Algebras of subnormal operators on the unit polydisc. (Q2760125)

A (commuting) tuple \(T = (T_1,\dots,T_n)\) of operators on a Hilbert space \(H\) is called reflexive if the collection of all operators \(S\) on \(H\) which leave invariant every closed subspace of \(H\), which is invariant under \(T_1,\dots, T_n\), coincides with the weakly closed algebra of operators generated by \(T_1,\dots, T_n\). Subnormal operators are reflexive, as was shown by \textit{R. Olin} and \textit{J. Thomson} [J. Funct. Anal. 37, 271--301 (1980; Zbl 0435.47034)]; the question of whether subnormal tuples are reflexive, remains open.NEWLINENEWLINE In the paper under review, the author shows that each completely non-unitary subnormal tuple \(T\), acting on a Hilbert space \(H\), with an isometric \(w^*\)-continuous \(H^{\infty}\)-functional calculus, is reflexive. This fact is shown to be a consequence of a strong factorisation result for \(w^*\)-continuous linear functionals on the dual algebra \(\mathcal{U}_T\) generated by \(T\); namely, it is proved that \(\mathcal{U}_T\) possesses property \(\mathbb{A}_{1, \aleph_0}\). As another consequence of this factorisation, it is shown that the vectors generating analytic invariant subspaces for \(T\) form a dense subset of \(H\).NEWLINENEWLINEFor the entire collection see [Zbl 0971.00063].