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

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