Maps preserving the inner local spectral radius zero of generalized product of operators

Abstract

Let $${\mathscr {B}}(X)$$ be the algebra of all bounded linear operators on a complex Banach space X. For an operator $$T\in {\mathscr {B}}(X)$$ , let $$\iota _{T}(x)$$ denote the inner local spectral radius of T at any vector $$x\in X$$ . For an integer $$k\ge 2$$ , let $$(i_1,\dots ,i_m)$$ be a finite sequence such that $$\{i_1,\dots ,i_m\}=\{1,\dots ,k\}$$ and at least one of the terms in $$(i_1,\dots ,i_m)$$ appears exactly once. The generalized product of k operators $$T_1,\dots ,T_k\in {\mathscr {B}}(X)$$ is defined by $$\begin{aligned} T_1*\cdots *T_k:=T_{i_1}T_{i_2}\dots T_{i_m}, \end{aligned}$$ and includes the usual product TS and the triple product TST. We show that a surjective map $$\varphi $$ on $${\mathscr {B}}(X)$$ satisfies $$\begin{aligned} \iota _{_{\varphi (T_1)*\cdots *\varphi (T_k)}}(x)=0 \Longleftrightarrow \iota _{ _{T_{1}*\cdots *T_{k}}}(x) = 0 \end{aligned}$$ for all $$x\in X$$ and all $$T_{1},\ldots ,T_{k} \in {\mathscr {B}}(X)$$ if and only if there exists a map $$\gamma : {\mathscr {B}}(X)\rightarrow {\mathbb {C}}\setminus \{0\} $$ such that $$\varphi (T)=\gamma (T) T$$ for all $$T\in {\mathscr {B}}(X)$$ .