Submodules of the Hardy module over polydisc

Abstract: We say that a submodule $\cls$ of $H^2(\mathbb{D}^n)$ ($n >1$) is co-doubly commuting if the quotient module $H^2(\mathbb{D}^n)/\cls$ is doubly commuting. We show that a co-doubly commuting submodule of $H^2(\mathbb{D}^n)$ is essentially doubly commuting if and only if the corresponding one variable inner functions are finite Blaschke products or that $n = 2$. In particular, a co-doubly commuting submodule $\cls$ of $H^2(\mathbb{D}^n)$ is essentially doubly commuting if and only if $n = 2$ or that $\cls$ is of finite co-dimension. We obtain an explicit representation of the Beurling-Lax-Halmos inner functions for those submodules of $H^2_{H^2(\mathbb{D}^{n-1})}(\mathbb{D})$ which are co-doubly commuting submodules of $H^2(\mathbb{D}^n)$. Finally, we prove that a pair of co-doubly commuting submodules of $H^2(\mathbb{D}^n)$ are unitarily equivalent if and only if they are equal.