Geometric Arveson-Douglas conjecture for the Hardy space and a related compactness criterion
type: Item
instance of: scholarly article
title: Geometric Arveson-Douglas conjecture for the Hardy space and a related compactness criterion
author: Yi Wang
author: Jingbo Xia
published in: Advances in Mathematics
publication date: 2021-07-26T00:00:00Z
review text: Suppose that \(\mathcal{N}\) is a either a submodule or a quotient module of a Hilbert module \(\mathcal{H}\). Let \(P_{\mathcal{N}}:\mathcal{H}\to\mathcal{N}\) be the orthogonal projection and define \(\mathcal{Z}_{\mathcal{N},j}= P_{\mathcal{N}}M_{z_j}|\mathcal{N}\) for \(j=1,\dots,n\). Recall that \(\mathcal{N}\) is said to be \(p\)-essentially normal if all commutators \([\mathcal{Z}_{\mathcal{N},i}^*,\mathcal{Z}_{\mathcal{N},j}]\) with \(1\le i,j\le n\) are in the Schatten class \(\mathcal{C}_p\). Let \(\mathbb{B}=\{z\in\mathbb{C}^n:|z|<1\}\) and \(\mathbb{S}=\{z\in\mathbb{C}^n:|z|=1\}\), where \(\mathbb{S}\) is equipped with the standard spherical measure \(d\sigma\). The Hardy space \(H^2(\mathbb{S})\) is the closure of the ring of analytic polynomials \(\mathbb{C}[z_1,\dots,z_n]\) in \(L^2(\mathbb{S},d\sigma)\). Let \(\Omega\) be a complex manifold. A set \(A\subset\Omega\) is called a complex analytic subset of \(\Omega\) if for each \(a\in A\) there are a neighborhood \(U\) of \(a\) and functions \(f_1,\dots,f_N\) analytic in this neighborhood such that \(A\cap U=\{z\in U:f_1(z)=\dots=f_N(z)=0\}\). Let \(\widetilde{M}\) be an analytic subset of an open neighborhood of \(\overline{\mathbb{B}}\) with \(1\le\dim_{\mathbb{C}}\widetilde{M}\le n-1\) and \(M=\overline{B}\cap\widetilde{M}\). Consider a submodule \(\mathcal{R}=\{f\in H^2(\mathbb{S}):f=0 \text{ on } M\}\) and the corresponding quotient module \(\mathcal{Q}=H^2(\mathbb{S})\ominus\mathcal{R}\). The authors proved the geometric Arveson-Douglas conjecture saying that the quotient module \(\mathcal{Q}\) is \(p\)-essentially normal for every \(p>d=\dim_{\mathbb{C}}\widetilde{M}\). Furhter, for a measure \(\mu\) on \(M\), one can define the Toeplitz operator \(T_\mu\) by \((T_\mu h)(z)=\int_M h(w)(1-\langle z,w\rangle)^{-n}d\mu(w)\). Let \(Q\) denote the orthogonal projection from\(L^2(\mathbb{S},d\sigma)\) onto \(\mathcal{Q}\). The second main result says that for constants \(0<c\le C<\infty\), there exis…
reviewed by: Oleksiy Karlovych
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47A13
Mathematics Subject Classification ID: 32A36
Mathematics Subject Classification ID: 47L80
zbMATH DE Number: 7374924
zbMATH Keywords: quotient module
zbMATH Keywords: essential normality
zbMATH Keywords: compactness criterion
MaRDI profile type: Publication
full work available at URL: j.aim.2021.107890
OpenAlex ID: W3185817897
Wikidata QID: Q113881021
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DOI: 10.1016/J.AIM.2021.107890
description: scientific article; zbMATH DE number 7374924