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

Let $\mathbb{D}^n$ be the open unit polydisc in $\mathbb{C}^n$, $n \geq 1$, 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 $(C_{z_1}, \ldots, C_{z_n})$ on the Beurling type quotient module $\mathcal{Q}_{\theta}$ is not essentially normal, where $$\mathcal{Q}_{\theta} = H^2(\mathbb{D}^n)/ \theta H^2(\mathbb{D}^n) \quad \mbox{and} \quad C_{z_j} = P_{\mathcal{Q}_{\theta}} M_{z_j}|_{\mathcal{Q}_{\theta}}\quad (j = 1, \ldots, n).$$ 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 $C^*$-algebras corresponding to different classes of quotient modules including doubly commuting quotient modules and homogeneous quotient modules.