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1997 | 125 | 3 | 255-270
Tytuł artykułu

Conical measures and properties of a vector measure determined by its range

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EN
Abstrakty
EN
We characterize some properties of a vector measure in terms of its associated Kluvánek conical measure. These characterizations are used to prove that the range of a vector measure determines these properties. So we give new proofs of the fact that the range determines the total variation, the σ-finiteness of the variation and the Bochner derivability, and we show that it also determines the (p,q)-summing and p-nuclear norm of the integration operator. Finally, we show that Pettis derivability is not determined by the range and study when every measure having the same range of a given measure has a Pettis derivative.
Twórcy
  • Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Aptdo. 1160, Sevilla 41080, Spain, piazza@cica.es
  • Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Aptdo. 1160, Sevilla 41080, Spain, mcromero@cica.es
Bibliografia
  • [AD] R. Anantharaman and J. Diestel, Sequences in the range of a vector measure, Comment. Math. Prace Mat. 30 (1991), 221-235.
  • [C] C. H. Choquet, Lectures on Analysis, Vols. I, II, III, Benjamin, New York, 1969.
  • [DJT] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Stud. Adv. Math. 43, Cambridge Univ. Press, 1995.
  • [DU] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc. Providence, R.I., 1977.
  • [E] G. Edgar, Measurability in a Banach space, Indiana Univ. Math. J. 26 (1977), 663-677.
  • [FT] D. Fremlin and M. Talagrand, A decomposition theorem for additive set functions and applications to Pettis integral and ergodic means, Math. Z. 168 (1979), 117-142.
  • [K] I. Kluvánek, Characterization of the closed convex hull of the range of a vector measure, J. Funct. Anal. 21 (1976), 316-329.
  • [L] D. R. Lewis, On integrability and summability in vector spaces, Illinois J. Math. 16 (1972), 294-307.
  • [M] K. Musiał, The weak Radon-Nikodym property in Banach spaces, Studia Math. 64 (1979), 151-174.
  • [P] A. Pietsch, Operator Ideals, North-Holland, Amsterdam, 1980.
  • [R1] L. Rodríguez-Piazza, The range of a vector measure determines its total variation, Proc. Amer. Math. Soc. 111 (1991), 205-214.
  • [R2] L. Rodríguez-Piazza, Derivability, variation and range of a vector measure, Studia Math. 112 (1995), 165-187.
  • [T] M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. 307 (1984).
  • [Th] E. Thomas, Integral representations in convex cones, Groningen University Report ZW-7703 (1977).
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv125i3p255bwm
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