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 $\phi ,\psi \in H^{\infty }(\mathbb{D})$ be two inner functions. If ${\mathcal{Q}}_{\phi }=H^2(\mathbb{D})/\phi H^2(\mathbb{D})$ and ${\mathcal{Q}}_{\psi }=H^2(\mathbb{D})/\psi H^2(\mathbb{D})$, 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 $\mathcal{S}=\Phi H^2({\mathbb{D}}^2)$ 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.

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