Maps Preserving the Spectrum of Skew Lie Product of Operators

Eman Alzedani, Mohamed Mabrouk

Let $\lh$ denote the algebra of all bounded linear operators acting on a complex Hilbert space $\h$. In this paper, we show that a surjective map $\vf$ on $\lh$ satisfies \[ igmaeft(ǎrphi(T)ǎrphi(S)-ǎrphi(S)ǎrphi(T)^*\right)=igmaeft(TS-ST^*\right),\quad T,Sıh, \] if and only if there exists a unitary operator $U\in \lh$ such that \[ǎrphi(T)=ambda UTU^{*}, \quad Tıh,\] where $\lambda\in\left\{-1, 1\right\}$.