(untitled) (Communications in Mathematical Research, 2023)
A closed subspace M of the Hardy space H^2(D^2) over the bidisk is called submodule if it is invariant under multiplication by coordinate functions z and w. Whether every finitely generated submodule is Hilbert-Schmidt is an unsolved problem. This paper proves that every finitely generated submodule M containing θ(z)−φ(w) is Hilbert-Schmidt, where θ(z), φ(w) are two finite Blaschke products.