Properties of Beurling-type submodules via Agler decompositions

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• 作者：Kelly Bickel^a ; ¹ ; ^{kelly.bickel@bucknell.edu" class="auth_mail" title="E-mail the corresponding author} ; Constanze Liaw^b ; ² ; ^{Constanze_Liaw@baylor.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：47A13 ; 47A20 ; 46E22
• 刊名：Journal of Functional Analysis
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：272
• 期：1
• 页码：83-111
• 全文大小：508 K

文摘

In this paper, we study operator-theoretic properties of the compressed shift operators S_z1$S_{z_1}$ and S_z2$S_{z_2}$ on complements of submodules of the Hardy space over the bidisk H²(D²)$H^2({\mathbb{D}}^2)$. Specifically, we study Beurling-type submodules – namely submodules of the form θH²(D²)$\theta H^2({\mathbb{D}}^2)$ for θ inner – using properties of Agler decompositions of θ   to deduce properties of S_z1$S_{z_1}$ and S_z2$S_{z_2}$ on model spaces H²(D²)⊖θH²(D²)$H^2({\mathbb{D}}^2)\ominus \theta H^2({\mathbb{D}}^2)$. Results include characterizations (in terms of θ  ) of when a commutator $[S_{z_j}^{⁎},S_{z_j}]$ has rank n   and when subspaces associated to Agler decompositions are reducing for S_z1$S_{z_1}$ and S_z2$S_{z_2}$. We include several open questions.