Reflexivity for Subnormal Systems With Dominating Spectrum in Product Domains

Abstract.

The question whether every subnormal tuple ${S}\,=\,({S}_1, \ldots , {S}_n)$ on a complex Hilbert space is reflexive is one of the major open problems in multivariable invariant subspace theory. Positive answers have been given for subnormal tuples with rich spectrum in the unit polydisc or the unit ball. The ball case has been extended by Didas [6] to strictly pseudoconvex domains. In the present note we extend the polydisc case by showing that every subnormal tuple with pure components and rich Taylor spectrum in a bounded polydomain $U = U_1 \times \ldots \times U_n \subset {\mathbb{C}}^{n}$ is reflexive.