Contractive Hilbert modules on quotient domains

TL;DR

This paper studies contractive operator tuples on quotient domains influenced by complex reflection groups, establishing their properties, inequivalence, and invariant subspace representations, with implications for Hardy and Bergman modules.

Contribution

It introduces and analyzes $oldsymbol heta_n$-contractions associated with complex reflection groups, demonstrating their inequivalence and providing a Beurling-Lax-Halmos type theorem.

Findings

$oldsymbol heta_n$-contractions are mutually unitarily inequivalent under mild conditions

Negative division results for Hardy and Bergman modules on the bidisc

Invariant subspace structure characterized by a Beurling-Lax-Halmos type theorem

Abstract

Let the complex reflection group $G(m,p,n)$ act on the unit polydisc $\mathbb D^n$ in $\mathbb C^n.$ A $\boldsymbol\Theta_n$-contraction is a commuting tuple of operators on a Hilbert space having $$\overline{\boldsymbol\Theta}_n:=\{\boldsymbol\theta(z)=(\theta_1(z),\ldots,\theta_n(z)):z\in\overline{\mathbb D}^n\}$$ as a spectral set, where $\{\theta_i\}_{i=1}^n$ is a homogeneous system of parameters associated to $G(m,p,n).$ A plethora of examples of $\boldsymbol\Theta_n$-contractions is exhibited. Under a mild hypothesis, it is shown that these $\boldsymbol\Theta_n$-contractions are mutually unitarily inequivalent. These inequivalence results are obtained concretely for the weighted Bergman modules under the action of the permutation groups and the dihedral groups. The division problem is shown to have negative answers for the Hardy module and the Bergman module on the bidisc. A Beurling-Lax-Halmos type representation for the invariant subspaces of $\boldsymbol\Theta_n$-isometries is obtained.