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Positivity

, Volume 11, Issue 3, pp 399–416 | Cite as

Optimal Domains for L0-valued Operators Via Stochastic Measures

  • Guillermo P. CurberaEmail author
  • Olvido Delgado
Article

Abstract

We study extension of operators T: EL0([0, 1]), where E is an F–function space and L0([0, 1]) the space of measurable functions with the topology of convergence in measure, to domains larger than E, and we study the properties of such domains. The main tool is the integration of scalar functions with respect to stochastic measures and the corresponding spaces of integrable functions.

Mathematics Subject Classification (2000)

46G10 47B38 46A16 46E30 

Keywords

F–spaces kernel operators lattices operators between function spaces optimal domains space of integrable functions space of measurable functions vector measures 

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References

  1. C. Aliprantis, O. Burkinshaw, Locally solid Riesz spaces, Academic Press, New York-London (1978)Google Scholar
  2. N. Aronszajn, P. Szeptycki, On general integral transformations, Math. Ann. 163 (1966), 127–154.Google Scholar
  3. R.G. Bartle, N. Dunford, J. Schwartz, Weak compactness and vector measures, Canad. J. Math. 7 (1955), 289–305.Google Scholar
  4. G. P. Curbera, Operators into L1 of a vector measure and applications to Banach lattices, Math. Ann. 293 (1992), 317–330.Google Scholar
  5. G. P. Curbera, When L1 of a vector measure is an AL–space, Pacific J. Math. 162 (1994), 287–303.Google Scholar
  6. G. P. Curbera, Banach space properties of L1 of a vector measure, Proc. Amer. Math. Soc. 123 (1995), 3797–3806.Google Scholar
  7. G. P. Curbera, W. J. Ricker, Optimal domains for kernel operators via interpolation, Math. Nachr. 244 (2002), 47–63.Google Scholar
  8. G. P. Curbera, W. J. Ricker, Optimal domains for the kernel operator associated with Sobolev’s inequality, Studia Math. 158 (2003) 131–152 and 170 (2005), 217–218.Google Scholar
  9. G. P. Curbera, W. J. Ricker, Banach lattices with the Fatou property and optimal domains of kernel operators, Indag. Math. (N.S.), 17 (2006), 187–204.Google Scholar
  10. G. P. Curbera, W. J. Ricker, Compactness properties of Sobolev imbeddings for rearrangement invariant norms, Trans. Amer. Math. Soc., 359 (2007), 287–294.Google Scholar
  11. O. Delgado, Banach function subspaces of L1 of a vector measure and related Orlicz spaces, Indag. Math. (N.S.) 15 (2004), 485–495.Google Scholar
  12. O. Delgado, Optimal domains for kernel operators on [0,∞) × [0,∞), Studia Math., to appear.Google Scholar
  13. J. Diestel, J. J. Jr. Uhl, Vector measures, Math. Surveys 15, Amer. Math. Soc., Providence, R.I., (1977)Google Scholar
  14. N. J. Kalton, N. T. Peck, J. W. Roberts, L0-valued vector measures are bounded, Proc. Amer. Math. Soc. 85 (1982), 575–582.Google Scholar
  15. N. J. Kalton, N. T. Peck, J. W. Roberts, An F-space sampler, Cambridge University Press, Cambridge (1984).Google Scholar
  16. D. R. Lewis, Integration with respect to vector measures, Pacific J. Math. 33 (1970), 157–165.Google Scholar
  17. D. R. Lewis, On integrability and summability in vector spaces, Illinois J. Math. 16 (1972), 294–307.Google Scholar
  18. J. Lindenstrauss, L. Tzafriri, Classical Banach spaces Vol. II, Springer-Verlag, Berlin, (1979).Google Scholar
  19. W. A. Luxemburg, A. C. Zaanen, Riesz spaces Vol. I, North-Holland, Amsterdam (1971).Google Scholar
  20. B. Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces Lp, Astérisque, No. 11, Société Mathématique de France, Paris (1974)Google Scholar
  21. B. Maurey, G. Pisier, Un théorème d’extrapolation et ses conséquences, C. R. Acad. Sci. Paris Sér. A-B 277 (1973), A39–A42.Google Scholar
  22. S. Okada, The dual space of \({\mathcal{L}}^1(\mu)\) for a vector measure μ, J. Math. Anal. Appl. 177 (1993), 583–599.Google Scholar
  23. S. Okada, W. J. Ricker, Optimal domains and integral representations of convolution operators in Lp(G), Integral Equations Oper. Theory 48 (2004), 525–546.Google Scholar
  24. W. J. Ricker, Compactness properties of extended Volterra operators in Lp([0,1]) for 1 ≤  p  ≤  ∞, Arch. Math. 66 (1996), 132–140.Google Scholar
  25. S. Rolewicz, Metric linear spaces, Monografie Matematyczne 56, PWN–Polish Scientific Publishers, Warsaw (1972).Google Scholar
  26. S. Rolewicz, Metric linear spaces, 2d ed., PWN–Polish Scientific Publishers, Warsaw (1984).Google Scholar
  27. S. Rolewicz, C. Ryll-Nardzewski, On unconditional convergence in linear metric spaces, Colloq. Math. 17 (1967), 327–331.Google Scholar
  28. C. Ryll-Nardzewski, W. A. Woyczyński, Bounded multiplier convergence in measure of random vector series, Proc. Amer. Math. Soc. 53 (1975), 96–98.Google Scholar
  29. L. Schwartz, Un théorème de convergence dans les Lp, 0≤ p≤ + ∞, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A704–A706.Google Scholar
  30. P. Szeptycki, Domains of integral transformations on general measure spaces, Math. Ann. 242 (1979), 267–271.Google Scholar
  31. M. Talagrand, Les mesures vectorielles à valeurs dans L0 sont bornées, Ann. Sci. École Norm. Sup. (4) 14 (1981), 445–452.Google Scholar
  32. E. G. F. Thomas, Vector integration, unpublished.Google Scholar
  33. Ph. Turpin, Une mesure vectorielle non bornée, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), A509–A511.Google Scholar
  34. Ph. Turpin, Intégration par rapport à une mesure à valeurs dans un espace vectoriel topologique non supposé localement convexe, Intégration vectorielle et multivoque (Colloq., Univ. Caen, Caen, 1975), Exp. No. 8, Dép. Math., U. E. R. Sci., Univ. Caen, Caen, (1975).Google Scholar
  35. Ph. Turpin, Convexités dans les espaces vectoriels topologiques généraux, Diss. Math. 131 (1976).Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 2007

Authors and Affiliations

  1. 1.Facultad de MatemáticasUniversidad de SevillaSevillaSpain
  2. 2.Departamento de Matemática Aplicada II, Escuela Universitaria PolitécnicaUniversidad de SevillaSevillaSpain

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