Characterizing derivations for any nest algebras on Banach spaces by their behaviors at an injective operator

Let NN be a nest on a complex Banach space X   and let AlgN be the associated nest algebra. We say that an operator Z∈AlgN is an all-derivable point of AlgN if every linear map δ   from AlgN into itself derivable at Z (i.e. δ   satisfies δ(A)B+Aδ(B)=δ(Z)δ(A)B+Aδ(B)=δ(Z) for any A,B∈AlgN with AB=ZAB=Z) is a derivation. In this paper, it is shown that every injective operator and every operator with dense range in AlgN are all-derivable points of AlgN without any additional assumption on the nest.