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

TL;DR

This paper investigates $N_{ ext{p}}$-type quotient modules on the torus, providing insights into their structure and connections with classical single-variable operator theories, which aids in understanding the complex structure of quotient modules in Hardy spaces.

Contribution

It introduces and analyzes $N_{ ext{p}}$-type quotient modules, highlighting their structure and relation to classical operator theories, offering new examples for studying quotient modules.

Findings

$N_{ ext{p}}$-type quotient modules have a rich structure.

These modules are closely connected to classical single-variable operator theories.

They serve as useful examples for understanding quotient modules in Hardy spaces.

Abstract

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