Espaces de séries de Dirichlet et leurs opérateurs de composition

Hervé Queffélec

doi:10.5802/ambp.351

Hervé Queffélec ¹

¹ Université Lille Nord de France UMR 8524 CNRS 59655 Villeneuve d’Ascq CEDEX, France

Annales mathématiques Blaise Pascal, Numéro spécial Ecole de Printemps d’Analyse, Tome 22 (2015) no. S2, pp. 267-344.

Résumé

Ce survol est divisé en trois chapitres : le premier porte sur les propriétés générales des séries de Dirichlet $\sum_{n = 1}^{\infty} a_{n} n^{- s}$ et de leur somme, et présente le point de vue de Bohr (relèvement). Le second étudie les espaces de Hardy-Dirichlet de telles séries sur un demi-plan vertical, avec une application aux systèmes de Riesz. Le troisième enfin porte sur les opérateurs de composition agissant sur ces espaces et leurs nombres d’approximation. Le comportement de ces nombres se révèle assez différent de ceux rencontrés dans le cas des espaces de Hardy classiques.

DOI : 10.5802/ambp.351

Classification : 47B33, 30B50, 30H10
Mots-clés : Dirichlet series, Composition operators, Approximation numbers

Affiliations des auteurs :

Hervé Queffélec ¹

¹ Université Lille Nord de France UMR 8524 CNRS 59655 Villeneuve d’Ascq CEDEX, France

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Hervé Queffélec. Espaces de séries de Dirichlet et leurs opérateurs de composition. Annales mathématiques Blaise Pascal, Numéro spécial Ecole de Printemps d’Analyse, Tome 22 (2015) no. S2, pp. 267-344. doi : 10.5802/ambp.351. https://ambp.centre-mersenne.org/articles/10.5802/ambp.351/

Bibliographie
Cité par

[1] Alexandru Aleman; Jan-Fredrik Olsen; Eero Saksman Fourier multipliers for Hardy spaces of Dirichlet series, Int. Math. Res. Not. IMRN (2014) no. 16, pp. 4368-4378 | MR | Zbl

[2] Tom M. Apostol Introduction to analytic number theory, Springer-Verlag, New York-Heidelberg, 1998 (Undergraduate Texts in Mathematics) | MR | Zbl

[3] Maxime Bailleul Espaces de Banach de séries de Dirichlet et leurs opérateurs de composition, Université d’Artois (France) (2014) (Ph. D. Thesis)

[4] Maxime Bailleul; Ole Fredrik Brevig Composition operators on Bohr-Bergman spaces of Dirichlet series (2014) (http://arxiv.org/abs/1409.3017v1)

[5] Maxime Bailleul; Pascal Lefèvre Some Banach spaces of Dirichlet series, Studia Math., Volume 226 (2015) no. 1, pp. 17-55 | DOI | MR

[6] R. Balasubramanian; B. Calado; H. Queffélec The Bohr inequality for ordinary Dirichlet series, Studia Math., Volume 175 (2006) no. 3, pp. 285-304 | DOI | MR | Zbl

[7] Paul T. Bateman; Harold G. Diamond Analytic number theory, Monographs in Number Theory, 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004, pp. xiv+360 (An introductory course) | MR | Zbl

[8] F. Bayart; A. Mouze Division et composition dans l’anneau des séries de Dirichlet analytiques, Ann. Inst. Fourier (Grenoble), Volume 53 (2003) no. 7, pp. 2039-2060 | Numdam | MR | Zbl

[9] Frédéric Bayart Hardy spaces of Dirichlet series and their composition operators, Monatsh. Math., Volume 136 (2002) no. 3, pp. 203-236 | DOI | MR | Zbl

[10] Frédéric Bayart Compact composition operators on a Hilbert space of Dirichlet series, Illinois J. Math., Volume 47 (2003) no. 3, pp. 725-743 http://projecteuclid.org/euclid.ijm/1258138190 | MR | Zbl

[11] Frédéric Bayart; Hervé Queffélec; Kristian Seip Approximation numbers of composition operators on $H^{p}$ spaces of Dirichlet series (à paraître dans Ann. Inst. Fourier)

[12] R. P. Boas A general moment problem, Amer. J. Math., Volume 63 (1941), pp. 361-370 | MR

[13] Harald Bohr Über die gleichmäßige Konvergenz Dirichletscher Reihen, J. Reine Angew. Math., Volume 143 (1913), pp. 203-211 | DOI | MR

[14] D. G. Bourgin; C. W. Mendel Orthonormal sets of periodic functions of the type ${f (n x)}$ , Trans. Amer. Math. Soc., Volume 57 (1945), pp. 332-363 | MR | Zbl

[15] J.F. Burnol, 2014 (Communication personnelle)

[16] Bernd Carl; Irmtraud Stephani Entropy, compactness and the approximation of operators, Cambridge Tracts in Mathematics, 98, Cambridge University Press, Cambridge, 1990, pp. x+277 | DOI | MR | Zbl

[17] E. D. Cashwell; C. J. Everett The ring of number-theoretic functions, Pacific J. Math., Volume 9 (1959), pp. 975-985 | MR | Zbl

[18] Carl C. Cowen; Barbara D. MacCluer Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995, pp. xii+388 | MR | Zbl

[19] Philip J. Davis Interpolation and approximation, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963, pp. xiv+393 | MR | Zbl

[20] Samuel E. Ebenstein Some $H^{p}$ spaces which are uncomplemented in $L^{p}$ , Pacific J. Math., Volume 43 (1972), pp. 327-339 | MR | Zbl

[21] Catherine Finet; Hervé Queffélec; Alexander Volberg Compactness of composition operators on a Hilbert space of Dirichlet series, J. Funct. Anal., Volume 211 (2004) no. 2, pp. 271-287 | DOI | MR | Zbl

[22] John B. Garnett Bounded analytic functions, Graduate Texts in Mathematics, 236, Springer, New York, 2007, pp. xiv+459 | MR | Zbl

[23] Julia Gordon; Håkan Hedenmalm The composition operators on the space of Dirichlet series with square summable coefficients, Michigan Math. J., Volume 46 (1999) no. 2, pp. 313-329 | DOI | MR | Zbl

[24] R. P. Gosselin; J. H. Neuwirth On Paley-Wiener bases, J. Math. Mech., Volume 18 (1968/69), pp. 871-879 | MR | Zbl

[25] G. H. Hardy; M. Riesz The general theory of Dirichlet’s series, Dover Phenix Editions, Second Edition, 2005

[26] G. H. Hardy; E. M. Wright An introduction to the theory of numbers, The Clarendon Press, Oxford University Press, New York, 1979, pp. xvi+426 | MR | Zbl

[27] Håkan Hedenmalm; Peter Lindqvist; Kristian Seip A Hilbert space of Dirichlet series and systems of dilated functions in $L^{2} (0, 1)$ , Duke Math. J., Volume 86 (1997) no. 1, pp. 1-37 | DOI | MR | Zbl

[28] Håkan Hedenmalm; Peter Lindqvist; Kristian Seip Addendum to : “A Hilbert space of Dirichlet series and systems of dilated functions in $L^{2} (0, 1)$ ”, Duke Math. J., Volume 99 (1999) no. 1, pp. 175-178 | DOI | MR | Zbl

[29] Henry Helson Hankel forms and sums of random variables, Studia Math., Volume 176 (2006) no. 1, pp. 85-92 | DOI | MR | Zbl

[30] Henry Helson Hankel forms, Studia Math., Volume 198 (2010) no. 1, pp. 79-84 | DOI | MR | Zbl

[31] Edmund Hlawka; Johannes Schoissengeier; Rudolf Taschner Geometric and analytic number theory, Universitext, Springer-Verlag, Berlin, 1991, pp. x+238 (Translated from the 1986 German edition by Charles Thomas) | DOI | MR | Zbl

[32] Brian Hollenbeck; Igor E. Verbitsky Best constants for the Riesz projection, J. Funct. Anal., Volume 175 (2000) no. 2, pp. 370-392 | DOI | MR | Zbl

[33] Jean-Pierre Kahane Some random series of functions, Cambridge Studies in Advanced Mathematics, 5, Cambridge University Press, Cambridge, 1985, pp. xiv+305 | MR | Zbl

[34] Jacob Korevaar Tauberian theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 329, Springer-Verlag, Berlin, 2004, pp. xvi+483 (A century of developments) | DOI | MR | Zbl

[35] D. Li, 2014 (Communication orale)

[36] Daniel Li; Hervé Queffélec Introduction à l’étude des espaces de Banach, Cours Spécialisés [Specialized Courses], 12, Société Mathématique de France, Paris, 2004, pp. xxiv+627 (Analyse et probabilités. [Analysis and probability theory]) | MR | Zbl

[37] Daniel Li; Hervé Queffélec; Luis Rodríguez-Piazza On approximation numbers of composition operators, J. Approx. Theory, Volume 164 (2012) no. 4, pp. 431-459 | DOI | MR | Zbl

[38] Peter Lindqvist; Kristian Seip Note on some greatest common divisor matrices, Acta Arith., Volume 84 (1998) no. 2, pp. 149-154 | EuDML | MR | Zbl

[39] Adam W. Marcus; Daniel A. Spielman; Nikhil Srivastava Interlacing families II : Mixed characteristic polynomials and the Kadison-Singer problem, Ann. of Math. (2), Volume 182 (2015) no. 1, pp. 327-350 | DOI | MR

[40] John E. McCarthy Hilbert spaces of Dirichlet series and their multipliers, Trans. Amer. Math. Soc., Volume 356 (2004) no. 3, p. 881-893 (electronic) | DOI | MR | Zbl

[41] A. V. Megretskiĭ; V. V. Peller; S. R. Treil The inverse spectral problem for self-adjoint Hankel operators, Acta Math., Volume 174 (1995) no. 2, pp. 241-309 | DOI | MR | Zbl

[42] H. L. Montgomery; R. C. Vaughan Hilbert’s inequality, J. London Math. Soc. (2), Volume 8 (1974), pp. 73-82 | MR | Zbl

[43] Jan-Fredrik Olsen; Kristian Seip Local interpolation in Hilbert spaces of Dirichlet series, Proc. Amer. Math. Soc., Volume 136 (2008) no. 1, p. 203-212 (electronic) | DOI | MR | Zbl

[44] Albrecht Pietsch Weyl numbers and eigenvalues of operators in Banach spaces, Math. Ann., Volume 247 (1980) no. 2, pp. 149-168 | DOI | MR | Zbl

[45] George Pólya; Gábor Szegő Problems and theorems in analysis. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin-New York, 1972 (Series, integral calculus, theory of functions, Translated from the German by D. Aeppli,) | MR

[46] H. Queffélec; C. Zuily Analyse pour l’Agrégation, Dunod, 2013

[47] Hervé Queffélec Composition operators in the Dirichlet series setting, Perspectives in operator theory (Banach Center Publ.), Volume 75, Polish Acad. Sci., Warsaw, 2007, pp. 261-287 | DOI | Zbl

[48] Hervé Queffélec; Martine Queffélec Diophantine approximation and Dirichlet series, Harish-Chandra Research Institute Lecture Notes, 2, Hindustan Book Agency, New Delhi, 2013, pp. xii+232 | MR

[49] Hervé Queffélec; Kristian Seip Approximation numbers of composition operators on the $H^{2}$ space of Dirichlet series, J. Funct. Anal., Volume 268 (2015) no. 6, pp. 1612-1648 | DOI | MR

[50] O. Ramaré, 2013 (Communication personnelle)

[51] E. Saksman, 2012 (Communication personnelle)

[52] Eero Saksman; Kristian Seip Integral means and boundary limits of Dirichlet series, Bull. Lond. Math. Soc., Volume 41 (2009) no. 3, pp. 411-422 | DOI | MR | Zbl

[53] K. Seip, 2014 (Communication personnelle)

[54] H. S. Shapiro; A. L. Shields On some interpolation problems for analytic functions, Amer. J. Math., Volume 83 (1961), pp. 513-532 | MR | Zbl

[55] Joel H. Shapiro Composition operators and classical function theory, Universitext : Tracts in Mathematics, Springer-Verlag, New York, 1993, pp. xvi+223 | DOI | MR | Zbl

[56] Dirk Werner Funktionalanalysis, Springer-Verlag, Berlin, 2007, pp. xii+501 | MR | Zbl

[57] Robert M. Young An introduction to nonharmonic Fourier series, Pure and Applied Mathematics, 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980, pp. x+246 | MR | Zbl

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