The essential normality of \(N_{\underline{\eta}}\)-type quotient module of Hardy module on the polydisc (Q2862179)

The Hardy space over the polydisc \(H^2({\mathbb D}^n)\) has a structure of a \({\mathbb C}[z_1,\dots,z_n]\)-module. A closed subspace \(M\) is called a a submodule if, for each polynomial \(p\), \(pM\subset M\). For a submodule \(M\), let \(N=H^2({\mathbb D}^n)/M\) be the quotient module. The quotient module can be identified with \(H^2({\mathbb D}^n)\ominus M\), equipped with the module structure NEWLINE\[NEWLINE p\cdot f=p(S_{z_1},\dots,S_{z_n})f \quad \text{for any} \;f\in M^\bot \;\text{and} \;p\in {\mathbb C}[z_1,\dots,z_n], NEWLINE\]NEWLINE where \(S_{z_i}=P_NM_{z_i}|_N\). A quotient module is called essentially normal if the self-commutators \([S_{z_i}^\ast,S_{z_i}]\) are all compact. Let \(\eta_i(z_i)\) be nonconstant inner functions in \(H^2_{z_i}({\mathbb D})\), the Hardy space on the unit disc with the independent variable \(z_i\). Let \(\underline{\eta}=(\eta_1(z_1),\dots,\eta_n(z_n))\) and let \(M_{\underline\eta}\) be the submodule generated by the elements \(\eta_i(z_i)-\eta_{i+1}(z_{i+1})\), \ \(i=1,\dots, n-1\). Let \(N_{\underline\eta}=M^\bot_{\underline\eta}\).NEWLINENEWLINE\textit{D. N. Clark} [Am. J. Math. 110, No. 6, 1119--1152 (1988; Zbl 0687.32008)] proved that \(N_{\underline\eta}\) is essentially normal if \(\eta_i\) are finite Blaschke products. The main result of the paper is the following.NEWLINENEWLINE{Theorem}. If \(\eta_i\) are nonconstant inner functions in \(H^2_{z_i}({\mathbb D})\), then the quotient module \(N_{\underline\eta}\) is essentially normal only if \(\eta_i\) are finite Blaschke products.