On some reflexive operator algebras constructed from two sets of closed operators and from a set of reflexive operator algebras
The paper considers a new method of constructing operator algebras \(\mathcal A\) from two given sets of closed operators \(\{F_ i\}^{n-1}_{i=1}\), \(\{G_ i\}^{n-1}_{i=1}\) and from a given set of reflexive operator algebras \(\{\mathcal T_ i\}^ n_{i=1}\) (1\(\leq n\leq \infty\)) in Hilbert spaces \(\{H_ i\}^ n_{i=1}\). For example, if \(n=2\), if \(F_1\) and \(G_1\) are closed operators from \(H_2\) into \(H_1\) and if \(\mathcal T_ i\), \(i=1,2\), are reflexive subalgebras of \(B(H_ i)\), then \[ \begin{aligned} \mathcal A = \{A=& \begin{pmatrix} T_ 1 & T_ F \\ 0 & T_2 \end{pmatrix} \in B(H_1 \oplus H_2): \\ &1) T_ i\in \mathcal T_ i, \quad T_2D(G_1) \subseteq D(G_1) \text{ and }T_2D(F_1) \subseteq D(F_1); \\ &2) T_1 G_1 |_{D(G_ 1)} = G_1 T_2 |_{D(G_1)}; \\ &3) T_ F |_{D(F_1)} = (F_1 T_2 - T_1F_1) |_{D(F_1)}\}. \end{aligned} \] Some sufficient conditions for the algebras \(\mathcal A\) to be reflexive are investigated and a detailed description of Lat \(\mathcal A\) is obtained. The method produces a wide variety of reflexive algebras for which Ad \(\mathcal A\neq \mathcal A + \mathcal A'\) (Ad \(\mathcal A\) consists of all bounded operators which generate derivations on \(\mathcal A\)).
scientific article; zbMATH DE number 3942299