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

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.