Espaces de séries de Dirichlet et leurs opérateurs de composition

Hervé Queffélec

doi:10.5802/ambp.351

Hervé Queffélec

Annales Mathématiques Blaise Pascal, 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 : https://doi.org/10.5802/ambp.351
Classification : 47B33, 30B50, 30H10
Mots clés : Dirichlet series, Composition operators, Approximation numbers

@article{AMBP_2015__22_S2_267_0,
     author = {Herv\'e Queff\'elec},
     title = {Espaces de s\'eries de Dirichlet et leurs op\'erateurs de composition},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {267--344},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {22},
     number = {S2},
     year = {2015},
     doi = {10.5802/ambp.351},
     language = {fr},
     url = {ambp.centre-mersenne.org/item/AMBP_2015__22_S2_267_0/}
}

Hervé Queffélec. Espaces de séries de Dirichlet et leurs opérateurs de composition. Annales Mathématiques Blaise Pascal, Tome 22 (2015) no. S2, pp. 267-344. doi : 10.5802/ambp.351. https://ambp.centre-mersenne.org/item/AMBP_2015__22_S2_267_0/

Bibliographie

[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 3250037 | Zbl 1303.42009

[2] Tom M. Apostol Introduction to analytic number theory, Springer-Verlag, New York-Heidelberg, 1998 (Undergraduate Texts in Mathematics) | MR 434929 | Zbl 0335.10001

[3] Maxime Bailleul Espaces de Banach de séries de Dirichlet et leurs opérateurs de composition (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 | Article | MR 3322601

[6] R. Balasubramanian; B. Calado; H. Queffélec The Bohr inequality for ordinary Dirichlet series, Studia Math., Volume 175 (2006) no. 3, pp. 285-304 | Article | MR 2261747 | Zbl 1110.30001

[7] Paul T. Bateman; Harold G. Diamond Analytic number theory, Monographs in Number Theory, Volume 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004, xiv+360 pages (An introductory course) | MR 2111739 | Zbl 1074.11001

[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 http://aif.cedram.org/item?id=AIF_2003__53_7_2039_0 | Numdam | MR 2044167 | Zbl 1077.32002

[9] Frédéric Bayart Hardy spaces of Dirichlet series and their composition operators, Monatsh. Math., Volume 136 (2002) no. 3, pp. 203-236 | Article | MR 1919645 | Zbl 1076.46017

[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 2007233 | Zbl 1059.47023

[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 Jr. A general moment problem, Amer. J. Math., Volume 63 (1941), pp. 361-370 | MR 3848

[13] Harald Bohr Über die gleichmäßige Konvergenz Dirichletscher Reihen, J. Reine Angew. Math., Volume 143 (1913), pp. 203-211 | Article | MR 1580881

[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 12158 | Zbl 0060.17107

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

[16] Bernd Carl; Irmtraud Stephani Entropy, compactness and the approximation of operators, Cambridge Tracts in Mathematics, Volume 98, Cambridge University Press, Cambridge, 1990, x+277 pages | Article | MR 1098497 | Zbl 0705.47017

[17] E. D. Cashwell; C. J. Everett The ring of number-theoretic functions, Pacific J. Math., Volume 9 (1959), pp. 975-985 | MR 108510 | Zbl 0092.04602

[18] Carl C. Cowen; Barbara D. MacCluer Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995, xii+388 pages | MR 1397026 | Zbl 0873.47017

[19] Philip J. Davis Interpolation and approximation, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963, xiv+393 pages | MR 157156 | Zbl 0329.41010

[20] Samuel E. Ebenstein Some $H^{p}$ spaces which are uncomplemented in $L^{p}$ , Pacific J. Math., Volume 43 (1972), pp. 327-339 | MR 318793 | Zbl 0281.42017

[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 | Article | MR 2056832 | Zbl 1070.47013

[22] John B. Garnett Bounded analytic functions, Graduate Texts in Mathematics, Volume 236, Springer, New York, 2007, xiv+459 pages | MR 2261424 | Zbl 1106.30001

[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 | Article | MR 1704209 | Zbl 0963.47021

[24] R. P. Gosselin; J. H. Neuwirth On Paley-Wiener bases, J. Math. Mech., Volume 18 (1968/69), pp. 871-879 | MR 410250 | Zbl 0177.16502

[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, xvi+426 pages | MR 568909 | Zbl 0058.03301

[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 | Article | MR 1427844 | Zbl 0887.46008

[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 | Article | MR 1700745 | Zbl 0953.46015

[29] Henry Helson Hankel forms and sums of random variables, Studia Math., Volume 176 (2006) no. 1, pp. 85-92 | Article | MR 2263964 | Zbl 1108.43003

[30] Henry Helson Hankel forms, Studia Math., Volume 198 (2010) no. 1, pp. 79-84 | Article | MR 2640082 | Zbl 1229.47042

[31] Edmund Hlawka; Johannes Schoissengeier; Rudolf Taschner Geometric and analytic number theory, Universitext, Springer-Verlag, Berlin, 1991, x+238 pages (Translated from the 1986 German edition by Charles Thomas) | Article | MR 1123023 | Zbl 0749.11001

[32] Brian Hollenbeck; Igor E. Verbitsky Best constants for the Riesz projection, J. Funct. Anal., Volume 175 (2000) no. 2, pp. 370-392 | Article | MR 1780482 | Zbl 0963.42006

[33] Jean-Pierre Kahane Some random series of functions, Cambridge Studies in Advanced Mathematics, Volume 5, Cambridge University Press, Cambridge, 1985, xiv+305 pages | MR 833073 | Zbl 0571.60002

[34] Jacob Korevaar Tauberian theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Volume 329, Springer-Verlag, Berlin, 2004, xvi+483 pages (A century of developments) | Article | MR 2073637 | Zbl 1056.40002

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

[36] Daniel Li; Hervé Queffélec Introduction à l’étude des espaces de Banach, Cours Spécialisés [Specialized Courses], Volume 12, Société Mathématique de France, Paris, 2004, xxiv+627 pages (Analyse et probabilités. [Analysis and probability theory]) | MR 2124356 | Zbl 1078.46001

[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 | Article | MR 2885418 | Zbl 1246.47007

[38] Peter Lindqvist; Kristian Seip Note on some greatest common divisor matrices, Acta Arith., Volume 84 (1998) no. 2, pp. 149-154 | EuDML 207140 | MR 1614259 | Zbl 0898.11007

[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 | Article | MR 3374963

[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) | Article | MR 1984460 | Zbl 1039.30001

[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 | Article | MR 1351320 | Zbl 0865.47015

[42] H. L. Montgomery; R. C. Vaughan Hilbert’s inequality, J. London Math. Soc. (2), Volume 8 (1974), pp. 73-82 | MR 337775 | Zbl 0281.10021

[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) | Article | MR 2350405 | Zbl 1146.30003

[44] Albrecht Pietsch Weyl numbers and eigenvalues of operators in Banach spaces, Math. Ann., Volume 247 (1980) no. 2, pp. 149-168 | Article | MR 568205 | Zbl 0428.47027

[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 580154

[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 | Article | Zbl 1127.47026

[48] Hervé Queffélec; Martine Queffélec Diophantine approximation and Dirichlet series, Harish-Chandra Research Institute Lecture Notes, Volume 2, Hindustan Book Agency, New Delhi, 2013, xii+232 pages | MR 3099268

[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 | Article | MR 3306358

[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 | Article | MR 2506825 | Zbl 1180.30002

[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 133446 | Zbl 0112.29701

[55] Joel H. Shapiro Composition operators and classical function theory, Universitext : Tracts in Mathematics, Springer-Verlag, New York, 1993, xvi+223 pages | Article | MR 1237406 | Zbl 0791.30033

[56] Dirk Werner Funktionalanalysis, Springer-Verlag, Berlin, 2007, xii+501 pages | MR 1787146 | Zbl 0831.46002

[57] Robert M. Young An introduction to nonharmonic Fourier series, Pure and Applied Mathematics, Volume 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980, x+246 pages | MR 591684 | Zbl 0493.42001