Inner local spectral radius preservers

Abstract

Let \({\mathcal {L}}(X)\) be the Banach algebra of all bounded linear operators on a complex Banach space X. For an operator \(T\in {\mathcal {L}}(X)\), let \(\iota _T(x)\) denote the inner local spectral radius of T at any vector x in X. We characterize maps \(\phi \) (not necessarily linear nor surjective) on \({\mathcal {L}}(X)\) which satisfy $$\begin{aligned} \iota _{T-S} (x)=0 \text{ if } \text{ and } \text{ only } \text{ if }\ \ \iota _{\phi (T)-\phi (S)}(x)=0 \end{aligned}$$ for every \(x\in X\) and \(T, S\in {\mathcal {L}}(X)\). We also describe surjective linear maps \(\phi \) on \({\mathcal {L}}(X)\) for which \(\phi (I)\) is invertible and either $$\begin{aligned} \iota _{T}(x)=0 \Longrightarrow \iota _{\phi (T)}(x)=0 \end{aligned}$$ for every \(x\in X\) and \(T\in {\mathcal {L}}(X)\), or $$\begin{aligned} \iota _{\phi (T)}(x)=0 \Longrightarrow \iota _{T}(x)=0 \end{aligned}$$ for every \(x\in X\) and \(T\in {\mathcal {L}}(X)\).