Maps preserving the local spectral subspace of product or Jordan triple product of operators

Abstract

Let $\mathcal {L}(X)$ be the algebra of all bounded linear operators on an infinite-dimensional Banach space X. Let $\lambda _0$ be a fixed complex scalar. Let $X_T(\{\lambda _0 \})$ denote the local spectral subspace of an operator $T\in \mathcal {L}(X)$ associated with $\{\lambda _0\}$. The purpose of this paper is to characterize maps $\phi$ on $\mathcal {L}(X)$ that satisfy

$$\begin{aligned} X_{\phi (T)\phi (S)}(\{\lambda _0\})=X_{TS}(\{ \lambda _0\}) \text { for all } T,S \in \mathcal {L}(X). \end{aligned}$$

We also characterize maps $\phi$ on $\mathcal {L}(X)$ for which the range contains all operators of rank at most four and

$$\begin{aligned} X_{\phi (T)\phi (S)\phi (T)}(\{\lambda _0\})=X_{TST}(\{ \lambda _0\}) \text { for all } T,S \in \mathcal {L}(X). \end{aligned}$$