Hilbert-Schmidtness of Submodules in $H^2 (\mathbb{D}^2 )$ Containing $θ(z)−\varphi (w)$

Abstract

A closed subspace $M$ of the Hardy space $H^2(\mathbb{D}^2)$ 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 $θ(z)−\varphi(w)$ is Hilbert-Schmidt, where $θ(z),$ $\varphi(w)$ are two finite Blaschke products.