Partial\(^*\)-algebras of closed linear operators in Hilbert space
By \textit{K. D. Kürsten}'s counterexamples in [Publ. Res. Inst. Math. Sci. 22, 151-175 (1986; Zbl 0607.47043)] the authors noticed the following facts: Let \({\mathcal D}\) be a dense subspace in a Hilbert space \({\mathcal H}\), \(\bar C({\mathcal D},{\mathcal H})\) (the set of all closed operators A in \({\mathcal H}\) such that \({\mathcal D}\subset {\mathcal D}(A)\cap {\mathcal D}(A^*)\) and \[ {\mathcal C}=\{\overline{A/{\mathcal D}};\quad A\in \bar C({\mathcal D},{\mathcal H})\}. \] 1. Equip \({\mathcal C}\) with the addition \({\hat +}\), the involution \(A\to A^{\neq}\) and the multiplication \(\cdot\) as follows: \[ A\hat +B=\overline{(A+B)/{\mathcal D}},\quad A^{\neq}=\overline{A^*/{\mathcal D}}, \] \[ A\cdot B=\overline{A(B/{\mathcal D})} \] whenever Range \((B/{\mathcal D})\subset {\mathcal D}(A^{\neq \neq})\) and Range \((A^{\neq}/{\mathcal D})\subset {\mathcal D}(B^{\neq}).\) Then the multiplication \(\cdot\) is not necessarily distributive. 2. Put \({\mathcal C}^*=\{A^*\); \(A\in {\mathcal C}\}\) and equip it with the sum \(\check +\), the involution \(\dag\) and the multiplication * as follows: \[ A\check +B=[(A^*+B^*)/{\mathcal D}]^*,\quad A^{\dag}=(A/{\mathcal D})^*, \] \[ A*B=[B^{\dag}(A^{\dag}/{\mathcal D})]^* \] whenever Range \((B/{\mathcal D})\subset {\mathcal D}(A^{\dag\dag})\) and Range \((A^{\dag}/{\mathcal D})\subset {\mathcal D}(B^{\dag}).\) Then the multiplication * on \({\mathcal C}^*\) is not necessarily associative. By these facts they have connected some results in their former paper [Publ. Res. Inst. Math. Sci. 21, 205-236 (1985; Zbl 0609.47058)].
partial \(^ *\)-algebras of closed linear operators in Hilbert space
scientific article; zbMATH DE number 4017707
scientific article