Additive maps derivable or Jordan derivable at zero point on nest algebras (Q967514)

Let \({\mathcal A}\) be an algebra. An additive map \(d:{\mathcal A}\to{\mathcal A}\) is said to be derivable at zero point if \( x d(y)+ d(x) y=0 \) for each pair \(x, y \in{\mathcal A}\) with \(xy=0\). The map \(d\) is said to be Jordan derivable at zero point if \( x d(y)+ d(x) y+ y d(x)+ d(y)x=0\) for each pair \(x, y \in{\mathcal A}\) with \(xy+yx=0\). Now let \(\mathcal N\) be a nest on a (real or complex) Banach space \(X\) and let \({\mathcal A}\) be the nest algebra associated to \(\mathcal N\). Let \(d:{\mathcal A}\to{\mathcal A}\) be an additive map. Assume further that, for every \(N \in \mathcal N\) satisfying \(N_-=N\), the subspace \(N\) is complemented. The authors prove that the following conditions are equivalent: {\parindent=6mm \begin{itemize}\item[(1)] \( d\) is derivable at zero point; \item[(2)] \(d \) is Jordan derivable at zero point; \item[(3)] there exists an additive derivation \( \tau\) and a scalar \(\lambda\) such that \(d (T)= \tau (T)+ \lambda T\) for all \(T \in{\mathcal A}\). \end{itemize}} Moreover, if \(X\) is infinite-dimensional, then the additive derivation \(\tau\) is inner (and hence, linear). The authors also study additive maps \(d:{\mathcal A}\to{\mathcal A}\) which are generalized derivable at zero point (that is, \(d(x)y+x d(y) -x d(I) y=0\) whenever \(xy=0\)). For the case of nest algebras on Hilbert spaces, see \textit{W.\,Jing}, \textit{S.-J.\thinspace Lu} and \textit{P.-T.\thinspace Li} [Bull.\ Aust.\ Math.\ Soc.\ 66, 227--232 (2002; Zbl 1035.47019)], and \textit{J.\,Zhu} and \textit{C.-P.\thinspace Xiong} [Acta Math.\ Sin.\ 45, 783--788 (2002; Zbl 1015.47052)].