TY - JOUR
T1 - Hilbert-Schmidtness of Submodules in $H^2 (\mathbb{D}^2 )$ Containing $θ(z)−\varphi (w)$
AU - Zu , Chao
AU - Yang , Yixin
AU - Lu , Yufeng
JO - Communications in Mathematical Research 
VL - 3
SP - 331
EP - 341
PY - 2023
DA - 2023/04
SN - 39
DO - http://doi.org/10.4208/cmr.2022-0034
UR - https://global-sci.org/intro/article_detail/cmr/21606.html
KW - Hardy space over the bidisk, Hilbert-Schmidt submodule, fringe operator, Fredholm index.
AB - <p style="text-align: justify;">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.</p>