In this article, on certain triangular algebra T , we completely characterize the forms of linear maps F1, F2, F3, F4 : T → T that satisfy F1(X)Y + XF2(Y) + F3(Y)X + YF4(X) = 0 for any X, Y ∈ T with XY = YX = 0. Then we apply the result to upper triangular matrix algebras, finite nest algebras and complex block upper triangular matrix algebras.
