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

Authors: MHAMED ELHODAIBI, SOMAYA SABER

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 $

for all $x \in X$ and $A,T \in \mathscr{B}(X)$.

Keywords: Nonlinear preservers, quasinilpotent part, local spectral radius, Jordan product

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