Beurling quotient modules on the polydisc

Abstract: Let $H^2(\mathbb{D}^n)$ denote the Hardy space over the polydisc $\mathbb{D}^n$, $n \geq 2$. 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 $C_{z_i} = P_{\mathcal{Q}} M_{z_i}|{\mathcal{Q}}$, $i=1, \ldots, n$. 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.