In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc
[À l’ombre de l’HR  : Vecteurs cycliques de l’espace de Hardy du multidisque hilbertien]
Annales de l'Institut Fourier, Tome 62 (2012) no. 5, pp. 1601-1626.

Il s’agit du problème de la complétude d’un système de dilatations (ϕ(nx)) n1 dans l’espace de Lebesgue L 2 (0,1)ϕ est une fonction impaire 2-périodique. Sans utiliser les séries de Dirichlet, on montre que le problème est équivalent à une question ouverte sur les vecteurs cycliques dans l’espace de Hardy H 2 (𝔻 2 ) du multidisque 𝔻 2 de Hilbert. Quelques conditions suffisantes de cyclicité sont établies, ce qui néanmoins inclut pratiquement tous les résultats précédents du sujet (ceux de Wintner ; Kozlov ; Neuwirth, Ginsberg, and Newman ; Hedenmalm, Lindquist, and Seip). Par exemple, chacune des conditions suivantes entraîne la cyclicité d’une fonction f dans H 2 (𝔻 2 ) : 1) f 1+ϵ H 2 (𝔻 2 ), f -ϵ H 2 (𝔻 2 ) ; 2) Re(f(z))0, z𝔻 2  ; 3) fHol((1+ϵ)𝔻 2 ) et f(z)0 sur 𝔻 2 . L’Hypothèse de Riemann sur les zéros de la fonction ζ d’Euler est équivalente à un problème semblable de la complétude des dilatations (B.Nyman).

Completeness of a dilation system (ϕ(nx)) n1 on the standard Lebesgue space L 2 (0,1) is considered for 2-periodic functions ϕ. We show that the problem is equivalent to an open question on cyclic vectors of the Hardy space H 2 (𝔻 2 ) on the Hilbert multidisc 𝔻 2 . Several simple sufficient conditions are exhibited, which include however practically all previously known results (Wintner; Kozlov; Neuwirth, Ginsberg, and Newman; Hedenmalm, Lindquist, and Seip). For instance, each of the following conditions implies cyclicity of a function fH 2 (𝔻 2 ): 1) f 1+ϵ H 2 (𝔻 2 ), f -ϵ H 2 (𝔻 2 ); 2) Re(f(z))0, z𝔻 2 ; 3) fHol((1+ϵ)𝔻 2 ) and f(z)0 on 𝔻 2 . The Riemann Hypothesis on zeros of the Euler ζ-function is known to be equivalent to a completeness of a similar but non-periodic dilation system (due to Nyman).

DOI : 10.5802/aif.2731
Classification : 32A35, 32A60, 42B30, 42C30, 47A16
Keywords: dilation semigroup, Hilbert’s multidisc, cyclic vector, outer function, completeness problem, Riemann hypothesis
Mot clés : semigroupe de dilatation, multidisque d’Hilbert, vecteurs cycliques, fonctions extérieure, problème de complétude, l’hypothèse de Riemann
Nikolski, Nikolai 1

1 Université de Bordeaux 1 UFR de Mathématiques et Informatique 351 cours de la Libération 33405 Talence France Steklov Institute of Mathematics 27 Fontanka 191023, St.Petersburg Russia
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Nikolski, Nikolai. In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc. Annales de l'Institut Fourier, Tome 62 (2012) no. 5, pp. 1601-1626. doi : 10.5802/aif.2731. http://archive.numdam.org/articles/10.5802/aif.2731/

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