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 $\varphi, \psi \in H^\infty(\mathbb{D})$ 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.