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, p. 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 : 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
@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, Tome 62 (2012) no. 5, pp. 1601-1626. doi : 10.5802/aif.2731. http://www.numdam.org/item/AIF_2012__62_5_1601_0/

[1] Agrawal, O.; Clark, D.; Douglas, R. Invariant subspaces in the polydisk, Pacific J. Math., Tome 121 (1986), pp. 1-11 | Article | MR 815027 | Zbl 0609.47012

[2] Báez-Duarte, L. A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis, Atti Acad. Naz. Lincei, Tome 14 (2003), pp. 5-11 | MR 2057270 | Zbl 1097.11041

[3] Báez-Duarte, L.; Balazard, M.; Landreau, B.; Saias, E. Notes sur la fonction ζ de Riemann, 3, Adv. Math., Tome 149 (2000) no. 1, pp. 130-144 | Article | MR 1742356 | Zbl 1008.11032

[4] Bagchi, B. On Nyman, Beurling and Baez-Duarte’s Hilbert space reformulation of the Riemann hypothesis, Proc. Indian Acad. Sci. (Math. Sci.), Tome 116 (2006) no. 2, pp. 137-146 | Article | MR 2226127 | Zbl 1125.11049

[5] Balazard, M. Completeness problems and the Riemann hypothesis: an annotated bibliography, Number theory for the millenium, (M.A.Bennett et al., eds), AK Peters, Boston (Proc. Millenial Conf. Number Theory, Urbana, IL, 2000) (2002), pp. 21-48 | MR 1956217 | Zbl 1044.11083

[6] Beurling, A. A closure problem related to the Riemann zeta-function, Proc. Nat. Acad. USA, Tome 41 (1955) no. 5, pp. 312-314 | Article | MR 70655 | Zbl 0065.30303

[7] Beurling, A. On the completeness of {ψ(nt)} on L 2 (0,1), Harmonic Analysis, Contemp. Mathematicians, Birkhaüser, Boston (The collected Works of Arne Beurling) Tome 2 (1989), pp. 378-380 | MR 1057614

[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), Tome 53 (1986), pp. 112-142 | Article | MR 842158 | Zbl 0624.46032

[10] Cotlar, M.; Sadosky, C. A polydisc version of Beurling’s characterization for invariant subspaces of finite multi-codimension, Contemp. Math., Tome 212 (1998), pp. 51-56 | Article | MR 1486589 | Zbl 0984.47003

[11] Gelca, R. Topological Hilbert Nullstellensatz for Bergman spaces, Int. Equations Operator Theory, Tome 28 (1997) no. 2, pp. 191-195 | Article | MR 1451500 | Zbl 0905.47004

[12] Ginsberg, J.; Neuwirth, J.; Newman, D. Approxiamtion by f(kx), J. Funct. Anal., Tome 5 (1970), pp. 194-203 | Article | MR 282114 | Zbl 0189.12902

[13] Green, B.; Tao, T. The primes contain arbitrarily long arithmetic progressions, Annals of Math., Tome 167 (2008), pp. 481-547 | Article | MR 2415379 | Zbl 1191.11025

[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., Tome 86 (1997), pp. 1-37 | Article | MR 1427844 | Zbl 0887.46008

[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., Tome 99 (1999), pp. 175-178 | Article | MR 1700745 | Zbl 0953.46015

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

[17] Krantz, S.; Parks, H. A Primer of Real Analytic Functions, Birkhaüser, Basel (1992) | MR 1182792 | Zbl 0767.26001

[18] Lojaciewicz, S. Sur le problème de la division, Studia Math., Tome 18 (1959), pp. 87-136 | MR 107168 | Zbl 0115.10203

[19] Lopez, J.; Ross, K. Sidon sets, N.Y., M.Dekker (1975) | MR 440298 | Zbl 0351.43008

[20] Mandrekar, V. The validity of Beurling theorems in polydiscs, Proc. Amer. Math. Soc., Tome 103 (1998) no. 1, pp. 145-148 | Article | MR 938659 | Zbl 0658.47033

[21] Nikolski, N. Selected problems of weighted approximation and spectral analysis, Trudy Math. Inst. Steklova, Moscow (Russian) Tome 120 (1974) (English transl.: Proc. Steklov Math. Inst., No 120 (1974), AMS, Providence, 1976) | MR 467270 | Zbl 0342.41028

[22] Nyman, B. On the one-dimensional translation group and semi-group in certain function spaces, Thesis, Uppsala Univ. (1950) | MR 36444 | Zbl 0037.35401

[23] Olofsson, A. On the shift semigroup on the Hardy space of Dirichlet series, Acta Math. Hungar. (2010) | MR 2671009 | Zbl 1230.47072

[24] Rudin, W. Function theory in polydiscs, W.A.Benjamin, Inc, N.Y. - Amsterdam (1969) | MR 255841 | Zbl 0177.34101

[25] Shamoyan, F. Weakly invertible elements in weighted anisothropic spaces of holomorphic functions in a polydisk, Mat. Sbornik, Tome 193:6 (1998), pp. 143-160 ((Russian); Engl. transl.) | MR 1957957 | Zbl 1064.32005

[26] Shamoyan, R.; Li, Haiying On weakly invertible functions in the unit ball and polydisk and related problems, J. Math. Analysis, Tome 1:1 (2010), pp. 8-19 | MR 2783539

[27] Vasyunin, V. On a biorthogonal system related with the Riemann hypothesis, Algebra i Analyz, Tome 7 (1995), pp. 118-135 ((Russian); English transl.: St.Petersburg Math. J. 7 (1996), 405-419) | MR 1353492 | Zbl 0851.11051

[28] Wintner, A. Diophantine approximation and Hilbert’s space, Amer. J. Math., Tome 66 (1944), pp. 564-578 | Article | MR 11497 | Zbl 0061.24902

[29] Zhu, K Spaces of Holomorphic Functions in the Unit Ball, Springer, New York (2005) | MR 2115155 | Zbl 1067.32005