Hilbert-Schmidtness of some finitely generated submodules in H2(D2)

A closed subspace M of the Hardy space H2(D2) over the bidisk is called a submodule if it is invariant under multiplication by coordinate functions z1 and z2. Whether every finitely generated submodule is Hilbert-Schmidt is an unsolved problem. This paper proves that every finitely generated submodule M containing z1−φ(z2) is Hilbert-Schmidt, where φ is any finite Blaschke product. Some other related topics such as fringe operator and Fredholm index are also discussed.