Let $\A$ be an algebra. In this paper we consider the problem of determining a linear map $\psi$ on $\A$ satisfying $a,b\in \A$, $ab=0 ==> \psi([a,b])=[\psi(a),b] \, (C1) $ or $ab=0 ==> \psi([a,b])=[a,\psi(b)] \, (C2)$. We first compare linear maps satisfying $(C1)$ or $(C2)$, commuting linear maps, and Lie centralizers with a variety of examples. In fact, we see that linear maps satisfying $(C1)$, $(C2)$ and commuting linear maps are different classes of each other. Then we introduce a class of operator algebras on Banach spaces such that if $\A$ is in this class, then any linear map on $\A$ satisfying $(C1)$ (or $(C2)$) is a commuting linear map. As an application of these results we characterize Lie centralizers and linear maps satisfying $(C1)$ (or $(C2)$) on nest algebras.
