Integral Equations and Operator Theory

, Volume 66, Issue 1, pp 79–112 | Cite as

Interpolation of Banach Lattices and Factorization of p-Convex and q-Concave Operators

Article

Abstract

We extend a result of Šestakov to compare the complex interpolation method [X0, X1]θ with Calderón-Lozanovskii’s construction \({{{{X^{1-\theta}_{0}X^{\theta}_{1}}}}}\), in the context of abstract Banach lattices. This allows us to prove that an operator between Banach lattices T : EF which is p-convex and q-concave, factors, for any \({{{{\theta \in (0, 1)}}}}\), as TT2T1, where T2 is (\({{\left({\frac{p}{{\theta + (1 - \theta)p}}} \right)}}\)-convex and T1 is \({{\left({\frac{q}{{1 - \theta }}} \right)}}\)-concave.

Mathematics Subject Classification (2010)

Primary 47B60 Secondary 46B42 46B70 

Keywords

Interpolation of Banach lattices method of Calderón-Lozanovskii factorization p-convex operator q-concave operator 

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References

  1. 1.
    Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory. Graduate Studies in Mathematics, vol. 50, American Mathematical Society, Providence, RI, 2002.Google Scholar
  2. 2.
    F. Albiac, and N. J. Kalton, Topics in Banach space theory. Graduate Texts in Mathematics, 233, Springer, New York, 2006.Google Scholar
  3. 3.
    J. Bergh, On the relation between the two complex methods of interpolation. Indiana Univ. Math. J. 28 (1979), no. 5, 775–778.Google Scholar
  4. 4.
    J. Bergh, and J. Löfström, Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, no. 223, Springer-Verlag, Berlin-New York, 1976.Google Scholar
  5. 5.
    A. V. Bukhvalov, Interpolation of linear operators in spaces of vector-valued functions and with mixed norm. (Russian) Sibirsk. Mat. Zh. 28 (1987), no. 1, i, 37–51. English translation: Siberian Math. J. 28 (1987), no. 1, 24–36.Google Scholar
  6. 6.
    A. P. Calderón, Intermediate spaces and interpolation, the complex method. Studia Math. 24 (1964), 113–190.MATHMathSciNetGoogle Scholar
  7. 7.
    T. Figiel, Uniformly convex norms on Banach lattices. Studia Math. 68 (1980), 215–247.MATHMathSciNetGoogle Scholar
  8. 8.
    N. Ghoussoub, andW. B. Johnson, Factoring operators through Banach lattices not containing C(0, 1). Math. Z. 194 (1987), 153-171.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators. Translations of Mathematical Monographs, 54, American Mathematical Society, 1982.Google Scholar
  10. 10.
    J. L. Krivine, Théorèmes de factorisation dans les espaces réticulés. (French) Séminaire Maurey-Schwartz 1973–1974: Espaces L p, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. 22 et 23. Centre de Math., École Polytech., Paris, 1974.Google Scholar
  11. 11.
    J. Lindenstrauss, and L. Tzafriri, Classical Banach Spaces II: Function Spaces. Springer-Verlag, Berlin-New York, 1979.MATHGoogle Scholar
  12. 12.
    G. Ya. Lozanovskii, Certain Banach lattices. (Russian) Sibirsk. Mat. Ž. 10 (1969), 584–599.MathSciNetGoogle Scholar
  13. 13.
    G. Ya. Lozanovskii, Certain Banach lattices IV. (Russian) Sibirsk. Mat. Ž. 14 (1973), 140–155.MathSciNetGoogle Scholar
  14. 14.
    P. Meyer-Nieberg, Kegel p-absolutsummierende und p-beschränkende Operatoren. (German) Nederl. Akad. Wetensch. Indag. Math. 40 (1978), no. 4, 479–490.Google Scholar
  15. 15.
    P. Meyer-Nieberg, Banach Lattices. Springer-Verlag, 1991.Google Scholar
  16. 16.
    G. Pisier, Some applications of the complex interpolation method to Banach lattices. J. Analyse Math. 35 (1979), 264–281.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    S. Reisner, Operators which factor through convex Banach lattices. Canad. J. Math. 32 (1980), 1482–1500.MATHMathSciNetGoogle Scholar
  18. 18.
    H. H. Schaefer, Banach lattices and positive operators. Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, 1974.Google Scholar
  19. 19.
    V. A. Šestakov, Complex interpolation in Banach spaces of measurable functions. (Russian) Vestnik Leningrad. Univ. No. 19 Mat. Meh. Astronom. Vyp. 4 (1974), 64–68, 171.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  1. 1.Institut de Mathématiques de Jussieu, CNRS and UPMC-Univ. Paris-06ParisFrance
  2. 2.Departamento de Análisis MatemáticoUniversidad Complutense de MadridMadridSpain

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