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 $\mbox{ Alg}{\mathcal N}$betheassociatednestalgebra\mathrm{.}Wesaythatanoperator$Z\in \mbox{ Alg}{\mathcal N}$isanall-derivablepointof$\mbox{ Alg}{\mathcal N}$ifeverylinearmap$\delta$from$\mbox{ Alg}{\mathcal N}$ into itself derivable at $Z$ (i.e. $\delta$ satisfies $\delta (A)B+A\delta (B)=\delta (Z)$ for any $A,B\in Alg\mathcal{N}$ with $AB=Z$) is a derivation. In this paper, it is shown that every injective operator and every operator with dense range in $Alg\mathcal{N}$ are all-derivable points of $Alg\mathcal{N}$ without any additional assumption on the nest.