Additive maps derivable or Jordan derivable at zero point on nest algebras

Let AlgN be a nest algebra associated with the nest N on a (real or complex) Banach space X. Assume that every N is an element of N is complemented whenever N_ = N. Let delta : AlgN --> AlgN be an additive map. It is shown that the following three conditions are equivalent: (1) delta is derivable at zero point, i.e., delta(AB) = delta(A)B + A delta(B) whenever AB = 0; (2) delta is Jordan derivable at zero point, i.e., delta(AB + BA) = delta(A)B + A delta(B) + B delta(A) + delta(B)A whenever AB + BA = 0; (3) delta has the form delta(A) = tau(A) + cA for some additive derivation tau and some scalar c. It is also shown that delta is generalized derivable at zero point, i.e., delta(AB) = delta(A)B + A delta(B) - A delta(I)B whenever AB = 0, if and only if delta is an additive generalized derivation. Finer characterizations of above maps are given for the case dim X = infinity. (C) 2009 Elsevier Inc. All rights reserved.