CHARACTERIZATIONS OF JORDAN DERIVATIONS ON STRONGLY DOUBLE TRIANGLE SUBSPACE LATTICE ALGEBRAS

Let D be a strongly double triangle subspace lattice on a nonzero complex reflexive Banach space X and let delta : Alg D -> Alg D be a linear mapping. We show that delta is Jordan derivable at zero, that is, delta(AB + BA) = delta(A)B + A delta(B) + delta(B)A + B delta(A) whenever AB + BA = 0 if and only if delta has the form delta(A) = tau(A) + lambda A for some derivation tau and some scalar lambda. We also show that if the dimension of X is greater than 2, then delta satisfies delta(AB + BA) = delta(A)B + A delta(B) + delta(B)A + B delta(A) whenever AB = 0 if and only if delta is a derivation.