On Quotient modules of $H^2({\mathbb{D}}^n)$: Essential Normality and Boundary Representations

Abstract

Let ${\mathbb{D}}^n$ be the open unit polydisc in ${\mathbb{C}}^n$, $n\ge 1$, and let $H^2({\mathbb{D}}^n)$ be the Hardy space over ${\mathbb{D}}^n$. For $n\ge 3$, we show that if $\theta \in H^{\infty }({\mathbb{D}}^n)$ is an inner function, then the $n$-tuple of commuting operators $(C_{z_1},\dots ,C_{z_n})$ on the Beurling type quotient module ${\mathcal{Q}}_{\theta }$ is not essentially normal, where \[\mathcal{Q}_θ = H^2(\mathbb{D}^n)/ θH^2(\mathbb{D}^n) \quad \mbox{and} \quad C_{z_j} = P_{\mathcal{Q}_θ} M_{z_j}|_{\mathcal{Q}_θ}\quad (j = 1, \ldots, n).\] Rudin's quotient modules of $H^2({\mathbb{D}}^2)$ are also shown to be not essentially normal. We prove several results concerning boundary representations of $C^{\ast }$-algebras corresponding to different classes of quotient modules including doubly commuting quotient modules and homogeneous quotient modules.

19 pages. To appear in Proceedings A of the Royal Society of Edinburgh

References in corpus (1)

• Submodules of the Hardy module over polydisc