Characterizations of all-derivable points in nest algebras (Q2839304)

Let \(\mathcal {A}\) be an operator algebra and \(G\) be an element of \(\mathcal {A}\). A linear map \(\varphi : \mathcal {A}\to \mathcal {A}\) is called a derivable mapping at \(G\) if \(\varphi (ST)=\varphi (S)T+S\varphi (T)\) for any \(S, T\in \mathcal {A}\) with \(ST=G\). Moreover, \(G\) is said to be an all-derivable point if every derivable linear mapping \(\varphi \) at \(G\) is a derivation. Suppose that \(\mathcal {N}\) is a nontrivial complete nest in Hilbert space \(H\) and the corresponding nest algebra is \(\mathrm{alg}\mathcal {N}\). In the paper under review, the authors show that \(G\in\mathrm{alg}\mathcal {N}\) is an all-derivable point if and only if \(G\not = 0\).