Lie centralizers at unit product on generalized matrix algebras

Let $\mathcal{G}$ be a generalized matrix algebra, and $\phi: \mathcal{G}\to\mathcal{G}$ a linear map. It is shown that, under some mild conditions, the following are equivalent: (i) $\phi([S, T])=[\phi(S), T]$, for all $S, T \in\mathcal{G}$ with $ST=I$; (ii) $\phi([S, T])=[S, \phi(T)]$, for all $S, T \in\mathcal{G} $ with $ST=I$; (iii) $\phi(A)=ZA+\gamma(A)$, for all $A \in \mathcal{G}$, where $Z\in\mathcal{Z}(\mathcal{G})$ and $\gamma: \mathcal{G}\to\mathcal{Z}(\mathcal{G})$ is a linear map vanishing on each commutator $[S, T]$ whenever $ST=I$. These results are then applied to matrix algebras and some operator algebras.