Invariant subspaces of the Cesàro operator
[Sous-espaces invariants de l’opérateur de Cesàro]
Gallardo-Gutiérrez, Eva A.1, 2 ; Partington, Jonathan R.3 ; Ross, William T.4
1Departamento de Análisis, Matemático y Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid (Spain)
2Instituto de Ciencias, Matemáticas ICMAT, (CSIC-UAM-UC3M-UCM), Madrid (Spain)
3School of Mathematics, University of Leeds, Leeds LS2 9JT (United Kingdom)
4Department of Mathematics, and Statistics, University of Richmond, Richmond, VA 23173 (USA)
Annales de l'Institut Fourier, Online first, 37 p.

This paper explores various classes of invariant subspaces of the classical Cesàro operator $C$ on the Hardy space $H^2$. We provide a characterization of the finite co-dimensional $C$-invariant subspaces, based on earlier work of the first two authors, and determine exactly which model spaces are $C$-invariant subspaces; using this, we describe the $C$-invariant subspaces contained in model spaces, which we show are all cyclic. Along the way, we re-examine an associated Hilbert space of analytic functions on the unit disk developed by Kriete and Trutt. We also make a connection between the adjoint of the Cesàro operator and certain composition operators on $H^2$ which have universal translates in the sense of Rota.

Cet article explore différentes classes de sous-espaces invariants de l’opérateur classique de Cesàro $C$ sur l’espace de Hardy $H^2$. Nous fournissons une caractérisation des sous-espaces $C$-invariants de codimension finie, basée sur des travaux antérieurs des deux premiers auteurs, et nous déterminons exactement quels espaces modèles sont des sous-espaces $C$-invariants ; en utilisant cela, nous décrivons les sous-espaces $C$-invariants contenus dans les espaces modèles, dont nous montrons qu’ils sont tous cycliques. En passant, nous réexaminons un espace de Hilbert associé de fonctions analytiques sur le disque unité développé par Kriete et Trutt. Nous faisons également un lien entre l’adjoint de l’opérateur de Cesàro et certains opérateurs de composition sur $H^2$ qui ont des translations universelles au sens de Rota.

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Révisé le :
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DOI : 10.5802/aif.3774
Classification : 47A15, 47A55, 47B15
Keywords: Cesàro operator, Hardy space, invariant subspace
Mots-clés : opérateur de Cesàro, espace de Hardy, sous-espace invariant

Gallardo-Gutiérrez, Eva A.  1 , 2   ; Partington, Jonathan R.  3   ; Ross, William T.  4

1 Departamento de Análisis, Matemático y Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid (Spain)
2 Instituto de Ciencias, Matemáticas ICMAT, (CSIC-UAM-UC3M-UCM), Madrid (Spain)
3 School of Mathematics, University of Leeds, Leeds LS2 9JT (United Kingdom)
4 Department of Mathematics, and Statistics, University of Richmond, Richmond, VA 23173 (USA) 
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Gallardo-Gutiérrez, Eva A.; Partington, Jonathan R.; Ross, William T. Invariant subspaces of the Cesàro operator. Annales de l'Institut Fourier, Online first, 37 p.

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