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
Article

Abstract

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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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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