$N_{\text{p}}$-type quotient modules on the torus

Structure of the quotient modules in $\text{hh}$ is very complicated. A good understanding of some special examples will shed light on the general picture. This paper studies the so-call $N_{\text{p}}$-type quotient modules, namely, quotient modules of the form $\text{hh}\ominus [z-\text{p}]$, where $\text{p}(w)$ is a function in the classical Hardy space $H^2(\text{G})$ and $[z-\text{p}]$ is the submodule generated by $z-\text{p}(w)$. This type of quotient modules serve as good examples in many studies. A notable feature of the $N_{\text{p}}$-type quotient module is its close connections with some classical single variable operator theories.