Mathematische Annalen

, Volume 342, Issue 3, pp 533–555

Bohr’s strip for vector valued Dirichlet series

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

DOI: 10.1007/s00208-008-0246-z

Cite this article as:
Defant, A., García, D., Maestre, M. et al. Math. Ann. (2008) 342: 533. doi:10.1007/s00208-008-0246-z

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 an 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 32A05Secondary 46B0746B0946G20

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