On certain representations of \(H^ \infty (G)\) and the reflexivity of associated operator algebras (Q1891811)

Let \(G\) be a bounded finitely connected domain in the complex plane, whose boundary consists of a disjoint union of a finite number of Jordan loops. The authors study a class of operators on a Hilbert space \(H\) admitting a so-called \(G\)-rational normal boundary dilation. The main application is the following result. Let \(T\) be a bounded operator on \(H\) with \(G\) as a spectral set and the boundary of \(G\) a subset of the spectrum of \(T\). Then either the weak\(*\) closed rational dual algebra generated by \(T\) is reflexive or \(T\) has a nontrivial hyperinvariant subspace.