Submodules of the Hardy module over the polydisc

Abstract

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