  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.
