Additive maps derivable at some points on J-subspace lattice algebras

Let L be a J-subspace lattice on a real or complex Banach space dim X with X > 2 and AlgL be the associated J-subspace lattice algebra. Let δ:AlgL→AlgL be an additive map. It is shown that, if δ is derivable at zero point, i.e., δ(AB)=δ(A)B+Aδ(B) whenever AB = 0, then δ(A)=τ(A)+λA, ∀A, where τ is an additive derivation and λ is a scalar; if δ is generalized derivable at zero point, i.e., δ(AB)=δ(A)B+Aδ(B)-Aδ(I)B whenever AB = 0, then δ is a generalized derivation. It is also shown that, if X is complex, then every linear map derivable at unit operator on AlgL is a derivation.