Rudin's submodules of $ {H}^{2}({\mathbb{D}}^{2})$

Das, B. Krishna; Sarkar, Jaydeb

doi:10.1016/j.crma.2014.10.005

Functional analysis

Rudin's submodules of

H^{2} (D^{2})

[Sous-modules de Rudin de

H^{2} (D^{2})

]

Présenté par : Gilles Pisier

Das, B. Krishna ¹ ; Sarkar, Jaydeb ¹

¹ Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India

Comptes Rendus. Mathématique, Tome 353 (2015) no. 1, pp. 51-55

Abstract (VO)
Résumé

Let ${α_{n}}_{n \geq 0}$ be a sequence of scalars in the open unit disc of $C$ , and let ${l_{n}}_{n \geq 0}$ be a sequence of natural numbers satisfying $\sum_{n = 0}^{\infty} (1 - l_{n} | α_{n} |) < \infty$ . Then the joint $(M_{z_{1}}, M_{z_{2}})$ invariant subspace

S_{Φ} = ⋁_{n = 0}^{\infty} (z_{1}^{n} \prod_{k = n}^{\infty} {(\frac{- {\bar{α}}_{k}}{| α_{k} |} \frac{z_{2} - α_{k}}{1 - {\bar{α}}_{k} z_{2}})}^{l_{k}} H^{2} (D^{2})),

is called a Rudin submodule. In this paper, we analyze the class of Rudin submodules and prove that

\dim (S_{Φ} ⊖ (z_{1} S_{Φ} + z_{2} S_{Φ})) = 1 + # {n \geq 0 : α_{n} = 0} < \infty .

In particular, this answers a question earlier raised by Douglas and Yang (2000) [4].

Soit ${α_{n}}_{n \geq 0}$ une suite de scalaires du disque unité ouvert de $C$ , et soit ${l_{n}}_{n \geq 0}$ une suite de nombres naturels vérifiant $\sum_{n = 0}^{\infty} (1 - l_{n} | α_{n} |) < \infty$ . Alors le sous-espace invariant $(M_{z_{1}}, M_{z_{2}})$

S_{Φ} = ⋁_{n = 0}^{\infty} (z_{1}^{n} \prod_{k = n}^{\infty} {(\frac{- \bar{α_{k}}}{| α_{k} |} \frac{z_{2} - α_{k}}{1 - {\bar{α}}_{k} z_{2}})}^{l_{k}} H^{2} (D^{2})),

est appelé sous-module de Rudin. Dans cette Note, on analyse la classe des sous-modules de Rudin et on démontre que

\dim (S_{Φ} ⊖ (z_{1} S_{Φ} + z_{2} S_{Φ})) = 1 + # {n \geq 0 : α_{n} = 0} < \infty .

En particulier, ce résultat répond à une question posée précédemment par Douglas et Yang (2000) [4].

Reçu le : 2014-05-06
Accepté le : 2014-10-10
Publié le : 2014-10-25

DOI : 10.1016/j.crma.2014.10.005

Affiliations des auteurs :

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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     title = {Rudin's submodules of $ {H}^{2}({\mathbb{D}}^{2})$},
     journal = {Comptes Rendus. Math\'ematique},
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Das, B. Krishna; Sarkar, Jaydeb. Rudin's submodules of $ {H}^{2}({\mathbb{D}}^{2})$. Comptes Rendus. Mathématique, Tome 353 (2015) no. 1, pp. 51-55. doi: 10.1016/j.crma.2014.10.005

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