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

B. Krishna Das; Jaydeb Sarkar

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

B. Krishna Das ¹ ; Jaydeb Sarkar ¹

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

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

Résumé
Abstract

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].

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].

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 :

B. Krishna Das ¹ ; Jaydeb Sarkar ¹

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

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     author = {B. Krishna Das and Jaydeb Sarkar},
     title = {Rudin's submodules of $ {H}^{2}({\mathbb{D}}^{2})$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {51--55},
     publisher = {Elsevier},
     volume = {353},
     number = {1},
     year = {2015},
     doi = {10.1016/j.crma.2014.10.005},
     language = {en},
}

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B. Krishna Das; Jaydeb Sarkar. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.10.005/

Bibliographie
Cité par

[1] A. Aleman; S. Richter; C. Sundberg Beurling's theorem for the Bergman space, Acta Math., Volume 177 (1996), pp. 275-310

[2] A. Chattopadhyay; B.K. Das; J. Sarkar; S. Sarkar Wandering subspaces of the Bergman space and the Dirichlet space over polydisc, Integral Equ. Oper. Theory, Volume 79 (2014) no. 4, pp. 567-577

[3] R. Douglas; V. Paulsen Hilbert Modules over Function Algebras, Research Notes in Mathematics Series, vol. 47, Longman, Harlow, UK, 1989

[4] R. Douglas; R. Yang Operator theory in the Hardy space over the bidisk (I), Integral Equ. Oper. Theory, Volume 38 (2000), pp. 207-221

[5] K.J. Izuchi; K.H. Izuchi; Y. Izuchi Ranks of invariant subspaces of the Hardy space over the bidisk, J. Reine Angew. Math., Volume 659 (2011), pp. 101-139

[6] T. Nakazi Szegö's theorem on a bidisk, Trans. Amer. Math. Soc., Volume 328 (1991), pp. 421-432

[7] W. Rudin Function Theory in Polydiscs, Benjamin, New York, 1969

[8] J. Sarkar; A. Sasane; B. Wick Doubly commuting submodules of the Hardy module over polydiscs, Stud. Math., Volume 217 (2013), pp. 179-192

[9] M. Seto A new proof that Rudin's module is not finitely generated, Recent Advances in Operator Theory and Applications, Operator Theory: Advances and Applications, vol. 187, 2009, pp. 195-197

[10] M. Seto Infinite sequences of inner functions and submodules in $H^{2} (D^{2})$ , J. Oper. Theory, Volume 61 (2009), pp. 75-86

[11] M. Seto A perturbation theory for core operators of Hilbert–Schmidt submodules, Integral Equ. Oper. Theory, Volume 70 (2011), pp. 379-394

[12] M. Seto; R. Yang Inner sequence based invariant subspaces in $H^{2} (D^{2})$ , Proc. Amer. Math. Soc., Volume 135 (2007), pp. 2519-2526

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