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Comptes Rendus Mathématique
Volume 335, n° 4
pages 325-328 (2002)
Doi : S1631-073X(02)02483-4
Received : 17 June 2002 ;  accepted : 24 June 2002
Image numérique et compacité d'opérateurs de composition sur un espace de Hilbert de séries de Dirichlet
Numerical range and compacity of some composition operators on a Hilbert space of Dirichlet series
 

Catherine Finet a , Hervé Queffélec b , Alexander Volberg c
a Institut de mathématique, Université de Mons-Hainaut, « Le Pentagone », avenue du Champ de Mars, 6, 7000 Mons, Belgique 
b UFR de mathématiques, Université de Lille 1, 59655 Villeneuve d'Ascq cedex, France 
c UFR de mathématiques, Université Pierre et Marie Curie - Paris 6, 75252 Paris cedex 05, France 

Note présentée par Jean-Pierre Kahane

Résumé

Dans cette Note, nous étudions l'image numérique et la compacité de certains opérateurs de composition, définis sur un espace de Hilbert de séries de Dirichlet introduit par Hedenmalm, Lindqvist et Seip. Nous montrons que dans la plupart des cas, zéro est un point intérieur à cette image numérique. L'étude de la compacité fait apparaître un phénomène rappelant le théorème de Polya sur les marches aléatoires en dimension d : si la longueur du symbole de l'opérateur est d +1, et l'image du symbole est non-triviale, alors l'opérateur est non compact si d =1 ; compact, non Hilbert-Schmidt, si d =2 ; Hilbert-Schmidt si d ⩾3. Pour citer cet article : C. Finet et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 325-328.

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Abstract

In this Note, we study the numerical range and compacity of some composition operators, defined on a Hilbert space of Dirichlet series introduced by Hedenmalm, Linqvist and Seip. We show that most often, zero is the interior of this numerical range. The study of compacity exhibits a phenomenon which recalls Polya's theorem on random walks in dimension d : if the length of the symbol of the operator is d +1, and if its image is non-trivial, then the operator is non-compact if d =1; compact, non Hilbert-Schmidt, if d =2; Hilbert-Schmidt if d ⩾3. To cite this article: C. Finet et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 325-328.

The full text of this article is available in PDF format.


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