Lie centralizers at the zero products on generalized matrix algebras

Let U be a 2-torsion free unital generalized matrix algebra, and phi : U -> U be a linear map satisfying S,T is an element of U, ST = 0 -> phi([S, T]) = [phi(S), T] = [S, phi(T)]. In this paper, we study the structure of phi and under some mild conditions on U we present the necessary and sufficient conditions for phi to be in terms of centralizers. We then provide the characterizations of Lie centralizers on U and our results generalize some of the previous results. We also refer to some applications of our results for triangular algebras and some operator algebras.