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

Let {αn}n0 be a sequence of scalars in the open unit disc of C, and let {ln}n0 be a sequence of natural numbers satisfying n=0(1ln|αn|)<. Then the joint (Mz1,Mz2) invariant subspace

SΦ=n=0(z1nk=n(α¯k|αk|z2αk1α¯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+#{n0:αn=0}<.
In particular, this answers a question earlier raised by Douglas and Yang (2000) [4].

Soit {αn}n0 une suite de scalaires du disque unité ouvert de C, et soit {ln}n0 une suite de nombres naturels vérifiant n=0(1ln|αn|)<. Alors le sous-espace invariant (Mz1,Mz2)

SΦ=n=0(z1nk=n(αk¯|αk|z2αk1α¯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+#{n0:αn=0}<.
En particulier, ce résultat répond à une question posée précédemment par Douglas et Yang (2000) [4].

Received:
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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