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

Nikolai Nikolski[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

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 5, page 1601-1626
  • ISSN: 0373-0956

Abstract

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Completeness of a dilation system ( ϕ ( n x ) ) n ≥ 1 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 f ∈ H 2 ( 𝔻 2 ∞ ) : 1) f 1 + ϵ ∈ H 2 ( 𝔻 2 ∞ ) , f - ϵ ∈ H 2 ( 𝔻 2 ∞ ) ; 2) R e ( f ( z ) ) ≥ 0 , z ∈ 𝔻 2 ∞ ; 3) f ∈ H o l ( ( 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).

How to cite

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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 62.5 (2012): 1601-1626. <http://eudml.org/doc/251082>.

@article{Nikolski2012,
abstract = {Completeness of a dilation system $ (\varphi (nx))_\{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\}(\mathbb\{D\}_\{2\}^\{\infty \})$ on the Hilbert multidisc $ \mathbb\{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\}(\mathbb\{D\}_\{2\}^\{\infty \})$: 1) $ f^\{1+\epsilon \}\in H^\{2\}(\mathbb\{D\}_\{2\}^\{\infty \})$, $ f^\{-\epsilon \}\in H^\{2\}(\mathbb\{D\}_\{2\}^\{\infty \})$; 2) $ Re(f(z))\ge 0$, $ z\in \mathbb\{D\}_\{2\}^\{\infty \}$; 3) $ f\in Hol((1+\epsilon )\mathbb\{D\}_\{2\}^\{\infty \})$ and $ f(z)\ne 0$ on $ \mathbb\{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).},
affiliation = {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},
author = {Nikolski, Nikolai},
journal = {Annales de l’institut Fourier},
keywords = {dilation semigroup; Hilbert’s multidisc; cyclic vector; outer function; completeness problem; Riemann hypothesis; cyclic vectors; Hardy spaces; Hilbert's multidisc},
language = {eng},
number = {5},
pages = {1601-1626},
publisher = {Association des Annales de l’institut Fourier},
title = {In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc},
url = {http://eudml.org/doc/251082},
volume = {62},
year = {2012},
}

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
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 5
SP - 1601
EP - 1626
AB - Completeness of a dilation system $ (\varphi (nx))_{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}(\mathbb{D}_{2}^{\infty })$ on the Hilbert multidisc $ \mathbb{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}(\mathbb{D}_{2}^{\infty })$: 1) $ f^{1+\epsilon }\in H^{2}(\mathbb{D}_{2}^{\infty })$, $ f^{-\epsilon }\in H^{2}(\mathbb{D}_{2}^{\infty })$; 2) $ Re(f(z))\ge 0$, $ z\in \mathbb{D}_{2}^{\infty }$; 3) $ f\in Hol((1+\epsilon )\mathbb{D}_{2}^{\infty })$ and $ f(z)\ne 0$ on $ \mathbb{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).
LA - eng
KW - dilation semigroup; Hilbert’s multidisc; cyclic vector; outer function; completeness problem; Riemann hypothesis; cyclic vectors; Hardy spaces; Hilbert's multidisc
UR - http://eudml.org/doc/251082
ER -

References

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  1. O. Agrawal, D. Clark, R. Douglas, Invariant subspaces in the polydisk, Pacific J. Math. 121 (1986), 1-11 Zbl0609.47012MR815027
  2. L. Báez-Duarte, A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis, Atti Acad. Naz. Lincei 14 (2003), 5-11 Zbl1097.11041MR2057270
  3. L. Báez-Duarte, M. Balazard, B. Landreau, E. Saias, Notes sur la fonction ζ de Riemann, 3, Adv. Math. 149 (2000), 130-144 Zbl1008.11032MR1742356
  4. B. Bagchi, On Nyman, Beurling and Baez-Duarte’s Hilbert space reformulation of the Riemann hypothesis, Proc. Indian Acad. Sci. (Math. Sci.) 116 (2006), 137-146 Zbl1125.11049MR2226127
  5. M. Balazard, Completeness problems and the Riemann hypothesis: an annotated bibliography, Number theory for the millenium (2002), 21-48, (M.A.Bennett et al., eds), AK Peters, Boston Zbl1044.11083MR1956217
  6. A. Beurling, A closure problem related to the Riemann zeta-function, Proc. Nat. Acad. USA 41 (1955), 312-314 Zbl0065.30303MR70655
  7. A. Beurling, On the completeness of { ψ ( n t ) } on L 2 ( 0 , 1 ) , Harmonic Analysis, Contemp. Mathematicians 2 (1989), 378-380, Birkhaüser, Boston MR1057614
  8. H. Bohr, Über die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichletschen Reien ∑ a n n s , (1913, A9), Nachr. Ges. Wiss. Göttingen. Math.-Phys. Kl. Zbl44.0306.01
  9. B. Cole, T. Gamelin, Representing measures and Hardy spaces for the infinite polydisk algebra, Proc. London Math. Soc. (3) 53 (1986), 112-142 Zbl0624.46032MR842158
  10. M. Cotlar, C. Sadosky, A polydisc version of Beurling’s characterization for invariant subspaces of finite multi-codimension, Contemp. Math. 212 (1998), 51-56 Zbl0984.47003MR1486589
  11. R. Gelca, Topological Hilbert Nullstellensatz for Bergman spaces, Int. Equations Operator Theory 28 (1997), 191-195 Zbl0905.47004MR1451500
  12. J. Ginsberg, J. Neuwirth, D. Newman, Approxiamtion by f ( k x ) , J. Funct. Anal. 5 (1970), 194-203 Zbl0189.12902MR282114
  13. B. Green, T. Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Math. 167 (2008), 481-547 Zbl1191.11025MR2415379
  14. H. Hedenmalm, P. Lindquist, K. Seip, A Hilbert space of Dirichlet series and sytems of dilated functions in L 2 ( 0 , 1 ) , Duke Math. J. 86 (1997), 1-37 Zbl0887.46008MR1427844
  15. H. Hedenmalm, P. Lindquist, K. Seip, Addendum to “A Hilbert space of Dirichlet series and sytems of dilated functions in L 2 ( 0 , 1 ) ”, Duke Math. J. 99 (1999), 175-178 Zbl0953.46015MR1700745
  16. D. Hilbert, Wesen und Ziele einer Analysis der unendlich vielen unabhängigen Variablen, Rend. Cir. Mat. Palermo 27 (1909), 59-74 Zbl40.0391.02
  17. S. Krantz, H. Parks, A Primer of Real Analytic Functions, (1992), Birkhaüser, Basel Zbl0767.26001MR1182792
  18. S. Lojaciewicz, Sur le problème de la division, Studia Math. 18 (1959), 87-136 Zbl0115.10203MR107168
  19. J. Lopez, K. Ross, Sidon sets, (1975), N.Y., M.Dekker Zbl0351.43008MR440298
  20. V. Mandrekar, The validity of Beurling theorems in polydiscs, Proc. Amer. Math. Soc. 103 (1998), 145-148 Zbl0658.47033MR938659
  21. N. Nikolski, Selected problems of weighted approximation and spectral analysis, 120 (1974), Trudy Math. Inst. Steklova, Moscow (Russian) Zbl0342.41028MR467270
  22. B. Nyman, On the one-dimensional translation group and semi-group in certain function spaces, Thesis, Uppsala Univ. (1950) Zbl0037.35401MR36444
  23. A. Olofsson, On the shift semigroup on the Hardy space of Dirichlet series, (2010), Acta Math. Hungar. Zbl1230.47072MR2671009
  24. W. Rudin, Function theory in polydiscs, (1969), W.A.Benjamin, Inc, N.Y. - Amsterdam Zbl0177.34101MR255841
  25. F. Shamoyan, Weakly invertible elements in weighted anisothropic spaces of holomorphic functions in a polydisk, Mat. Sbornik 193:6 (1998), 143-160 Zbl1064.32005MR1957957
  26. R. Shamoyan, Haiying Li, On weakly invertible functions in the unit ball and polydisk and related problems, J. Math. Analysis 1:1 (2010), 8-19 Zbl1312.32003MR2783539
  27. V. Vasyunin, On a biorthogonal system related with the Riemann hypothesis, Algebra i Analyz 7 (1995), 118-135 Zbl0851.11051MR1353492
  28. A. Wintner, Diophantine approximation and Hilbert’s space, Amer. J. Math. 66 (1944), 564-578 Zbl0061.24902MR11497
  29. K Zhu, Spaces of Holomorphic Functions in the Unit Ball, (2005), Springer, New York Zbl1067.32005MR2115155

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