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Jordan product and local spectrum preservers

Volume 234 / 2016

Abdellatif Bourhim, Mohamed Mabrouk Studia Mathematica 234 (2016), 97-120 MSC: Primary 47B49; Secondary 47A10, 47A11. DOI: 10.4064/sm8240-6-2016 Published online: 23 August 2016


Let $X$ and $Y$ be two infinite-dimensional complex Banach spaces, and fix two nonzero vectors $x_0\in X$ and $y_0\in Y$. Let ${\mathscr B}(X)$ (resp. ${\mathscr B}(Y)$) denote the algebra of all bounded linear operators on $X$ (resp. on $Y$). We show that a map $\varphi $ from ${\mathscr B}(X)$ onto ${\mathscr B}(Y)$ satisfies \[ \sigma _{\varphi (T)\varphi (S)+\varphi (S)\varphi (T)}(y_0) =\sigma _{TS+ST}(x_0)\ \hskip 1em (T,S\in {\mathscr B}(X)) \] if and only if there exists a bijective bounded linear mapping $A$ from $X$ into $Y$ such that $Ax_0=y_0$ and either $\varphi (T)= ATA^{-1}$ for all $T\in {\mathscr B}(X)$ or $\varphi (T)=- ATA^{-1}$ for all $T\in {\mathscr B}(X)$.


  • Abdellatif BourhimDepartment of Mathematics
    Syracuse University
    215 Carnegie Building
    Syracuse, NY 13244, U.S.A.
  • Mohamed MabroukDepartment of Mathematics
    College of Applied Sciences
    P.O. Box 715
    Makkah 21955, KSA
    Department of Mathematics
    Faculty of Sciences of Gabès
    University of Gabès
    Cité Erriadh
    6072 Zrig, Gabès, Tunisia

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