On Maps that Preserve the Lie Products Equal to Fixed Elements

Abstract

This work characterizes the general form of a bijective linear map $\Psi:\mathscr{M}_n(\mathbb{C}) \to \mathscr{M}_n(\mathbb{C})$ such that $[\Psi(A_1),~\Psi(A_2)]=D_2$ whenever $[A_1,~A_2]=D_1$ where $D_1~\text{and}~D_2$ are fixed matrices. Additionally, let $\mathscr{H}_1$ and $\mathscr{H}_2$ be the infinite-dimensional complex Hilbert spaces. We characterize the bijective linear map $\Psi: \mathscr{B}(\mathscr{H}_1) \to \mathscr{B}(\mathscr{H}_2)$ where $\Psi(A_1) \circ ~\Psi(A_2)=D_2$ whenever $A_1\circ ~A_2=D_1$ and $D_1~\text{and}~D_2$ are fixed operators.