, Volume 20, Issue 2, pp 369–383 | Cite as

Improving integrability via absolute summability: a general version of Diestel’s Theorem

  • D. Pellegrino
  • P. RuedaEmail author
  • E. A. Sánchez-Pérez


A classical result by J. Diestel establishes that the composition of a summing operator with a (strongly measurable) Pettis integrable function gives a Bochner integrable function. In this paper we show that a much more general result is possible regarding the improvement of the integrability of vector valued functions by the summability of the operator. After proving a general result, we center our attention in the particular case given by the \((p,\sigma )\)-absolutely continuous operators, that allows to prove a lot of special results on integration improvement for selected cases of classical Banach spaces—including C(K), \(L^p\) and Hilbert spaces—and operators—p-summing, (qp)-summing and p-approximable operators.


Absolutely summing operator Absolutely continuous operator Pettis integrable function Bochner integrable function 

Mathematics Subject Classification

Primary 46E40 Secondary 47B10 



D. Pellegrino acknowledges with thanks the support of CNPq Grant 401735/2013-3 (Brazil). P. Rueda acknowledges with thanks the support of the Ministerio de Economía y Competitividad (Spain) MTM2011-22417. E.A. Sánchez Pérez acknowledges with thanks the support of the Ministerio de Economía y Competitividad (Spain) MTM2012-36740-C02-02.


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Copyright information

© Springer Basel 2015

Authors and Affiliations

  • D. Pellegrino
    • 1
  • P. Rueda
    • 2
    Email author
  • E. A. Sánchez-Pérez
    • 3
  1. 1.Departamento de MatemáticaUniversidade Federal da ParaíbaJoão PessoaBrazil
  2. 2.Departamento de Análisis MatemáticoUniversidad de ValenciaValenciaSpain
  3. 3.Instituto Universitario de Matemática Pura y AplicadaUniversitat Politècnica de ValènciaValenciaSpain

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