Vector Measures, $c_0$, and (sb) Operators

Tom 54 / 2006

Elizabeth M. Bator, Paul W. Lewis, Dawn R. Slavens Bulletin Polish Acad. Sci. Math. 54 (2006), 63-73 MSC: 46B20, 46B28, 46G10, 28B05. DOI: 10.4064/ba54-1-6

Streszczenie

Emmanuele showed that if ${\mit\Sigma}$ is a $\sigma$-algebra of sets, $X$ is a Banach space, and $\mu : {\mit\Sigma} \to X$ is countably additive with finite variation, then $\mu ({\mit\Sigma} )$ is a Dunford–Pettis set. An extension of this theorem to the setting of bounded and finitely additive vector measures is established. A new characterization of strongly bounded operators on abstract continuous function spaces is given. This characterization motivates the study of the set of (sb) operators. This class of maps is used to extend results of P. Saab dealing with unconditionally converging operators. A characterization of the existence of a countably additive, non-strongly bounded representing measure in terms of $c_0$ is presented. This characterization resolves a question posed in 1970.

Autorzy

  • Elizabeth M. BatorDepartment of Mathematics
    University of North Texas
    Denton, TX 76203-1430, U.S.A.
    e-mail
  • Paul W. LewisDepartment of Mathematics
    University of North Texas
    Denton, TX 76203-1430, U.S.A.
    e-mail
  • Dawn R. SlavensDepartment of Mathematics
    Midwestern State University
    Wichita Falls, TX 76308, U.S.A.
    e-mail

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