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Comptes Rendus Mathématique
Volume 353, n° 1
pages 51-55 (janvier 2015)
Doi : 10.1016/j.crma.2014.10.005
Received : 6 May 2014 ;  accepted : 10 October 2014
Rudin's submodules of  
Sous-modules de Rudin de  
 

B. Krishna Das , Jaydeb Sarkar
 Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India 

Abstract

Let   be a sequence of scalars in the open unit disc of  , and let   be a sequence of natural numbers satisfying  . Then the joint   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 thatdim(SΦ⊖(z1SΦ+z2SΦ))=1+#{n≥0:αn=0}<∞. In particular, this answers a question earlier raised by Douglas and Yang (2000) [[4]].

The full text of this article is available in PDF format.
Résumé

Soit   une suite de scalaires du disque unité ouvert de  , et soit   une suite de nombres naturels vérifiant  . Alors le sous-espace invariant  
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 quedim(SΦ⊖(z1SΦ+z2SΦ))=1+#{n≥0:αn=0}<∞. En particulier, ce résultat répond à une question posée précédemment par Douglas et Yang (2000) [[4]].

The full text of this article is available in PDF format.


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