Essential normality of homogenous quotient modules over the polydisc: distinguished variety case

Polydisc version of Arveson's conjecture

In the present paper, we solve the polydisc-version of Arveson Conjecture by giving a complete criteria for essential normality of homogeneous quotient modules of the Hardy module over the polydisc, and it turns out that our method applies to quotient modules of the weighted Bergman modules $A^2_s(\mathbb D^d)$.

$H^2(\boldsymbol{D}^2)$中商模的本质正规性与子模的希尔伯特-施密特性

Essential normality of quotient modules vs. Hilbert-Schmidtness of submodules in $H^2(\mathbb D^2)$

本文针对$H^2(\boldsymbol{D}^2)$中含特殊齐次多项式的有限生成子模，证明其关联商模的本质正规性与子模希尔伯特-施密特性等价，部分解决了Yang猜想。

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 $θ\in H^\infty(\mathbb{D}^n)$ is an inner function, then the $n$-tuple of commuting operators $(C_{z_1}, \ldots, C_{z_n})$ on the Beurling type quotient module $\mathcal{Q}_θ$ 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 s…