Beurling quotient modules on the polydisc

Abstract

Let $H^2({\mathbb{D}}^n)$ denote the Hardy space over the polydisc ${\mathbb{D}}^n$, $n\ge 2$. A closed subspace $\mathcal{Q}\subseteq H^2({\mathbb{D}}^n)$ is called Beurling quotient module if there exists an inner function $\theta \in H^{\infty }({\mathbb{D}}^n)$ such that $\mathcal{Q}=H^2({\mathbb{D}}^n)/\theta H^2({\mathbb{D}}^n)$. We present a complete characterization of Beurling quotient modules of $H^2({\mathbb{D}}^n)$: Let $\mathcal{Q}\subseteq H^2({\mathbb{D}}^n)$ be a closed subspace, and let $C_{z_i}=P_{\mathcal{Q}}{M}_{z_i}{|}_{\mathcal{Q}}$, $i=1,\dots ,n$. Then $\mathcal{Q}$ is a Beurling quotient module if and only if \[ (I_{\mathcal{Q}} - C_{z_i}^* C_{z_i}) (I_{\mathcal{Q}} - C_{z_j}^* C_{z_j}) = 0 \qquad (i \neq j). \] We present two applications: first, we obtain a dilation theorem for Brehmer $n$-tuples of commuting contractions, and, second, we relate joint invariant subspaces with factorizations of inner functions. All results work equally well for general vector-valued Hardy spaces.

15 pages