Additive derivations and Jordan derivations on algebras of unbounded operators (Q2754965)

Let \(\mathcal D\) be a dense subspace in a Hilbert space, and it is said to have the property (B) if the following holds:NEWLINENEWLINENEWLINE(B) There exists an infinite orthonormal system \((\varphi_n)\) in \(\mathcal D\) with the following two properties:NEWLINENEWLINENEWLINEi) there is a sequence \((t_n),t_n \neq 0,t_n \in \mathbb F (\mathbb R \text{ or } \mathbb C)\) such that \(\sum t_n\varphi_n \in \mathcal D\),NEWLINENEWLINENEWLINEii) for all \((s_n),s_n \in \mathbb F\) and \(|s_n|\leq |t_n|, \sum s_n\varphi_n\) belongs also to \(\mathcal D\). NEWLINENEWLINENEWLINE\noindent Let \(\mathcal L^\dag(\mathcal D)\) be the \(\ast\)-algebra of all linear operators \(X\) from \(\mathcal D\) to \(\mathcal D\) such that \(\mathcal D(X^\ast) \supset \mathcal D\) and \(X^\ast\mathcal D \subset \mathcal D\), and \(\mathcal F(\mathcal D)\) the \(\ast\)-ideal of \(\mathcal L^\dag(\mathcal D)\) consisting of all finite rank operators is \(\mathcal L^\dag(\mathcal D)\). A \(\ast\)-subalgebra \(\mathcal A\) of \(\mathcal L^\dag(\mathcal D)\) containing \(\mathcal F(\mathcal D)\) is said to be a standard operator algebra on \(\mathcal D\). The author has shown that if \(\mathcal A\) is a standard operator algebra on a dense domain \(\mathcal D\) with property (B) and \(D\) is an additive derivation of \(\mathcal A\), that is, it is an additive mapping of \(\mathcal A\) into \(\mathcal L^\dag(\mathcal D)\) such that \(D(XY)=D(X)Y+XD(Y)\) for each \(X,Y \in \mathcal A\), then there exists an element \(T\) of \(\mathcal L^\dag(\mathcal D)\) such that \(D(X)=TX-XT\) for all \(X\in \mathcal A\). Furthermore, he has shown that a similar result is valid for Jordan \((\ast)\)-derivations.