Functional analysis
Rudin's submodules of $H2(D2)$
Comptes Rendus. Mathématique, Volume 353 (2015) no. 1, pp. 51-55.

Let ${αn}n≥0$ be a sequence of scalars in the open unit disc of $C$, and let ${ln}n≥0$ be a sequence of natural numbers satisfying $∑n=0∞(1−ln|αn|)<∞$. Then the joint $(Mz1,Mz2)$ invariant subspace

 $SΦ=⋁n=0∞(z1n∏k=n∞(−α¯k|αk|z2−αk1−α¯kz2)lkH2(D2)),$
is called a Rudin submodule. In this paper, we analyze the class of Rudin submodules and prove that
 $dim(SΦ⊖(z1SΦ+z2SΦ))=1+#{n≥0:αn=0}<∞.$
In particular, this answers a question earlier raised by Douglas and Yang (2000) [4].

Soit ${αn}n≥0$ une suite de scalaires du disque unité ouvert de $C$, et soit ${ln}n≥0$ une suite de nombres naturels vérifiant $∑n=0∞(1−ln|αn|)<∞$. Alors le sous-espace invariant $(Mz1,Mz2)$

 $SΦ=⋁n=0∞(z1n∏k=n∞(−αk¯|αk|z2−αk1−α¯kz2)lkH2(D2)),$
est appelé sous-module de Rudin. Dans cette Note, on analyse la classe des sous-modules de Rudin et on démontre que
 $dim(SΦ⊖(z1SΦ+z2SΦ))=1+#{n≥0:αn=0}<∞.$
En particulier, ce résultat répond à une question posée précédemment par Douglas et Yang (2000) [4].

Accepted:
Published online:
DOI: 10.1016/j.crma.2014.10.005
Das, B. Krishna 1; Sarkar, Jaydeb 1

1 Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India
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Das, B. Krishna; Sarkar, Jaydeb. Rudin's submodules of ${H}^{2}({\mathbb{D}}^{2})$. Comptes Rendus. Mathématique, Volume 353 (2015) no. 1, pp. 51-55. doi : 10.1016/j.crma.2014.10.005. http://www.numdam.org/articles/10.1016/j.crma.2014.10.005/

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