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Compact operators in Banach lattices

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Abstract

Disjoint sequence methods from the theory of Riesz spaces are used to study compact operators on Banach lattices. A principal new result of the paper is that each positive map from a Banach latticeE to a Banach latticeF with compact majorant is itself compact provided the norms onE′ andF are order continuous.

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References

  1. T. Andô,On compactness of integral operators, Indag. Math.24 (1962), 235–239.

    Google Scholar 

  2. J. Bourgain, D. H. Fremlin and M. Talagrand,Pointwise compact sets of Baire measurable functions, Amer. J. Math. (to appear).

  3. O. Burkinshaw and P. Dodds,Disjoint sequences, compactness and semi-reflexivity in locally convex Riezz spaces. Illinois J. Math.21 (1977), 759–775.

    MATH  MathSciNet  Google Scholar 

  4. P. G. Dodds,O-weakly compact mappings in Riesz space, Trans. Amer. Math. Soc.214 (1975), 389–402.

    Article  MATH  MathSciNet  Google Scholar 

  5. P. G. Dodds,Sequential convergence in the order duals of certain classes of Riesz spaces, Trans. Amer. Math. Soc.203 (1975), 391–403.

    Article  MATH  MathSciNet  Google Scholar 

  6. P. G. Dodds,Indices for Banach lattices, Proc. Acad. Sci. AmsterdamA 80 (1977), 73–86.

    MathSciNet  Google Scholar 

  7. D. H. Fremlin,Topological Riesz Spaces and Measure Theory, Cambridge University Press, 1974.

  8. D. H. Fremlin,A positive compact operator, Manuscripta Math.15 (1975), 323–327.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. J. Grobler,Compactness conditions for integral operators in Banach function space, Proc. Acad. Sci. AmsterdamA83 (1970), 287–294.

    MathSciNet  Google Scholar 

  10. J. J. Grobler,Indices for Banach function spaces, Math. Z.145 (1975), 99–109.

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Grothendieck,Sur les applications linéaires faiblement compactes d'espaces du type C(K), Canad. J. Math.5 (1953), 129–173.

    MATH  MathSciNet  Google Scholar 

  12. M. A. Krasnoselskii, P. O. Zabreiko, E. I. Pustylnik and P. E. Sobolevskii,Integral Operators in Spaces of Summable Functions, Noordhoff I. P., Leyden, 1976.

    Google Scholar 

  13. W. A. J. Luxemburg and A. C. Zaanen,Compactness of integral operators in Banach function spaces, Math. Ann.149 (1963), 150–180.

    Article  MATH  MathSciNet  Google Scholar 

  14. W. A. J. Luxemburg and A. C. Zaanen,Notes on Banach function spaces, Note VIA 66 (1963), 665–681.

    Google Scholar 

  15. W. A. J. Luxemburg and A. C. Zaanen,Riesz Spaces I, North-Holland Mathematical Library, Amsterdam, 1972.

    Google Scholar 

  16. B. Maurey,Type et cotype dans les espaces munis de structure locales inconditionelles, Séminaire Maurey-Schwartz, 1973–1974.

  17. P. Meyer-Nieberg,Zur schwachen Kompaktheit in Banachverbänden, Math. Z.134 (1973), 303–315.

    Article  MATH  MathSciNet  Google Scholar 

  18. P. Meyer-Nieberg,Über Klassen schwach kompakter Operatoren in Banachverbänden, Math. Z.138 (1974), 145–159.

    Article  MATH  MathSciNet  Google Scholar 

  19. P. Meyer-Nieberg,Kegel p-absolutsummierende und p-beschränkende Operatoren (to appear).

  20. R. J. Nagel and U. Schlotterbeck,Integralderstellung regulärer Operatoren auf Banachverbänden, Math. Z.127 (1972), 293–300.

    Article  MATH  MathSciNet  Google Scholar 

  21. R. J. Nagel and U. Schlotterbeck,Kompaktheit von Integral operatoren auf Banachverbänden, Math. Ann.202 (1973), 301–306.

    Article  MATH  MathSciNet  Google Scholar 

  22. R. E. A. C. Paley,Some theorems on abstract spaces, Bull. Amer. Math. Soc. (1936), 235–240.

  23. H. R. Pitt,A note on bilinear forms, J. London Math. Soc.11 (1936), 174–180.

    Article  MATH  Google Scholar 

  24. H. P. Rosenthal,On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from L p (μ) to L v (v), J. Functional Analysis2 (1969), 176–214.

    Article  Google Scholar 

  25. H. H. Schaeffer,Banach Lattices and Positive Operators, Springer-Verlag, Berlin-Heidelberg-New York, 1974.

    Google Scholar 

  26. P. C. Shields,Weakly compact operators on spaces of summable functions, Proc. Amer. Math. Soc.37 (1973), 456–458.

    Article  MATH  MathSciNet  Google Scholar 

  27. T. Shimogaki,Exponents of norms in semi-ordered linear spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom Phys.13 (1965), 135–140.

    MATH  MathSciNet  Google Scholar 

  28. S. Simons,On the Dunford-Pettis property and Banach spaces that contain c 0 , Math. Ann.216 (1975), 225–231.

    Article  MATH  MathSciNet  Google Scholar 

  29. A. C. Zaanen,Integral transformations and their resolvents in orlicz and Lebesgue spaces, Compositio Math.10 (1952), 56–94.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. The Flinders University of South Australia, 5042, Bedford Park, S.A.

    P. G. Dodds & D. H. Fremlin

  2. The University of Essex, Colchester, England

    P. G. Dodds & D. H. Fremlin

Authors
  1. P. G. Dodds
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  2. D. H. Fremlin
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Dodds, P.G., Fremlin, D.H. Compact operators in Banach lattices. Israel J. Math. 34, 287–320 (1979). https://doi.org/10.1007/BF02760610

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  • DOI: https://doi.org/10.1007/BF02760610

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