Rank of a co-doubly commuting submodule is 2

Abstract

We prove that the rank of a non-trivial co-doubly commuting submodule is $2$. More precisely, let $φ, ψ\in H^\infty(\mathbb{D})$ be two inner functions. If $\mathcal{Q}_φ = H^2(\mathbb{D})/ φH^2(\mathbb{D})$ and $\mathcal{Q}_ψ = H^2(\mathbb{D})/ ψH^2(\mathbb{D})$, then \[ \mbox{rank~}(\mathcal{Q}_φ \otimes \mathcal{Q}_ψ)^\perp = 2. \] 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 $\mathcal{S} = ΦH^2(\mathbb{D}^2)$ for some one variable inner function $Φ\in H^\infty(\mathbb{D}^2)$. This answers a question posed by R. G. Douglas and R. Yang.

7 pages, revised. To appear in Proceedings of American Math Society