Two problems on submodules of $H^2(\mathbb{D}^n)$

Abstract

Given any shift-invariant closed subspace $\mathcal{S}$ (aka submodule) of the Hardy space over the unit polydisc $H^2(\mathbb{D}^n)$ (where $n \geq 2$), let $R_{z_j}:=M_{z_j}|_{\mathcal{S}}$, and $E_{z_j}:=P_{\mathcal{S}}\circ ev_{z_j}$, for each $j \in \{1,\ldots,n\}$. Here, $ev_{z_j}$ is the operator evaluating at $0$ in the $z_j$-th variable. In this article, we prove that given any subset $\Lambda \subseteq \{1,\ldots,n\}$, there exists a collection of one-variable inner functions $\{\phi_\lambda (z_\lambda)\}_{\lambda \in \Lambda}$ on $\mathbb{D}^n$, such that \[ \mathcal{S} = \sum_{\lambda \in \Lambda} \phi_\lambda (z_\lambda)H^2(\mathbb{D}^n), \] if and only if the conditions $ (I_{\mathcal{S}}-E_{z_k}E_{z_k}^*)(I_{\mathcal{S}}-R_{z_k}R_{z_k}^*)=0$ for all $k \in \{1,\dots,n\} \setminus \Lambda$, and $(I_{\mathcal{S}}-E_{z_{i}}E_{z_{i}}^*)(I_{\mathcal{S}}-R_{z_{i}}R_{z_{i}}^*)(I_{\mathcal{S}}-E_{z_{j}}E_{z_{j}}^*)(I_{\mathcal{S}}-R_{z_{j}}R_{z_{j}}^*)=0$ for all distinct $i,j \in \Lambda$ are satisfied. Following this, we study R.G. Douglas's question on the commutativity of orthogonal projections onto the corresponding quotient modules.