Cyclicity of bicyclic operators

Evgeny Abakumov; Aharon Atzmon; Sophie Grivaux

doi:10.1016/j.crma.2007.02.008

Functional Analysis

Cyclicity of bicyclic operators

Presented by: Gilles Pisier

Evgeny Abakumov ¹; Aharon Atzmon ²; Sophie Grivaux ³

¹ Laboratoire d'Analyse et de Mathématiques Appliquées, UMR 8050, Université de Marne-la-Vallée, 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée, France
² School of Mathematical Sciences, Tel Aviv University, Ramat-Aviv 69978, Israel
³ Laboratoire Paul-Painlevé, UMR 8524, Université des sciences et technologies de Lille, Cité Scientifique, bâtiment M2, 59655 Villeneuve d'Ascq cedex, France

Comptes Rendus. Mathématique, Volume 344 (2007) no. 7, pp. 447-452.

Abstract
Résumé

We study cyclicity of injective operators on separable Banach spaces which admit a bicyclic vector such that the norms of its images under the iterates of the operator satisfy certain growth conditions. Our results apply in particular to the shift operator acting on the weighted spaces of sequences $ℓ_{ω}^{2} (Z)$ . We also prove completeness results of translates in certain Banach spaces of functions on $R$ .

Nous étudions la cyclicité de certains opérateurs bicycliques agissant sur des espaces de Banach séparables. Nos résultats s'appliquent en particulier aux opérateurs de décalage sur les espaces de suites pondérés $ℓ_{ω}^{2} (Z)$ . Nous prouvons également des résultats concernant la complétude des translatés d'une fonction dans certains espaces de Banach de fonctions sur $R$ .

Received: 2007-01-15
Accepted: 2007-01-30
Published online: 2007-03-21

DOI: 10.1016/j.crma.2007.02.008

Author's affiliations:

Evgeny Abakumov ¹; Aharon Atzmon ²; Sophie Grivaux ³

¹ Laboratoire d'Analyse et de Mathématiques Appliquées, UMR 8050, Université de Marne-la-Vallée, 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée, France
² School of Mathematical Sciences, Tel Aviv University, Ramat-Aviv 69978, Israel
³ Laboratoire Paul-Painlevé, UMR 8524, Université des sciences et technologies de Lille, Cité Scientifique, bâtiment M2, 59655 Villeneuve d'Ascq cedex, France

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     title = {Cyclicity of bicyclic operators},
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Evgeny Abakumov; Aharon Atzmon; Sophie Grivaux. Cyclicity of bicyclic operators. Comptes Rendus. Mathématique, Volume 344 (2007) no. 7, pp. 447-452. doi : 10.1016/j.crma.2007.02.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.02.008/

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