All-Derivable Subsets For Nest Algebras On Banach Spaces

Let N be a nest on a real or complex Banach space X and let AlgN be the associated nest algebra.δ is a linear map from AlgN into itself. We say that δ is derivable at Z∈AlgN if δ(A)B+Aδ(B)=δ(Z) for any A,B∈AlgN with AB=Z; δ is derivable on a subset S (？) AlgN if δ is derivable at every point in S. We say that S is an all-derivable subset of AlgN if every linear map δ derivable on S is a derivation; particularly, in the case that an all-derivable subset S={Z} be a single-point subset, we say that Z is an all-derivable point of AlgN. In this paper, we show that S is an all-derivable subset of AlgN if span{ran(Z):Z∈S} is dense in X or∩{kerZ:Z∈S}={0}. Particularly, every injective operator and every operator with dense range in AlgN are all-derivable points of AlgN.