Mappings on some reflexive algebras characterized by action on zero products or Jordan zero products

Let L be a subspace lattice on a Banach space X and let delta: Alg L -> B(X) be a linear mapping. If V{L is an element of L : L- not superset of L} = X or Lambda{L- : L is an element of L, L- not superset of L} = (0), we show that the following three conditions are equivalent: (1) delta(AB) = delta(A)B + A delta(B) whenever AB = 0; (2) delta(AB + BA) = delta(A)B + A delta(B) + delta(B)A + B delta(A) whenever AB + BA = 0; (3) delta is a generalized derivation and delta(I) is an element of (Alg L)' If V{L is an element of L : L- not superset of L} = X or Lambda{L- : L is an element of L, L- not superset of L} = (0) and delta satisfies delta(AB + BA) = delta(A)B + A delta(B) + delta(B)A+ B delta(A) whenever AB = 0, we show that delta is a generalized derivation and delta(I)A is an element of (Alg L)' for every A is an element of Alg L. We also prove that if V{L is an element of L : L- not superset of L} = X and Lambda{L- : L is an element of L, L- not superset of L} = (0), then delta is a local generalized derivation if and only if delta is a generalized derivation.