\(p\)-essential normality of quasi-homogeneous Drury-Arveson submodules (Q2840173)

Fix a positive integer \(n\geq 1\). For a real number \(t>-n-1\), let \(\mathcal{H}_{t}\) denote the Hilbert space of analytic functions on the unit ball in \(\mathbb{C}^n\) with the reproducing kernel NEWLINE\[NEWLINEK_z(\zeta) = \dfrac{1}{(1-\langle\zeta,z\rangle)^{n+1+t}}.NEWLINE\]NEWLINE The space \(\mathcal{H}_{t}\) admits the natural \(\mathbb{C}[z_1,\dots,z_n]\)-module structure that comes from multiplication by polynomials. It is clear that \(\mathcal{H}_0\) is the unweighted Bergman module over the unit ball, \(\mathcal{H}_{-1}\) is the Hardy module over the unit sphere, and \(\mathcal{H}_{-n}\) is the Drury-Arveson module.NEWLINENEWLINELet \(M\) be a (closed) submodule of \(\mathcal{H}_{t}\). For \(i=1,\dots, n\), let \(R_i\) be the restriction to \(M\) of the coordinate multiplication operator \(M_{z_i}\). We say that \(M\) is \(p\)-essentially normal if all commutators \([R_i,R_j]\), \(1\leq i,j\leq n\), are in the Schatten-von Neumann \(p\)-class of compact operators with \(p\)-summable singular values. \textit{W. B. Arveson} [J. Oper. Theory 54, No. 1, 101--117 (2005; Zbl 1107.47006)] showed that any submodule of the Drury-Arveson module \(\mathcal{H}_{-n}\) generated by monomials is \(p\)-essentially normal for \(p>n\) and he conjectured that this remains true for all homogeneous submodules of \(\mathcal{H}_{-n}\). This problem is related to C\(^{*}\)-extension theory, index theory and algebraic geometry. Since Arveson's work, researchers have been interested in the study of \(p\)-essential normality of submodules of \(\mathcal{H}_{t}\). Several results have been obtained by various authors under certain additional conditions.NEWLINENEWLINEThe paper under review focuses on quasi-homogeneous principal modules. Let \(K=(K_1,\dots, K_n)\) be an \(n\)-tuple of non-negative integers and let \(m\geq 0\) be an integer. A polynomial \(q\) in \(n\) complex variables is called \(K\)-quasi-homogeneous of degree \(m\) if \(q\) is a linear combination of monomials \(z^{\alpha} = z_1^{\alpha_1}\dots z_n^{\alpha_n}\) with \(\alpha_1 K_1+\dots+\alpha_n K_n=m\). The main result in the paper under review is the following theorem.NEWLINENEWLINE{Theorem}. If \(M=[q]\) is a principal submodule of \(\mathcal{H}_{t}\) generated by a quasi-homogeneous polynomial \(q\), then \(M\) is \(p\)-essentially normal for \(p>n\).NEWLINENEWLINEIn the case \(t>-2\), this result is a special case of the results obtained by \textit{R. G. Douglas} and \textit{K. Wang} [J. Funct. Anal. 261, No. 11, 3155--3180 (2011; Zbl 1254.47004)] for principal submodules of the Bergman module (\(t=0\)), and \textit{Q.-L. Fang} and \textit{J.-B. Xia} [J. Funct. Anal. 265, No. 12, 2991--3008 (2013; Zbl 1319.47001)] for other values \(t>-2\). The above theorem covers the case \(t\leq -2\) as well. As a consequence, the authors show that in dimensions \(n=2,3\), every quasi-homogeneous submodule of \(\mathcal{H}_{t}\) is \(p\)-essentially normal for \(p>n\). They also determine the related \(K\)-homology.