Jordan product and local spectrum preservers
Volume 234 / 2016
Studia Mathematica 234 (2016), 97-120
MSC: Primary 47B49; Secondary 47A10, 47A11.
DOI: 10.4064/sm8240-6-2016
Published online: 23 August 2016
Abstract
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 $T\in {\mathscr B}(X)$.
