Hilbert-Schmidtness of some finitely generated submodules in $H^2(\mathbb{D}^2)$

TL;DR

This paper investigates the Hilbert-Schmidt property of finitely generated submodules in the Hardy space over the bidisk, proving that those containing a specific type of element are Hilbert-Schmidt, advancing understanding of submodule structure.

Contribution

It proves that finitely generated submodules containing a certain element are Hilbert-Schmidt, addressing an open problem in the theory of Hardy space submodules.

Findings

Finitely generated submodules with $z_1 - ext{Blaschke product}$ are Hilbert-Schmidt.

Discussion of fringe operator and Fredholm index related to these submodules.

Provides partial answers to the open problem on Hilbert-Schmidt property of submodules.

Abstract

A closed subspace $\mathcal{M}$ of the Hardy space $H^2(\mathbb{D}^2)$ over the bidisk is called a submodule if it is invariant under multiplication by coordinate functions $z_1$ and $z_2$. Whether every finitely generated submodule is Hilbert-Schmidt is an unsolved problem. This paper proves that every finitely generated submodule $\mathcal{M}$ containing $z_1 - \varphi(z_2)$ is Hilbert-Schmidt, where $\varphi$ is any finite Blaschke product. Some other related topics such as fringe operator and Fredholm index are also discussed.