In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc

Nikolski, Nikolai

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 ${\left(\varphi \left(nx\right)\right)}_{n\ge 1}$ 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 $H^2\left({𝔻}_2^{\infty }\right)$ 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) $Re\left(f\right(z\left)\right)\ge 0$, $z\in {𝔻}_2^{\infty }$ ; 3) $f\in Hol\left((1+ϵ){𝔻}_2^{\infty }\right)$ 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).

Completeness of a dilation system ${\left(\varphi \left(nx\right)\right)}_{n\ge 1}$ 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 $H^2\left({𝔻}_2^{\infty }\right)$ 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) $Re\left(f\right(z\left)\right)\ge 0$, $z\in {𝔻}_2^{\infty }$; 3) $f\in Hol\left((1+ϵ){𝔻}_2^{\infty }\right)$ 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).

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