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Volume 39, Issue 3
Hilbert-Schmidtness of Submodules in $H^2 (\mathbb{D}^2 )$ Containing $θ(z)−\varphi (w)$

Chao Zu, Yixin Yang & Yufeng Lu

DOI: 10.4208/cmr.2022-0034

Commun. Math. Res., 39 (2023), pp. 331-341.

Published online: 2023-04

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  • 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.

  • Keywords

Hardy space over the bidisk, Hilbert-Schmidt submodule, fringe operator, Fredholm index.

  • AMS Subject Headings

Primary 43A65, 47A13

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-39-331, author = {Zu , ChaoYang , Yixin and Lu , Yufeng}, title = {Hilbert-Schmidtness of Submodules in $H^2 (\mathbb{D}^2 )$ Containing $θ(z)−\varphi (w)$}, journal = {Communications in Mathematical Research }, year = {2023}, volume = {39}, number = {3}, pages = {331--341}, 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.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2022-0034}, url = {http://global-sci.org/intro/article_detail/cmr/21606.html} }
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 -

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.

Chao Zu, Yixin Yang & Yufeng Lu. (2023). Hilbert-Schmidtness of Submodules in $H^2 (\mathbb{D}^2 )$ Containing $θ(z)−\varphi (w)$. Communications in Mathematical Research . 39 (3). 331-341. doi:10.4208/cmr.2022-0034
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