Compactness of the integration operator associated with a vector measure

Tom 150 / 2002

S. Okada, W. J. Ricker, L. Rodríguez-Piazza Studia Mathematica 150 (2002), 133-149 MSC: Primary 28B05, 46G10, 47B05. DOI: 10.4064/sm150-2-3

Streszczenie

A characterization is given of those Banach-space-valued vector measures $m$ with finite variation whose associated integration operator $I_m:f \mapsto \int f \kern .16667em dm$ is compact as a linear map from $L^1(m)$ into the Banach space. Moreover, in every infinite-dimensional Banach space there exist nontrivial vector measures $m$ (with finite variation) such that $I_m$ is compact, and other $m$ (still with finite variation) such that $I_m$ is not compact. If $m$ has infinite variation, then $I_m$ is never compact.

Autorzy

  • S. OkadaSchool of Mathematics
    The University of New South Wales
    Sydney, NSW 2052, Australia
    Current address:
    Department of Mathematics
    Macquarie University
    Sydney, NSW 2109, Australia
    e-mail
  • W. J. RickerSchool of Mathematics
    The University of New South Wales
    Sydney, NSW 2052, Australia
    current address
    Math.-Geogr. Fakutät
    Katholitsche Universität Eichstätt
    D-85071 Eichstätt, Germany
    e-mail
  • L. Rodríguez-PiazzaDepartamento de Anáisis Matemático
    Facultad de Matemáticas
    Universidad de Sevilla
    Aptdo. 1160
    41080 Sevilla, Spain
    e-mail

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