Operator ideal properties of vector measures with finite variation

Volume 205 / 2011

Susumu Okada, Werner J. Ricker, Luis Rodríguez-Piazza Studia Mathematica 205 (2011), 215-249 MSC: 28B05, 46B15, 46B42, 47B10, 47L20. DOI: 10.4064/sm205-3-2

Abstract

Given a vector measure $m$ with values in a Banach space $X$, a desirable property (when available) of the associated Banach function space $L^1 (m)$ of all $m$-integrable functions is that $L^1 (m)= L^1(|m|)$, where $|m|$ is the $[0,\infty ]$-valued variation measure of $m$. Closely connected to $m$ is its $X$-valued integration map $ I_m : f \mapsto \int f \, d m $ for $f \in L^1 (m)$. Many traditional operators from analysis arise as integration maps in this way. A detailed study is made of the connection between the property $L^1 (m)= L^1(|m|)$ and the membership of $I_m$ in various classical operator ideals (e.g., the compact, $p$-summing, completely continuous operators). Depending on which operator ideal is under consideration, the geometric nature of the Banach space $X$ may also play a crucial role. Of particular importance in this regard is whether or not $X$ contains an isomorphic copy of the classical sequence space $\ell ^1$. The compact range property of $X$ is also relevant.

Authors

  • Susumu Okada112 Marconi Crescent
    Kambah, ACT 2902, Australia
    e-mail
  • Werner J. RickerMath.-Geogr. Fakultät
    Katholische Universität Eichstätt-Ingolstadt
    D-85072 Eichstätt, Germany
    e-mail
  • Luis Rodríguez-PiazzaFacultad de Matemáticas
    Universidad de Sevilla
    Aptdo 1160
    E-41080 Sevilla, Spain
    e-mail

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