On the Maurey–Pisier and Dvoretzky–Rogers Theorems

Abstract

A famous theorem due to Maurey and Pisier asserts that for an infinite dimensional Banach space E, the infumum of the q such that the identity map \(id_{E}\) is absolutely \(\left( q,1\right) \)-summing is precisely \(\cot E\). In the same direction, the Dvoretzky–Rogers Theorem asserts \(id_{E}\) fails to be absolutely \(\left( p,p\right) \)-summing, for all \(p\ge 1\). In this note, among other results, we unify both theorems by charactering the parameters q and p for which the identity map is absolutely \(\left( q,p\right) \)-summing. We also provide a result that we call strings of coincidences that characterize a family of coincidences between classes of summing operators. We illustrate the usefulness of this result by extending a classical result of Diestel, Jarchow and Tonge and the coincidence result of Kwapień.

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Correspondence to Gustavo Araújo.

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J. Santos was supported by Conselho Nacional de Desenvolvimento Cientfico e Tecnolgico (CNPq Grant 309466/2018-0).

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Araújo, G., Santos, J. On the Maurey–Pisier and Dvoretzky–Rogers Theorems. Bull Braz Math Soc, New Series 51, 1–9 (2020). https://doi.org/10.1007/s00574-019-00140-5

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Keywords

  • Absolutely summing operators
  • Maurey–Pisier theorem
  • Dvoretzky–Rogers theorem

Mathematics Subject Classification

  • 46A32
  • 47H60