Estimate for Norm of a Composition Operator on the Hardy–Dirichlet Space

  • Perumal Muthukumar
  • Saminathan Ponnusamy
  • Hervé Queffélec


By using the Schur test, we give some upper and lower estimates on the norm of a composition operator on \(\mathcal {H}^2\), the space of Dirichlet series with square summable coefficients, for the inducing symbol \(\varphi (s)=c_1+c_{q}q^{-s}\) where \(q\ge 2\) is a fixed integer. We also give an estimate on the approximation numbers of such an operator.


Composition operator Hardy space Dirichlet series Schur test Zeta function 

Mathematics Subject Classification

Primary: 47B33 47B38 Secondary: 11M36 37C30 



The authors thank the referee for many useful comments. The first author thanks the Council of Scientific and Industrial Research (CSIR), India, for providing financial support in the form of a SPM Fellowship to carry out this research. The third author thanks the Indian Statistical Institute of Chennai for providing good and friendly working conditions in December 2015, when this collaboration was initiated. The second author is currently at ISI Chennai.


  1. 1.
    Appel, M.J., Bourdon, P.S., Thrall, J.J.: Norms of composition operators on the Hardy space. Exp. Math. 5(2), 111–117 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bayart, F.: Compact composition operators on a Hilbert space of Dirichlet series. Ill. J. Math. 47(3), 725–743 (2003)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bayart, F.: Composition operators on the polydisk induced by affine maps. J. Funct. Anal. 260(7), 1969–2003 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brevig, O.F.: Sharp norm estimates for composition operators and Hilbert-type inequalities. Bull. Lond. Math. Soc. 49(6), 965–978 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Carl, B., Stephani, I.: Entropy, Compactness and the Approximation of Operators. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle Scholar
  6. 6.
    Clifford, J.H., Dabkowski, M.G.: Singular values and Schmidt pairs of composition operators on the Hardy space. J. Math. Anal. Appl. 305(1), 183–196 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cowen, C.C.: Linear fractional composition operators on $H^2$. Integral Equ. Oper. Theory 11(2), 151–160 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cowen, C.C., MacCluer, B.D.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995)zbMATHGoogle Scholar
  9. 9.
    Gordon, J., Hedenmalm, H.: The composition operators on the space of Dirichlet series with square summable coefficients. Mich. Math. J. 46(2), 313–329 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Halmos, P.R.: A Hilbert Space Problem Book, 2nd edn. Springer, New York (1982)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hammond, C.: The norm of a composition operator with linear symbol acting on the Dirichlet space. J. Math. Anal. Appl. 303(2), 499–508 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hedenmalm, H.: Dirichlet series and functional analysis. In: Laudal, O.A., Piene, R. (eds.) The legacy of Niels Henrik Abel. Springer, Berlin, Heidelberg (2004)Google Scholar
  13. 13.
    Hedenmalm, H., Lindqvist, P., Seip, K.: A Hilbert space of Dirichlet series and systems of dilated functions in $L^{2}(0,1)$. Duke Math. J. 86(1), 1–37 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hlawka, E., Schoissengeier, J., Taschner, R.: Geometric and Analytic Number Theory. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  15. 15.
    Hurst, P.R.: Relating composition operators on different weighted Hardy spaces. Arch. Math. (Basel) 68(6), 503–513 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lavrik, A.F.: On the main term of the divisor’s problem and the power series of the Riemann’s zeta function in a neighbourhood of its pole. Trudy Mat. Inst. Akad. Nauk. SSSR 142, 165–173 (1976). (in Russian) MathSciNetGoogle Scholar
  17. 17.
    Ponnusamy, S.: Foundations of Mathematical Analysis. Springer, New York (2012)CrossRefzbMATHGoogle Scholar
  18. 18.
    Queffélec, H.: Norms of composition operators with affine symbols. J. Anal. 20, 47–58 (2012)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Queffélec, H., Seip, K.: Approximation numbers of composition operators on the $H^2$ space of Dirichlet series. J. Funct. Anal. 268(6), 1612–1648 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhu, K.: Operator Theory in Function Spaces. Mathematical Surveys and Monographs, vol. 138, 2nd edn. American Mathematical Society, Providence (2007)CrossRefzbMATHGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Stat-Math UnitIndian Statistical Institute (ISI), Chennai CentreChennaiIndia
  2. 2.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia
  3. 3.F-59 000 Lille, USTL, Laboratoire Paul Painlevé U.M.R. CNRS 8524Univ Lille Nord de FranceVilleneuve D’ascq CedexFrance

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