Two problems on submodules of H² (Dⁿ)

Given any shift-invariant closed subspace S (aka submodule) of the Hardy space over the unit polydisc H² (Dⁿ) (where n 2), let Rₙ䲛: =Mₙ䲛|ₒ, and Eₙ䲛: =Pₒ evₙ䲛, for each j \1, , n\. Here, evₙ䲛 is the operator evaluating at 0 in the zⱼ-th variable. In this article, we prove that given any subset \1, , n\, there exists a collection of one-variable inner functions \_ (z_) \ on Dⁿ, such that \ S = _ (z_) H² (Dⁿ), \ if and only if the conditions (Iₒ-Eₙ䂵Eₙ䂵^*) (Iₒ-Rₙ䂵Rₙ䂵^*) =0 for all k \1, , n\, and (Iₒ-Eₙ_₈Eₙ_₈^*) (Iₒ-Rₙ_₈Rₙ_₈^*) (Iₒ-Eₙ_₉Eₙ_₉^*) (Iₒ-Rₙ_₉Rₙ_₉^*) =0 for all distinct i, j are satisfied. Following this, we study R. G. Douglas's question on the commutativity of orthogonal projections onto the corresponding quotient modules.