Characterizations of derivations of Banach space nest algebras: All-derivable points

Let N be a nest on a complex Banach space X with N∈N complemented in X whenever N_-=N, 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(A)B+Aδ(B)=δ(Z) for any A,B∈A with AB=Z) is a derivation. In this paper, it is shown that if Z∈AlgN is an injective operator or an operator with dense range, or an idempotent operator with ran(Z)∈N, then Z is an all-derivable point of AlgN. Particularly, if N is a nest on a complex Hilbert space, then every idempotent operator with range in N, every injective operator as well as every operator with dense range in AlgN is an all-derivable point of the nest algebra AlgN.