Rudin's Submodules of $H^2(\mathbb{D}^2)$

TL;DR

This paper studies Rudin submodules of the Hardy space on the bidisc, characterizing their structure and dimension, and answering an open question about their properties.

Contribution

The paper provides a detailed analysis of Rudin submodules, including a finite-dimensionality result and a specific dimension formula, addressing a question posed by Douglas and Yang.

Findings

Dimension of the orthogonal complement is finite.

Explicit formula for the dimension involving zeroes of the sequence.

Answer to an open question by Douglas and Yang.

Abstract

Let $\{\alpha_n\}_{n\geq 0}$ be a sequence of scalars in the open unit disc of $\mathbb{C}$, and let $\{l_n\}_{n\geq 0}$ be a sequence of natural numbers satisfying $\sum_{n=0}^\infty (1 - l_n|\alpha_n|) <\infty$. Then the joint $(M_{z_1}, M_{z_2})$ invariant subspace \[\mathcal{S}_{\Phi} = \vee_{n=0}^\infty \Big( z_1^n \prod_{k=n}^\infty \left(\frac{-\bar{\alpha}_k}{|\alpha_k|} \frac{z_2 - \alpha_k}{1 - \bar{\alpha}_k z_2}\right)^{l_k} H^2(\mathbb{D}^2)\Big),\] is called a Rudin submodule. In this paper we analyze the class of Rudin submodules and prove that \[ \text{dim} (\mathcal{S}_{\Phi}\ominus (z_1 \mathcal{S}_{\Phi}+ z_2\mathcal{S}_{\Phi}))= 1+\#\{n\ge 0: \alpha_n=0\}<\infty. \]In particular, this answer a question earlier raised by Douglas and Yang (2000).