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

ABSTRACT

Let $\mathcal{B}\left(X\right)$ be the algebra of all bounded linear operators on a complex Banach space X. For an operator $T\in \mathcal{B}\left(X\right)$, let $r_T\left(x\right):=\underset{n\to \mathrm{\infty }}{lim sup}\parallel T^{n}x{\parallel }^{1/n}$ be the 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 $T_1,\dots ,T_k\in \mathcal{B}\left(X\right)$ is defined by $T_1\ast \dots \ast T_k:=T_{i_1}{T}_{i_2}\dots T_{i_m},$ and includes the usual product TS and the triple product TST. We show that a surjective map ϕ on $\mathcal{B}\left(X\right)$ satisfies $r_{{}_{\varphi \left(T_1\right)\ast \cdots \ast \varphi \left(T_k\right)}}\left(x\right)=0⟺r_{{}_{T_1\ast \cdots \ast T_k}}\left(x\right)=0$ for all $x\in X$ and all $T_1,\dots ,T_k\in \mathcal{B}\left(X\right)$ if and only if there exists a map $\gamma :\mathcal{B}\left(X\right)\to \mathbb{C}\setminus \left\{0\right\}$ such that $\varphi \left(T\right)=\gamma \left(T\right)T$ for all $T\in \mathcal{B}\left(X\right)$.

KEYWORDS:

• Nonlinear preservers
• quasi-nilpotent part of an operator
• local spectral radius
• generalized products

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• Primary 47B49
• Secondary 47A10
• 47A11

Acknowledgments

Thanks are due to the referee for his/her careful reading of the manuscript and some helpful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Zine El Abidine Abdelali  http://orcid.org/0000-0003-1372-588X