Cyclicity of composition operators on the Paley–Wiener spaces

Pham Viet Hai; Waleed Noor; Osmar Reis Severiano

doi:10.5802/crmath.765

Article de recherche - Analyse fonctionnelle

Cyclicity of composition operators on the Paley–Wiener spaces
[Cyclicité des opérateurs de composition sur les espaces de Paley–Wiener]

Pham Viet Hai ¹ ; Waleed Noor ² ; Osmar Reis Severiano ²

¹ Faculty of Mathematics and Informatics, Hanoi University of Science and Technology, Khoa Toan-Tin, Dai hoc Bach khoa Hanoi, 1 Dai Co Viet, Hanoi, Vietnam
² IMECC, Universidade Estadual de Campinas, Campinas-SP, Brazil

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 879-886

Abstract (VO)
Résumé

In this article we characterize the cyclicity of bounded composition operators $C_\phi f=f\circ \phi $ on the Paley–Wiener spaces of entire functions $B^2_\sigma $ for $\sigma >0$. We show that $C_\phi $ is cyclic precisely when $\phi (z)=z+b$ where either $b\in \mathbb{C}\setminus \mathbb{R}$ or $b\in \mathbb{R}$ with $0<\vert b \vert \le \pi /\sigma $. We also describe when the reproducing kernels of $B^2_\sigma $ are cyclic vectors for $C_\phi $ and see that this is related to a question of completeness of exponential sequences in $L^2[-\sigma ,\sigma ]$. The interplay between cyclicity and complex symmetry plays a key role in this work.

Dans cet article, nous caractérisons la cyclicité des opérateurs de composition bornée $C_\phi f=f\circ \phi $ sur les espaces de Paley–Wiener des fonctions entières $B^2_\sigma $ pour $\sigma >0$. Nous montrons que $C_\phi $ est cyclique précisément lorsque $\phi (z)=z+b$ où $b\in \mathbb{C}\setminus \mathbb{R}$ ou $b\in \mathbb{R}$ avec $0<\vert b \vert \le \pi /\sigma $. Nous décrivons également lorsque les noyaux reproducteurs de $B^2_\sigma $ sont des vecteurs cycliques pour $C_\phi $ et voyons que cela est lié à une question de complétude des suites exponentielles dans $L^2[-\sigma ,\sigma ]$. L’interaction entre cyclicité et symétrie complexe joue un rôle clé dans ce travail.

Reçu le : 2025-05-04
Accepté le : 2025-06-07
Publié le : 2025-07-21

DOI : 10.5802/crmath.765

Keywords: Cyclic operator, composition operator, Paley–Wiener space
Mots-clés : Opérateur cyclique, opérateur de composition, espace Paley–Wiener

Affiliations des auteurs :

Pham Viet Hai ¹ ; Waleed Noor ² ; Osmar Reis Severiano ²

¹ Faculty of Mathematics and Informatics, Hanoi University of Science and Technology, Khoa Toan-Tin, Dai hoc Bach khoa Hanoi, 1 Dai Co Viet, Hanoi, Vietnam
² IMECC, Universidade Estadual de Campinas, Campinas-SP, Brazil

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Pham Viet Hai and Waleed Noor and Osmar Reis Severiano},
     title = {Cyclicity of composition operators on the {Paley{\textendash}Wiener} spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {879--886},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     doi = {10.5802/crmath.765},
     language = {en},
}

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Pham Viet Hai; Waleed Noor; Osmar Reis Severiano. Cyclicity of composition operators on the Paley–Wiener spaces. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 879-886. doi: 10.5802/crmath.765

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