Characterization of Jordan derivations on J-subspace lattice algebras

Let L be a J-subspace lattice on a Banach space X and Alg L the associated J-subspace lattice algebra. Assume that delta : Alg L -> Alg L is an additive map. It is shown that delta satisfies delta(AB + BA) = delta(A)B + A delta(B) + delta(B)A + B delta(A) for any A, B is an element of Alg L with AB + BA = 0 if and only if delta(A) = tau(A) + delta(I)A for all A, where tau is an additive derivation; if X is complex with dim X >= 3 and if delta is linear, then delta satisfies delta(AB + BA) = delta(A)B + A delta(B) + delta(B)A + B delta(A) for any A, B is an element of Alg L with AB + BA = I if and only if delta is a derivation.