# Rank of a co-doubly commuting submodule is 2

## TL;DR

This paper proves that non-trivial co-doubly commuting submodules of the Hardy space over the bidisk have rank 2, providing a precise characterization and answering a question by Douglas and Yang.

## Contribution

It establishes the exact rank of co-doubly commuting submodules and characterizes rank-one submodules as those generated by one-variable inner functions.

## Key findings

- Rank of non-trivial co-doubly commuting submodules is 2.
- Rank-one co-doubly commuting submodules are generated by one-variable inner functions.
- Answers a question posed by Douglas and Yang regarding submodule ranks.

## Abstract

We prove that the rank of a non-trivial co-doubly commuting submodule is $2$. More precisely, let $\varphi, \psi \in H^\infty(\mathbb{D})$ be two inner functions. If $\mathcal{Q}_{\varphi} = H^2(\mathbb{D})/ \varphi H^2(\mathbb{D})$ and $\mathcal{Q}_{\psi} = H^2(\mathbb{D})/ \psi H^2(\mathbb{D})$, then \[ \mbox{rank~}(\mathcal{Q}_{\varphi} \otimes \mathcal{Q}_{\psi})^\perp = 2. \] An immediate consequence is the following: Let $\mathcal{S}$ be a co-doubly commuting submodule of $H^2(\mathbb{D}^2)$. Then $\mbox{rank~} \mathcal{S} = 1$ if and only if $\mathcal{S} = \Phi H^2(\mathbb{D}^2)$ for some one variable inner function $\Phi \in H^\infty(\mathbb{D}^2)$. This answers a question posed by R. G. Douglas and R. Yang.