Composition operators on spaces of double Dirichlet series


We study composition operators on spaces of double Dirichlet series, focusing our interest on the characterization of the composition operators of the space of bounded double Dirichlet series \({\mathcal {H}}^\infty ({\mathbb {C}}_+^2)\). We also show how the composition operators of this space of Dirichlet series are related to the composition operators of the corresponding spaces of holomorphic functions. Finally, we give a characterization of the superposition operators in \({\mathcal {H}}^\infty ({\mathbb {C}}_+)\) and in the spaces \({\mathcal {H}}^p\).

This is a preview of subscription content, access via your institution.


  1. 1.

    Aron, R.M., Bayart, F., Gauthier, P.M., Maestre, M., Nestoridis, V.: Dirichlet approximation and universal Dirichlet series. Proc. Am. Math. Soc. 145(10), 4449–4464 (2017)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bayart, F.: Hardy spaces of Dirichlet series and their composition operators. Monatsh. Math. 136(3), 203–236 (2002)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Cámera, G.A.: Nonlinear superposition on spaces of analytic functions. In: Harmonic Analysis and Operator Theory (Caracas, 1994), Volume 189 of Contemporary Mathematics, pp. 103–116. American Mathematical Society, Providence, RI (1995)

  4. 4.

    Castillo-Medina, J., García, D., Maestre, M.: Isometries between spaces of multiple Dirichlet series. J. Math. Anal. Appl. 472(1), 526–545 (2019)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Defant, A., García, D., Maestre, M., Sevilla-Peris, P.: Dirichlet series from the infinite dimensional point of view. Funct. Approx. Comment. Math. 59(2), 285–304 (2018)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Defant, A., García, D., Maestre, M., Sevilla-Peris, P.: Dirichlet Series and Holomorphic Funcions in High Dimensions, Volume 37 of New Mathematical Monographs. Cambridge University Press, Cambridge (2019)

    Google Scholar 

  7. 7.

    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)

    MathSciNet  Article  Google Scholar 

  8. 8.

    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)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Queffélec, H., Queffélec, M.: Diophantine Approximation and Dirichlet Series, Volume 2 of Harish–Chandra Research Institute Lecture Notes. Hindustan Book Agency, New Delhi (2013)

    Google Scholar 

  10. 10.

    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)

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Jaime Castillo-Medina.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first author was partially supported by the Grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front). The last four authors were supported by MINECO and FEDER Project MTM2017-83262-C2-1-P. The second author was also supported by Grant FPU14/04365 and MICINN. The third and fourth authors were also supported by project Prometeo/2017/102 of the Generalitat Valenciana.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bayart, F., Castillo-Medina, J., García, D. et al. Composition operators on spaces of double Dirichlet series. Rev Mat Complut 34, 215–237 (2021).

Download citation


  • Double Dirichlet series
  • Composition operator
  • Superposition operator

Mathematics Subject Classification

  • 30B50
  • 47B33
  • 46J15
  • 32A10