sigma _{T}\left( x_{0}\right) \Longleftrightarrow 0\in \sigma _{\varphi \left( T\right) }\left( x_{0}\right) \qquad \left( T\in \mathcal {L}\left( X\right) \right) . \end{aligned}$$ .