Let \({\mathcal {A}}\) be an algebra. In this paper, we consider the problem of determining a linear map \(\psi \) on \({\mathcal {A}}\) satisfying \(a,b\in {\mathcal {A}}\), \(ab=0 \Longrightarrow \psi ([a,b])=[\psi (a),b] \, (C1) \) or \(ab=0 \Longrightarrow \psi ([a,b])=[a,\psi (b)] \, (C2)\). 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 \({\mathcal {A}}\) is in this class, then any linear map on \({\mathcal {A}}\) 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.

令\（{\ mathcal {A}} \）为代数。在本文中，我们考虑确定线性映射的问题\（\ PSI \）上\（{\ mathcal {A}} \）满足\（A，B \在{\ mathcal {A}} \） ，\ （ab = 0 \ Longrightarrow \ psi（[a，b]）= [\ psi（a），b] \，（C1）\）或\（ab = 0 \ Longrightarrow \ psi（[a，b]）= [a，\ psi（b）] \，（C2）\）。我们首先通过各种示例比较满足（C 1）或（C 2）的线性图，换向线性图和Lie扶正器。实际上，我们看到线性映射满足（C 1），（C2）和通勤线性图是彼此不同的类。然后，我们引入一类算子代数的Banach空间，使得如果\（{\ mathcal {A}} \）是在这个类中，然后在任何线性地图\（{\ mathcal {A}} \）满足（Ç 1）（或（C 2））是通勤线性图。作为这些结果的应用，我们刻画了嵌套代数上满足（C 1）（或（C 2））的李定心器和线性映射。