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 [Integral Equations Operator Theory 38(2000), pp207–221]