Additive maps derivable at some points on image-subspace lattice algebras Original Research Article

Let image be a image-subspace lattice on a real or complex Banach space dim X with X > 2 and image be the associated image-subspace lattice algebra. Let image 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, for allA, 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 image is a derivation.