Jordan product and local spectrum preservers

Abdellatif Bourhim ¹   ; Mohamed Mabrouk ²

¹ Department of Mathematics Syracuse University 215 Carnegie Building Syracuse, NY 13244, U.S.A.
² Department of Mathematics College of Applied Sciences P.O. Box 715 Makkah 21955, KSA and Department of Mathematics Faculty of Sciences of Gabès University of Gabès Cité Erriadh 6072 Zrig, Gabès, Tunisia

Studia Mathematica, Tome 234 (2016) no. 2, pp. 97-120

Résumé

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 $T\in {\mathscr B}(X)$.

Détail

DOI : 10.4064/sm8240-6-2016

Keywords: infinite dimensional complex banach spaces fix nonzero vectors mathscr resp mathscr denote algebra bounded linear operators resp map varphi mathscr mathscr satisfies sigma varphi varphi varphi varphi sigma hskip mathscr only there exists bijective bounded linear mapping either varphi ata mathscr varphi ata mathscr