On Maps that Preserve the Lie Products Equal to Fixed Elements

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