Approximation numbers of composition operators on H p spaces of Dirichlet series
[Nombres d’approximation des opérateurs de composition sur les espaces H p des séries de Dirichlet]
Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 551-588.

Un théorème du premier auteur affirme que ϕ définit un opérateur de composition borné sur l’espace de Hardy p des séries de Dirichlet (1p<) dès lors que ϕ(s)=c 0 s+ψ(s), où c 0 est un entier positif ou nul et ψ est une série de Dirichlet qui envoie le demi-plan droit sur le demi-plan Res>1/2 lorsque c 0 =0 et soit est identiquement nulle, soit envoie le demi-plan droit dans lui-même si c 0 >0. Nous prouvons que le n-ième nombre d’approximation de ces opérateurs de composition est minoré, à une constante multiplicative près, par r n , 0<r<1 si c 0 =0 et par n -A , A>0, si c 0 >0. Ces minorations sont optimales et reposent sur une combinaison d’outils venant à la fois de la théorie des espaces de Banach (type et cotype, inégalités de Weyl, bases de Schauder) et sur une méthode d’interpolation pour 2 utilisant des estimations des solutions d’une équation ¯. Un principe de transfert avec les espaces H p du disque est discuté, conduisant à des exemples explicites d’opérateurs de composition ayant des nombres d’approximation avec divers types de décroissance sous-exponentielle. Enfin, une nouvelle formule de Littlewood-Paley est établie, conduisant à une condition suffisante de compacité pour un opérateur de composition sur p .

By a theorem of the first named author, ϕ generates a bounded composition operator on the Hardy space p of Dirichlet series (1p<) only if ϕ(s)=c 0 s+ψ(s), where c 0 is a nonnegative integer and ψ a Dirichlet series with the following mapping properties: ψ maps the right half-plane into the half-plane Res>1/2 if c 0 =0 and is either identically zero or maps the right half-plane into itself if c 0 is positive. It is shown that the nth approximation numbers of bounded composition operators on p are bounded below by a constant times r n for some 0<r<1 when c 0 =0 and bounded below by a constant times n -A for some A>0 when c 0 is positive. Both results are best possible. Estimates rely on a combination of soft tools from Banach space theory (s-numbers, type and cotype of Banach spaces, Weyl inequalities, and Schauder bases) and a certain interpolation method for 2 , developed in an earlier paper, using estimates of solutions of the ¯ equation. A transference principle from H p of the unit disc is discussed, leading to explicit examples of compact composition operators on 1 with approximation numbers decaying at a variety of sub-exponential rates. Finally, a new Littlewood–Paley formula is established, yielding a sufficient condition for a composition operator on p to be compact.

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DOI : 10.5802/aif.3019
Classification : 47B33, 30B50, 30H10, 47B07
Keywords: Dirichlets series, composition operators, approximation numbes
Mot clés : Séries de Dirichlet, opérateurs de composition, nombres d’approximation
Bayart, Frédéric 1, 2 ; Queffélec, Hervé 3 ; Seip, Kristian 4

1 Clermont Université, Université Blaise Pascal, Laboratoire de Mathématiques, BP 10448 63000 Clermont-Ferrand (France)
2 CNRS, UMR 6620 Laboratoire de Mathématiques 63177 Aubière (France)
3 Université Lille Nord de France, USTL Laboratoire Paul Painlevé UMR. CNRS 8524, 59 655 Villeneuve d’Ascq Cedex (France)
4 Department of Mathematical Sciences Norwegian University of Science and Technology 7491 Trondheim (Norway)
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     title = {Approximation numbers of composition operators  on $H^p$ spaces of {Dirichlet} series},
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Bayart, Frédéric; Queffélec, Hervé; Seip, Kristian. Approximation numbers of composition operators  on $H^p$ spaces of Dirichlet series. Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 551-588. doi : 10.5802/aif.3019. http://www.numdam.org/articles/10.5802/aif.3019/

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

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

[3] Bayart, Frédéric 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

[4] Berndtsson, Bo; Chang, Sun-Yung A.; Lin, Kai-Ching Interpolating sequences in the polydisc, Trans. Amer. Math. Soc., Volume 302 (1987) no. 1, pp. 161-169 | DOI

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

[6] Cole, Brian J.; Gamelin, T. W. Representing measures and Hardy spaces for the infinite polydisk algebra, Proc. London Math. Soc. (3), Volume 53 (1986) no. 1, pp. 112-142 | DOI

[7] Diestel, Joe; Jarchow, Hans; Tonge, Andrew Absolutely summing operators, Cambridge Studies in Advanced Mathematics, 43, Cambridge University Press, Cambridge, 1995, xvi+474 pages | DOI

[8] Ebenstein, Samuel E. Some H p spaces which are uncomplemented in L p , Pacific J. Math., Volume 43 (1972), pp. 327-339 | DOI

[9] Gordon, Julia; Hedenmalm, Håkan 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

[10] Hedenmalm, Håkan; Lindqvist, Peter; Seip, Kristian 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

[11] Helson, Henry Conjugate series and a theorem of Paley, Pacific J. Math., Volume 8 (1958), pp. 437-446 | DOI

[12] Johnson, W. B.; König, H.; Maurey, B.; Retherford, J. R. Eigenvalues of p-summing and l p -type operators in Banach spaces, J. Funct. Anal., Volume 32 (1979) no. 3, pp. 353-380 | DOI

[13] Koosis, Paul Introduction to H p spaces, Cambridge Tracts in Mathematics, 115, Cambridge University Press, Cambridge, 1998, xiv+289 pages (With two appendices by V. P. Havin [Viktor Petrovich Khavin])

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

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

[16] Maurey, Bernard; Pisier, Gilles Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math., Volume 58 (1976) no. 1, pp. 45-90

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

[18] Olsen, Jan-Fredrik Local properties of Hilbert spaces of Dirichlet series, J. Funct. Anal., Volume 261 (2011) no. 9, pp. 2669-2696 | DOI

[19] Olsen, Jan-Fredrik; Saksman, Eero On the boundary behaviour of the Hardy spaces of Dirichlet series and a frame bound estimate, J. Reine Angew. Math., Volume 663 (2012), pp. 33-66 | DOI

[20] Pietsch, Albrecht s-numbers of operators in Banach spaces, Studia Math., Volume 51 (1974), pp. 201-223

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

[22] Pietsch, Albrecht Eigenvalues and s-numbers, Cambridge Studies in Advanced Mathematics, 13, Cambridge University Press, Cambridge, 1987, 360 pages

[23] Pisier, G. Sur les espaces de Banach K-convexes, Seminar on Functional Analysis, 1979–1980 (French), École Polytech., Palaiseau, 1980, Exp. No. 11, 15 pages

[24] Queffélec, Hervé; Seip, Kristian 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

[25] Queffélec, Hervé; Seip, Kristian Decay rates for approximation numbers of composition operators, J. Anal. Math., Volume 125 (2015), pp. 371-399 | DOI

[26] Rudin, Walter Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962, ix+285 pages

[27] Saksman, Eero Private communication (Centre for Advanced Study, Oslo, 2012.)

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

[29] Seip, Kristian Interpolation by Dirichlet series in H , Linear and complex analysis (Amer. Math. Soc. Transl. Ser. 2), Volume 226, Amer. Math. Soc., Providence, RI, 2009, pp. 153-164

[30] Seip, Kristian Zeros of functions in Hilbert spaces of Dirichlet series, Math. Z., Volume 274 (2013) no. 3-4, pp. 1327-1339 | DOI

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

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

[33] Yamashita, Shinji Criteria for functions to be of Hardy class H p , Proc. Amer. Math. Soc., Volume 75 (1979) no. 1, pp. 69-72 | DOI

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