2024/06/13 by Ramlal Debnath, Debnath, Ramlal, Srijan Sarkar +1
Computer Science · Mathematics · #30H05 #30H10 #30J05 #30J10 #32A10 #47A13 #47A15 #47A20 #Algebraic and Geometric Analysis #Coding theory and cryptography #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2406.09245
openalex publication_date 2024/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given any shift-invariant closed subspace S (aka submodule) of the Hardy space over the unit polydisc H2(\mathbbDn) (where n ≥ 2), let Rzj:=Mzj|S, and Ezj:=PS∘ evzj, for each j ∈ \1,…,n\. Here, evzj is the operator evaluating at 0 in the zj-th variable. In this article, we prove that given any subset Λ⊆ \1,…,n\, there exists a collection of one-variable inner functions \ϕλ(zλ)\λ∈ Λ on \mathbbDn, such that S = ∑λ∈ Λ ϕλ(zλ)H2(\mathbbDn), if and only if the conditions (IS-EzkEzk^*)(IS-RzkRzk^*)=0 for all k ∈ \1,…,n\ ∖ Λ, and (IS-E_ziE_zi^*)(IS-R_ziR_zi^*)(IS-E_zjE_zj^*)(IS-R_zjR_zj^*)=0 for all distinct i,j ∈ Λ are satisfied. Following this, we study R.G. Douglas's question on the commutativity of orthogonal projections onto the corresponding quotient modules.