Hilbert-Schmidtness of some finitely generated submodules in [formula omitted

Abstract

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.