On determining the domain of the adjoint operator (Q408272)

Given a pair \((A, A_{0})\) of operators in a Hilbert space \(\mathfrak{K}\) with \(A\) closable and densely defined and \(A_{0}\subseteq A^{*}\), the author gives a condition for the equality \(\overline{A_{0}}=A^{*}\), which generalizes another one he had obtained earlier in [Bull. Lond. Math. Soc. 40, No. 5, 807--816 (2008; Zbl 1151.47045)]. Namely, if \((T_n)\) is a sequence of bounded operators in \(\mathfrak{K}\) which converges to \(I\) in the weak operator topology, \(\sup\| \overline{ A}T_n - T_n \overline{ A}\| < \infty\) and, for every \(n\in\mathbb{N}\), one has that \(\mathfrak{R}(T_n )\subseteq \mathfrak{D}(\overline{A})\), \(\mathfrak{R}(T^{*}_n )\subseteq \mathfrak{D}(\overline{A_{0}})\), then \(\overline{A_{0}}=A^{*}\). This serves as a criterion for selfadjointness of a symmetric operator, for normality of a formally normal operator or for \(H\)-selfadjointness of an \(H\)-symmetric operator. Differential operators and operators given by an infinite matrix are considered as examples.