Geometric Arveson-Douglas conjecture for the Hardy space and a related compactness criterion (Q2042027)

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 exists a measure \(\mu\) on \(M\) such that \(cQ\le T_\mu\le CQ\) on \(L^2(\mathbb{S},d\sigma)\). For each \(f\in L^\infty(\mathbb{S},d\sigma)\), we define \(Q_f=QM_f|\widetilde{Q}\). Let \(\mathcal{TQ}\) be the \(C^*\)-algebra generated by \(\{Q_f:f\in L^\infty(\mathbb{S},d\sigma)\}\). The third main result of the paper says that if \(A\in\mathcal{TQ}\) and \(\lim_{z\in M,|z|\to 1}\langle Ak_z,k-z\rangle=0\), where \(k_z\) is the reproducing kernel for \(H^2(\mathbb{S})\), then \(A\) is a compact operator.