Submodules of the Hardy module over the polydisc

Abstract

We say that a submodule S of H²(Dⁿ) (n > 1) is co-doubly commuting if the quotient module H²(ⁿ)/S is doubly commuting. We show that a co-doubly commuting submodule of H²(Dⁿ) 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 H²(Dⁿ) 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 H² _H²(D^n-1)(D) which are co-doubly commuting submodules of H²(Dⁿ). Finally, we prove that a pair of co-doubly commuting submodules of H²(Dⁿ) are unitarily equivalent if and only if they are equal.