Preservers of the local spectral radius zero of Jordan product of operators

Abstract

Let $\mathscr{B}(X)$ be the algebra of all bounded linear operators on an infinite-dimensional complex Banach space $X$, and denote by $r_{T}(x)$ the local spectral radius of any operator $T \in \mathscr{B}(X)$ at any vector $x \in X$. In this paper, we characterize surjective maps $\phi$ on $ \mathscr{B}(X)$ satisfying $ r_{\phi(T)\phi(A) + \phi(A)\phi(T)}(x)=0$ if and only if $ r_{TA+AT}(x)=0 $