Characterizations of all-derivable points in nest algebras

Let $\mathcal{A}$ be an operator algebra on a Hilbert space. We say that an element $G\in {\mathcal{A}}$ is an all-derivable point of ${\mathcal{A}}$ if every derivable linear mapping $ϕ$ at $G$ (i.e. $ϕ(ST)=ϕ(S)T+Sϕ(T)$ for any $S,T\in alg{\mathcal{N}}$ with $ST=G$) is a derivation. Suppose that $\mathcal{N}$ is a nontrivial complete nest on a Hilbert space $H$. We show in this paper that $G\in {alg\mathcal{N}}$ is an all-derivable point if and only if $G\neq0$.

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