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://www.numdam.org/articles/10.5802/aif.2731/

[1] Agrawal, O.; Clark, D.; Douglas, R. Invariant subspaces in the polydisk, Pacific J. Math., Volume 121 (1986), pp. 1-11 | DOI | MR | Zbl

[2] Báez-Duarte, L. A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis, Atti Acad. Naz. Lincei, Volume 14 (2003), pp. 5-11 | MR | Zbl

[3] Báez-Duarte, L.; Balazard, M.; Landreau, B.; Saias, E. Notes sur la fonction ζ de Riemann, 3, Adv. Math., Volume 149 (2000) no. 1, pp. 130-144 | DOI | MR | Zbl

[4] Bagchi, B. On Nyman, Beurling and Baez-Duarte’s Hilbert space reformulation of the Riemann hypothesis, Proc. Indian Acad. Sci. (Math. Sci.), Volume 116 (2006) no. 2, pp. 137-146 | DOI | MR | Zbl

[5] Balazard, M. Completeness problems and the Riemann hypothesis: an annotated bibliography, Number theory for the millenium (Proc. Millenial Conf. Number Theory, Urbana, IL, 2000), (M.A.Bennett et al., eds), AK Peters, Boston, 2002, pp. 21-48 | MR | Zbl

[6] Beurling, A. A closure problem related to the Riemann zeta-function, Proc. Nat. Acad. USA, Volume 41 (1955) no. 5, pp. 312-314 | DOI | MR | Zbl

[7] Beurling, A. On the completeness of {ψ(nt)} on L 2 (0,1), Harmonic Analysis, Contemp. Mathematicians (The collected Works of Arne Beurling), Volume 2, Birkhaüser, Boston, 1989, pp. 378-380 | MR

[8] Bohr, H. Über die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichletschen Reien a n n s , Nachr. Ges. Wiss. Göttingen. Math.-Phys. Kl., 1913, A9

[9] Cole, B.; Gamelin, T. Representing measures and Hardy spaces for the infinite polydisk algebra, Proc. London Math. Soc. (3), Volume 53 (1986), pp. 112-142 | DOI | MR | Zbl

[10] Cotlar, M.; Sadosky, C. A polydisc version of Beurling’s characterization for invariant subspaces of finite multi-codimension, Contemp. Math., Volume 212 (1998), pp. 51-56 | DOI | MR | Zbl

[11] Gelca, R. Topological Hilbert Nullstellensatz for Bergman spaces, Int. Equations Operator Theory, Volume 28 (1997) no. 2, pp. 191-195 | DOI | MR | Zbl

[12] Ginsberg, J.; Neuwirth, J.; Newman, D. Approxiamtion by f(kx), J. Funct. Anal., Volume 5 (1970), pp. 194-203 | DOI | MR | Zbl

[13] Green, B.; Tao, T. The primes contain arbitrarily long arithmetic progressions, Annals of Math., Volume 167 (2008), pp. 481-547 | DOI | MR | Zbl

[14] Hedenmalm, H.; Lindquist, P.; Seip, K. A Hilbert space of Dirichlet series and sytems of dilated functions in L 2 (0,1), Duke Math. J., Volume 86 (1997), pp. 1-37 | DOI | MR | Zbl

[15] Hedenmalm, H.; Lindquist, P.; Seip, K. Addendum to “A Hilbert space of Dirichlet series and sytems of dilated functions in L 2 (0,1), Duke Math. J., Volume 99 (1999), pp. 175-178 | DOI | MR | Zbl

[16] Hilbert, D. Wesen und Ziele einer Analysis der unendlich vielen unabhängigen Variablen, Rend. Cir. Mat. Palermo, Volume 27 (1909), pp. 59-74 | DOI

[17] Krantz, S.; Parks, H. A Primer of Real Analytic Functions, Birkhaüser, Basel, 1992 | MR | Zbl

[18] Lojaciewicz, S. Sur le problème de la division, Studia Math., Volume 18 (1959), pp. 87-136 | MR | Zbl

[19] Lopez, J.; Ross, K. Sidon sets, N.Y., M.Dekker, 1975 | MR | Zbl

[20] Mandrekar, V. The validity of Beurling theorems in polydiscs, Proc. Amer. Math. Soc., Volume 103 (1998) no. 1, pp. 145-148 | DOI | MR | Zbl

[21] Nikolski, N. Selected problems of weighted approximation and spectral analysis, 120, Trudy Math. Inst. Steklova, Moscow (Russian), 1974 English transl.: Proc. Steklov Math. Inst., No 120 (1974), AMS, Providence, 1976 | MR | Zbl

[22] Nyman, B. On the one-dimensional translation group and semi-group in certain function spaces, Thesis, Uppsala Univ., 1950 | MR | Zbl

[23] Olofsson, A. On the shift semigroup on the Hardy space of Dirichlet series, Acta Math. Hungar., 2010 | MR | Zbl

[24] Rudin, W. Function theory in polydiscs, W.A.Benjamin, Inc, N.Y. - Amsterdam, 1969 | MR | Zbl

[25] Shamoyan, F. Weakly invertible elements in weighted anisothropic spaces of holomorphic functions in a polydisk, Mat. Sbornik, Volume 193:6 (1998), pp. 143-160 (Russian); Engl. transl. | MR | Zbl

[26] Shamoyan, R.; Li, Haiying On weakly invertible functions in the unit ball and polydisk and related problems, J. Math. Analysis, Volume 1:1 (2010), pp. 8-19 | MR

[27] Vasyunin, V. On a biorthogonal system related with the Riemann hypothesis, Algebra i Analyz, Volume 7 (1995), pp. 118-135 (Russian); English transl.: St.Petersburg Math. J. 7 (1996), 405-419 | MR | Zbl

[28] Wintner, A. Diophantine approximation and Hilbert’s space, Amer. J. Math., Volume 66 (1944), pp. 564-578 | DOI | MR | Zbl

[29] Zhu, K Spaces of Holomorphic Functions in the Unit Ball, Springer, New York, 2005 | MR | Zbl

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