Comptes Rendus
no. 19-20
Complex Analysis/Functional Analysis
Composition operators on Hilbert spaces of entire Dirichlet series
[Opérateurs de composition sur les espaces de Hilbert de séries entières de Dirichlet]

Présenté par : the Editorial Board

Xiaolu Hou1 ; Bingyang Hu1 ; Le Hai Khoi1
1 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University (NTU), 637371, Singapore
Comptes Rendus. Mathématique, Volume 350 (2012) no. 19-20, pp. 875-878.

Dans cette Note, nous introduisons des espaces de Hilbert de séries entières de Dirichlet (à fréquences réelles) et considérons des opérateurs de composition sur ces espaces. Nous donnons des conditions nécessaires et suffisantes sur de telles séries pour avoir un ordre de Ritt égal à zéro, ainsi quʼun ordre logarithmique fini. Nous obtenons des critères dʼaction, de continuité, de compacité pour de tels opérateurs ou leurs différences.

In this Note, we introduce Hilbert spaces of entire Dirichlet series (with real frequencies) and consider composition operators on these spaces. We establish necessary and sufficient conditions for such series to have Ritt order zero, as well as a finite logarithmic order. Criteria for action, boundedness, compactness and compact difference of such operators are obtained.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.10.012
Xiaolu Hou 1 ; Bingyang Hu 1 ; Le Hai Khoi 1

1 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University (NTU), 637371, Singapore 
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Xiaolu Hou; Bingyang Hu; Le Hai Khoi. Composition operators on Hilbert spaces of entire Dirichlet series. Comptes Rendus. Mathématique, Volume 350 (2012) no. 19-20, pp. 875-878. doi : 10.1016/j.crma.2012.10.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.10.012/

[1] C. Cowen; B. MacCluer Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995

[2] X. Hou; L.H. Khoi Some properties of composition operators on entire Dirichlet series with real frequencies, C. R. Acad. Sci. Paris, Ser. I, Volume 350 (2012) no. 3–4, pp. 149-152

[3] L.H. Khoi Hilbert spaces of holomorphic Dirichlet series and applications to convolution equations, J. Math. Anal. Appl., Volume 206 (1997) no. 1, pp. 10-24

[4] G. Polya On an integral function of an integral function, J. Lond. Math. Soc., Volume 1 (1926) no. 1, p. 12

[5] A.R. Reddy On entire Dirichlet series of zero order, Tôhoku Math. J. (2), Volume 18 (1966), pp. 144-155

[6] J.F. Ritt On certain points in the theory of Dirichlet series, Amer. J. Math., Volume 50 (1928) no. 1, pp. 73-86

[7] J.H. Shapiro Compositions Operators and Classical Function Theory, Springer-Verlag, New York, 1993

[8] A.L. Shields Weighted shift operators and analytic function theory, Topics in Operator Theory, Math. Surveys, vol. 13, Amer. Math. Soc., Providence, RI, 1974, pp. 49-128

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