In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc
Annales de l'Institut Fourier, Volume 62 (2012) no. 5, p. 1601-1626

Completeness of a dilation system ${\left(\varphi \left(nx\right)\right)}_{n\ge 1}$ on the standard Lebesgue space ${L}^{2}\left(0,1\right)$ 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 ${H}^{2}\left({𝔻}_{2}^{\infty }\right)$ on the Hilbert multidisc ${𝔻}_{2}^{\infty }$. 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 $f\in {H}^{2}\left({𝔻}_{2}^{\infty }\right)$: 1) ${f}^{1+ϵ}\in {H}^{2}\left({𝔻}_{2}^{\infty }\right)$, ${f}^{-ϵ}\in {H}^{2}\left({𝔻}_{2}^{\infty }\right)$; 2) $Re\left(f\left(z\right)\right)\ge 0$, $z\in {𝔻}_{2}^{\infty }$; 3) $f\in Hol\left(\left(1+ϵ\right){𝔻}_{2}^{\infty }\right)$ and $f\left(z\right)\ne 0$ on ${𝔻}_{2}^{\infty }$. 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).

Il s’agit du problème de la complétude d’un système de dilatations ${\left(\varphi \left(nx\right)\right)}_{n\ge 1}$ dans l’espace de Lebesgue ${L}^{2}\left(0,1\right)$$\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 ${H}^{2}\left({𝔻}_{2}^{\infty }\right)$ du multidisque ${𝔻}_{2}^{\infty }$ 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 ${H}^{2}\left({𝔻}_{2}^{\infty }\right)$ : 1) ${f}^{1+ϵ}\in {H}^{2}\left({𝔻}_{2}^{\infty }\right)$, ${f}^{-ϵ}\in {H}^{2}\left({𝔻}_{2}^{\infty }\right)$ ; 2) $Re\left(f\left(z\right)\right)\ge 0$, $z\in {𝔻}_{2}^{\infty }$ ; 3) $f\in Hol\left(\left(1+ϵ\right){𝔻}_{2}^{\infty }\right)$ et $f\left(z\right)\ne 0$ sur ${𝔻}_{2}^{\infty }$. 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).

DOI : https://doi.org/10.5802/aif.2731
Classification:  32A35,  32A60,  42B30,  42C30,  47A16
Keywords: dilation semigroup, Hilbert’s multidisc, cyclic vector, outer function, completeness problem, Riemann hypothesis
@article{AIF_2012__62_5_1601_0,
author = {Nikolski, Nikolai},
title = {In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {62},
number = {5},
year = {2012},
pages = {1601-1626},
doi = {10.5802/aif.2731},
mrnumber = {3025149},
zbl = {1267.30108},
language = {en},
url = {http://www.numdam.org/item/AIF_2012__62_5_1601_0}
}

Nikolski, Nikolai. In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc. Annales de l'Institut Fourier, Volume 62 (2012) no. 5, pp. 1601-1626. doi : 10.5802/aif.2731. http://www.numdam.org/item/AIF_2012__62_5_1601_0/

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