Advertisement

Positivity

, 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
Article

Abstract

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.

Keywords

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

Mathematics Subject Classification

Primary 46E40 Secondary 47B10 

Notes

Acknowledgments

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.

References

  1. 1.
    Botelho, G., Pellegrino, D., Rueda, P.: A unified Pietsch domination theorem. J. Math. Anal. Appl. 365(1), 269–276 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Defant, A., Floret, K.: Tensor norms and operator ideals. North-Holland, Amsterdam (1992)zbMATHGoogle Scholar
  3. 3.
    Diestel, J.: An elementary characterization of absolutely summing operators. Math. Ann. 196, 101–105 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Diestel, J., Jarchow, H., Tonge, A.: Absolutely summing operators. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  5. 5.
    Farmer, J., Johnson, W.B.: Lipschitz p-summing operators. Proc. Amer. Math. Soc. 137, 2989–2995 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Jarchow, H.: Localy convex, spaces. Teubner, Stuttgart (1981)CrossRefzbMATHGoogle Scholar
  7. 7.
    López Molina, J.A., Sánchez Pérez, E.A.: Ideales de operadores absolutamente continuos, Ciencias Exactas, Físicas y Naturales, Madrid. Rev. Real Acad. 87, 349–378 (1993)Google Scholar
  8. 8.
    López Molina, J.A., Sánchez Pérez, E.A.: The associated tensor norm to \((q, p)\)-absolutely summing operators on \(C(K)\)-spaces. Czec. Math. J. 47(4), 627–631 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    López, J.A., Molina, Sánchez-Pérez, E.A.: On operator ideals related to \((p,\sigma )\)-absolutely continuous operator. Studia Math. 131(8), 25–40 (2000)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Matter, U.: Absolute continuous operators and super-reflexivity. Math. Nachr. 130, 193–216 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Pellegrino, D., Santos, J.: A general Pietsch domination theorem. J. Math. Anal. Appl. 375(1), 371–374 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pellegrino, D., Santos, J., Seoane-Sepúlveda, J.B.: Some techniques on nonlinear analysis and applications. Adv. Math. 229, 1235–1265 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pietsch, A.: Operator Ideals. Deutsch. Verlag Wiss., Berlin, 1978; North-Holland, Amsterdam-London-New York-Tokyo (1980)Google Scholar
  14. 14.
    Pisier, G.: Factorization of operators through \(L_{p\infty }\) or \( L_{p1}\) and noncommutative generalizations. Math. Ann. 276(1), 105–136 (1986)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Rodríguez, J.: Absolutely summing operators and integration of vector-valued functions. J. Math. Anal. Appl. 316(2), 579–600 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations