Subnormality of the quotients of $\mathbb T^d$-invariant Hilbert modules

Abstract

In this paper, we investigate $\mathbb T^d$-invariant Hilbert modules $\mathscr H$ over the polynomial ring $\mathbb C[z_1, \ldots, z_d]$ and their quotients, with primary emphasis on the classification of subnormal quotient modules of the form $\mathscr H/[p],$ where $p$ is a homogeneous polynomial in $d$ complex variables. The motivation for this classification arises from the case $p(z_1, z_2)=z_1-z_2,$ in which the subnormality of the quotient module $\widehat{\mathscr H_{κ_1} \otimes \mathscr H_{κ_2}}/[p]$ is equivalent to that of the module tensor product $\mathscr H_{κ_1} \otimes_{\mathbb C[z]} \mathscr H_{κ_2}$ of $\mathbb T$-invariant Hilbert modules $\mathscr H_{κ_1}$ and $\mathscr H_{κ_2}$, a problem first considered by N. Salinas. In addition to general structural results on principal homogeneous submodules $[p]$ of $\mathscr H$, we prove that if $\mathscr H/[p]$ is subnormal, then $p$ must be square-free. Furthermore, when $\mathscr H$ is either $H^2(\mathbb D^d)$ or $H^2(\mathbb B^d),$ $d \ge 1,$ the subnormality of the quotient module $\mathscr H/[p]$ implies that $\mathrm{deg}\,p \le 1.$ We further show that $H^2(\mathbb D^2)/[p]$ (resp. $H^2(\mathbb B^2)/[p]$) is subnormal if and only if $\mathrm{deg} \,p \le 1.$ If $H^2_d$ denotes the Drury-Arveson module in $d$ dimensions, then $H^2_2/[p]$ is subnormal if and only if $p$ is nonzero and $\mathrm{deg} \,p \le 1$. This is surprising, especially since $H^2_d$ is not a subnormal Hilbert module for $d \ge 2.$ Moreover, the phenomenon above does not occur for the Dirichlet module $D_2(\mathbb B^2)$. Finally, we present an example demonstrating that a $\mathcal U_d$-invariant subnormal Hilbert module $\mathscr H$ may have a subnormal quotient module $\mathscr H/[p]$ even when $\mathrm{deg}\, p = 2.$