Properties of Beurling-type submodules via Agler decompositions

摘要

In this paper, we study operator-theoretic properties of the compressed shift operators S-z1 and S-z2 on complements of submodules of the Hardy space over the bidisk H-2(D-2). Specifically, we study Beurling-type submodules - namely submodules of the form theta H-2(D-2) for theta inner - using properties of Agler decompositions of theta to deduce properties of Szi and S-z2 on model spaces H-2(D-2) circle minus theta H-2(D-2). Results include characterizations (in terms of theta) of when a commutator S*(zj), S-zj has rank n and when subspaces associated to Agler decompositions are reducing for S-z1 and S-z2.We include several open questions. (C) 2016 Elsevier Inc. All rights reserved.