Comptes Rendus
no. 3-4
Complex Analysis/Functional Analysis
Some properties of composition operators on entire Dirichlet series with real frequencies
[Quelques propriétés des opérateurs de composition sur les séries de Dirichlet entières à fréquences réelles]

Présenté par : Jean-Pierre Demailly

Xiaolu Hou1 ; 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. 3-4, pp. 149-152.

Dans cette Note nous considérons quelques problèmes concernant les opérateurs de composition sur une classe de séries de Dirichlet entières à fréquences réelles dans le plan complexe, dont lʼordre de Ritt est zéro, et dont les ordres logarithmiques sont finis. Nous donnons des critères dʼaction et de continuité pour de tels opérateurs.

In this Note we consider some problems for composition operators on a class of entire Dirichlet series with real frequencies in the complex plane whose Ritt order is zero and logarithmic orders are finite. Criteria for action and boundedness of such operators are given.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2012.01.023
Xiaolu Hou 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; Le Hai Khoi. Some properties of composition operators on entire Dirichlet series with real frequencies. Comptes Rendus. Mathématique, Volume 350 (2012) no. 3-4, pp. 149-152. doi : 10.1016/j.crma.2012.01.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.01.023/

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[4] G.H. Hardy; M. Riesz The General Theory of Dirichlet Series, Stechert–Hafner, Inc., New York, 1964

[5] S. Mandelbrojt Séries de Dirichlet. Principes et Méthodes, Monographies Internationales de Mathématiques Modernes, vol. 11, Gauthier–Villars, Paris, 1969

[6] G. Polya On an integral function of an integral function, J. London Math. Soc., Volume 1 (1926) no. 2, p. 12

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

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

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

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