Mathematische Annalen

, Volume 342, Issue 3, pp 533–555 | Cite as

Bohr’s strip for vector valued Dirichlet series

  • Andreas Defant
  • Domingo García
  • Manuel Maestre
  • David Pérez-García


Bohr showed that the width of the strip (in the complex plane) on which a given Dirichlet series \({\sum a_n/ n^s, \, s \in \mathbb{C}}\), converges uniformly but not absolutely, is at most 1/2, and Bohnenblust-Hille that this bound in general is optimal. We prove that for a given infinite dimensional Banach space Y the width of Bohr’s strip for a Dirichlet series with coefficients a n in Y is bounded by 1 - 1/Cot (Y), where Cot (Y) denotes the optimal cotype of Y. This estimate even turns out to be optimal, and hence leads to a new characterization of cotype in terms of vector valued Dirichlet series.

Mathematics Subject Classification (2000)

Primary 32A05 Secondary 46B07 46B09 46G20 


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  1. 1.
    Boas H.P.: The football player and the infinite series. Not. AMS 44, 1430–1435 (1997)MATHMathSciNetGoogle Scholar
  2. 2.
    Bombal F., Pérez-García D., Villanueva I.: Multilinear extensions of Grothendieck’s theorem. Q. J. Math. 55, 441–450 (2004)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bohnenblust H.F., Hille E.: On the absolute convergence of Dirichlet series. Ann. Math. 32(2), 600–622 (1934)MathSciNetGoogle Scholar
  4. 4.
    Bohr, H.: Über die gleichmässige Konverenz Dirichletscher Reihen. J. Reine Angew. Math. 143, 203–211 (1913)MATHGoogle Scholar
  5. 5.
    Bohr H.: Über die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichletschen Reihen \({\sum \frac{a_n}{n^s}}\). Nachr. Ges. Wiss. Gött. Math. Phys. Kl. 4, 441–488 (1913)Google Scholar
  6. 6.
    Bohr H.: A theorem concerning power series. Proc. Lond. Math. Soc. 13(2), 1–5 (1914)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Defant, A., García, D., Maestre, M.: New strips of convergence for Dirichlet series. Preprint (2008)Google Scholar
  8. 8.
    Defant, A., Maestre, M., Prengel, C.: Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables. J. Reine Angew. Math. (2008) (to appear)Google Scholar
  9. 9.
    Defant A., Prengel C.: Harald Bohr meets Stefan Banach. Lond. Math. Soc. Lect. Note Ser. 337, 317–339 (2006)MathSciNetGoogle Scholar
  10. 10.
    Dineen S.: Complex Analysis on Infinite Dimensional Spaces. Springer, London (1999)MATHGoogle Scholar
  11. 11.
    Diestel, J., Jarchow, H., Tonge, A.: Absolutely summing operators. Cambridge Stud. Adv. Math., vol. 43. Cambridge University Press, Cambridge (1995)Google Scholar
  12. 12.
    Hedenmalm, H.: Dirichlet series and functional analysis, The legacy of Niels Henrik Abel, pp. 673–684. Springer, Berlin (2004)Google Scholar
  13. 13.
    Lindenstrauss J., Tzafriri L.: Classical Banach Spaces I. Springer, Berlin (1977)MATHGoogle Scholar
  14. 14.
    Lindenstrauss J., Tzafriri L.: Classical Banach Spaces II. Springer, Berlin (1979)MATHGoogle Scholar
  15. 15.
    Maurey B., Pisier G.: Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach. Stud. Math. 58, 45–90 (1976)MATHMathSciNetGoogle Scholar
  16. 16.
    Mujica, J.: Complex analysis in Banach spaces. Math. Studies, vol. 120. North Holland, Amsterdam (1986)Google Scholar
  17. 17.
    Queffélec H.: H. Bohr’s vision of ordinary Dirichlet series; old and new results. J. Anal. 3, 43–60 (1995)MATHMathSciNetGoogle Scholar
  18. 18.
    Toeplitz, O.: Über eine bei Dirichletreihen auftretende Aufgabe aus der Theorie der Potenzreihen unendlich vieler Veraenderlichen. Nachr. Ges. Wiss. Gött. Math. Phys. Kl. 417–432 (1913)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Andreas Defant
    • 1
  • Domingo García
    • 2
  • Manuel Maestre
    • 2
  • David Pérez-García
    • 3
  1. 1.Institute of MathematicsCarl von Ossietzky UniversityOldenburgGermany
  2. 2.Departamento de Análisis MatemáticoUniversidad de ValenciaBurjasot (Valencia)Spain
  3. 3.Departamento de Análisis MatemáticoUniversidad Complutense de MadridMadridSpain

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