Properties of Beurling-type submodules via Agler decompositions

In this paper, we study operator-theoretic properties of the compressed shift operators Sz1Sz1 and Sz2Sz2 on complements of submodules of the Hardy space over the bidisk H2(D2)H2(D2). Specifically, we study Beurling-type submodules – namely submodules of the form θH2(D2)θH2(D2) for θ inner – using properties of Agler decompositions of θ   to deduce properties of Sz1Sz1 and Sz2Sz2 on model spaces H2(D2)⊖θH2(D2)H2(D2)⊖θH2(D2). Results include characterizations (in terms of θ  ) of when a commutator [Szj⁎,Szj] has rank n   and when subspaces associated to Agler decompositions are reducing for Sz1Sz1 and Sz2Sz2. We include several open questions.