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

Abstract:

The purpose of this note is to study the essential normality of $N_{\underline {\eta }}$-type quotient modules, generated by $\{\eta _i(z_i)-\eta _{i+1}(z_{i+1}) | i=1,\cdots ,n-1\}$, of the Hardy module over the polydisc for inner functions $\eta _i$. In 1988, Clark studied the structure of $N_{\underline {\eta }}$, for which $\eta _i$ are finite Blaschke products, and as a consequence, $N_{\underline {\eta }}$ is essentially normal if $\eta _i$ are finite Blaschke products. In this note, we will prove that to obtain the essential normality of $N_{\underline \eta }$, the condition that $\eta _i$ are finite Blaschke products is necessary.