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

B. Krishna Das 1; Jaydeb Sarkar 1

1 Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India
@article{CRMATH_2015__353_1_51_0,
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},
}
TY  - JOUR
AU  - B. Krishna Das
AU  - Jaydeb Sarkar
TI  - Rudin's submodules of ${H}^{2}({\mathbb{D}}^{2})$
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 51
EP  - 55
VL  - 353
IS  - 1
PB  - Elsevier
DO  - 10.1016/j.crma.2014.10.005
LA  - en
ID  - CRMATH_2015__353_1_51_0
ER  - 
%0 Journal Article
%A B. Krishna Das
%A Jaydeb Sarkar
%T Rudin's submodules of ${H}^{2}({\mathbb{D}}^{2})$
%J Comptes Rendus. Mathématique
%D 2015
%P 51-55
%V 353
%N 1
%I Elsevier
%R 10.1016/j.crma.2014.10.005
%G en
%F CRMATH_2015__353_1_51_0
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/

[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 $H2(D2)$, 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 $H2(D2)$, Proc. Amer. Math. Soc., Volume 135 (2007), pp. 2519-2526

Cited by Sources: