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]

Nikolski, Nikolai

Annales de l'Institut Fourier, Tome 62 (2012), p. 1601-1626 / Harvested from Numdam

Résumé

Il s’agit du problème de la complétude d’un système de dilatations $ (\varphi (nx))_{n\ge 1}$ dans l’espace de Lebesgue $ L^{2}(0,1)$ où $ \varphi $ 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}(\mathbb{D}_{2}^{\infty })$ du multidisque $ \mathbb{D}_{2}^{\infty }$ 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}(\mathbb{D}_{2}^{\infty })$ : 1) $ f^{1+\epsilon }\in H^{2}(\mathbb{D}_{2}^{\infty })$, $ f^{-\epsilon }\in H^{2}(\mathbb{D}_{2}^{\infty })$ ; 2) $ Re(f(z))\ge 0$, $ z\in \mathbb{D}_{2}^{\infty }$ ; 3) $ f\in Hol((1+\epsilon )\mathbb{D}_{2}^{\infty })$ et $ f(z)\ne 0$ sur $ \mathbb{D}_{2}^{\infty }$. L’Hypothèse de Riemann sur les zéros de la fonction $ \zeta $ d’Euler est équivalente à un problème semblable de la complétude des dilatations (B.Nyman).

Abstract

Completeness of a dilation system $ (\varphi (nx))_{n\ge 1}$ on the standard Lebesgue space $ L^{2}(0,1)$ is considered for 2-periodic functions $ \varphi $. We show that the problem is equivalent to an open question on cyclic vectors of the Hardy space $ H^{2}(\mathbb{D}_{2}^{\infty })$ on the Hilbert multidisc $ \mathbb{D}_{2}^{\infty }$. 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 $ f\in H^{2}(\mathbb{D}_{2}^{\infty })$: 1) $ f^{1+\epsilon }\in H^{2}(\mathbb{D}_{2}^{\infty })$, $ f^{-\epsilon }\in H^{2}(\mathbb{D}_{2}^{\infty })$; 2) $ Re(f(z))\ge 0$, $ z\in \mathbb{D}_{2}^{\infty }$; 3) $ f\in Hol((1+\epsilon )\mathbb{D}_{2}^{\infty })$ and $ f(z)\ne 0$ on $ \mathbb{D}_{2}^{\infty }$. The Riemann Hypothesis on zeros of the Euler $ \zeta $-function is known to be equivalent to a completeness of a similar but non-periodic dilation system (due to Nyman).

Détail

Publié le : 2012-01-01

MR 3025149 | Zbl 1267.30108

DOI : https://doi.org/10.5802/aif.2731
Classification:  32A35,  32A60,  42B30,  42C30,  47A16
Mots clés: semigroupe de dilatation, multidisque d’Hilbert, vecteurs cycliques, fonctions extérieure, problème de complétude, l’hypothèse de Riemann

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