Operator ideal properties of vector measures with finite variation

Susumu Okada ¹   ; Werner J. Ricker ²   ; Luis Rodríguez-Piazza ³

¹ 112 Marconi Crescent Kambah, ACT 2902, Australia
² Math.-Geogr. Fakultät Katholische Universität Eichstätt-Ingolstadt D-85072 Eichstätt, Germany
³ Facultad de Matemáticas Universidad de Sevilla Aptdo 1160 E-41080 Sevilla, Spain

Studia Mathematica, Tome 205 (2011) no. 3, pp. 215-249

Résumé

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.

Détail

DOI : 10.4064/sm205-3-2

Keywords: given vector measure values banach space desirable property available associated banach function space m integrable functions where infty valued variation measure closely connected its x valued integration map mapsto int many traditional operators analysis arise integration maps detailed study made connection between property membership various classical operator ideals compact p summing completely continuous operators depending which operator ideal under consideration geometric nature banach space may play crucial role particular importance regard whether contains isomorphic copy classical sequence space ell compact range property relevant