A characterization of generalized derivations of JSL algebras

Let AlgL be a J-subspace lattice algebra on a Banach space X and M be an operator in AlgL. We prove that if δ : Alg L → B(X) is a linear mapping satisfying δ(AB) = δ(A)B + Aδ(B) for all A,B ∈ AlgL with AMB = 0, then δ is a generalized derivation. This result can be applied to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras.