Characterizing centralizers and generalized derivations on triangular algebras by acting on zero product

Let U = Tri(A,M, B) be a triangular ring, where A and B are unital rings, and M is a faithful (A,B)-bimodule. It is shown that an additive map . on U is centralized at zero point (i.e., .(A)B = A.(B) = 0 whenever AB = 0) if and only if it is a centralizer. Let .:U .U be an additive map. It is also shown that the following four conditions are equivalent: (1) . is specially generalized derivable at zero point, i.e., .(AB) = .(A)B +A.(B).A.(I)B whenever AB = 0; (2) . is generalized derivable at zero point, i.e., there exist additive maps . 1 and . 2 on U derivable at zero point such that .(AB) = .(A)B + A. 1(B) = . 2(A)B + A.(B) whenever AB = 0; (3) . is a special generalized derivation; (4) . is a generalized derivation. These results are then applied to nest algebras of Banach space.