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

TL;DR

This paper characterizes certain shift-invariant subspaces of the Hardy space over the polydisc using inner functions and operator conditions, and explores the commutativity of projections in quotient modules.

Contribution

It provides necessary and sufficient conditions for representing submodules as sums of one-variable inner function multiples, and investigates projection commutativity in quotient modules.

Findings

Characterization of submodules via inner functions and operator conditions.

Conditions for submodules to be sums of one-variable inner function multiples.

Analysis of the commutativity of orthogonal projections in quotient modules.

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.