Il s’agit du problème de la complétude d’un système de dilatations dans l’espace de Lebesgue où 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 du multidisque 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 dans : 1) , ; 2) , ; 3) et sur . L’Hypothèse de Riemann sur les zéros de la fonction d’Euler est équivalente à un problème semblable de la complétude des dilatations (B.Nyman).
Completeness of a dilation system on the standard Lebesgue space is considered for 2-periodic functions . We show that the problem is equivalent to an open question on cyclic vectors of the Hardy space on the Hilbert multidisc . 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 : 1) , ; 2) , ; 3) and on . The Riemann Hypothesis on zeros of the Euler -function is known to be equivalent to a completeness of a similar but non-periodic dilation system (due to Nyman).
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
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TY - JOUR AU - Nikolski, Nikolai TI - In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc JO - Annales de l'Institut Fourier PY - 2012 SP - 1601 EP - 1626 VL - 62 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2731/ DO - 10.5802/aif.2731 LA - en ID - AIF_2012__62_5_1601_0 ER -
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.2731/
[1] Invariant subspaces in the polydisk, Pacific J. Math., Volume 121 (1986), pp. 1-11 | DOI | MR | Zbl
[2] A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis, Atti Acad. Naz. Lincei, Volume 14 (2003), pp. 5-11 | MR | Zbl
[3] Notes sur la fonction de Riemann, 3, Adv. Math., Volume 149 (2000) no. 1, pp. 130-144 | DOI | MR | Zbl
[4] 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] 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] 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] On the completeness of on , Harmonic Analysis, Contemp. Mathematicians (The collected Works of Arne Beurling), Volume 2, Birkhaüser, Boston, 1989, pp. 378-380 | MR
[8] Über die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichletschen Reien , Nachr. Ges. Wiss. Göttingen. Math.-Phys. Kl., 1913, A9
[9] 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] 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] Topological Hilbert Nullstellensatz for Bergman spaces, Int. Equations Operator Theory, Volume 28 (1997) no. 2, pp. 191-195 | DOI | MR | Zbl
[12] Approxiamtion by , J. Funct. Anal., Volume 5 (1970), pp. 194-203 | DOI | MR | Zbl
[13] The primes contain arbitrarily long arithmetic progressions, Annals of Math., Volume 167 (2008), pp. 481-547 | DOI | MR | Zbl
[14] A Hilbert space of Dirichlet series and sytems of dilated functions in , Duke Math. J., Volume 86 (1997), pp. 1-37 | DOI | MR | Zbl
[15] Addendum to “A Hilbert space of Dirichlet series and sytems of dilated functions in ”, Duke Math. J., Volume 99 (1999), pp. 175-178 | DOI | MR | Zbl
[16] Wesen und Ziele einer Analysis der unendlich vielen unabhängigen Variablen, Rend. Cir. Mat. Palermo, Volume 27 (1909), pp. 59-74 | DOI
[17] A Primer of Real Analytic Functions, Birkhaüser, Basel, 1992 | MR | Zbl
[18] Sur le problème de la division, Studia Math., Volume 18 (1959), pp. 87-136 | MR | Zbl
[19] Sidon sets, N.Y., M.Dekker, 1975 | MR | Zbl
[20] The validity of Beurling theorems in polydiscs, Proc. Amer. Math. Soc., Volume 103 (1998) no. 1, pp. 145-148 | DOI | MR | Zbl
[21] 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] On the one-dimensional translation group and semi-group in certain function spaces, Thesis, Uppsala Univ., 1950 | MR | Zbl
[23] On the shift semigroup on the Hardy space of Dirichlet series, Acta Math. Hungar., 2010 | MR | Zbl
[24] Function theory in polydiscs, W.A.Benjamin, Inc, N.Y. - Amsterdam, 1969 | MR | Zbl
[25] 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] 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] 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] Diophantine approximation and Hilbert’s space, Amer. J. Math., Volume 66 (1944), pp. 564-578 | DOI | MR | Zbl
[29] Spaces of Holomorphic Functions in the Unit Ball, Springer, New York, 2005 | MR | Zbl
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