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.
Categories: math.FA math.CV