Compactness of the integration operator associated with a vector measure

S. Okada¹ ; W. J. Ricker² ; L. Rodríguez-Piazza³

¹ School of Mathematics The University of New South Wales Sydney, NSW 2052, Australia Current address: Department of Mathematics Macquarie University Sydney, NSW 2109, Australia
² School of Mathematics The University of New South Wales Sydney, NSW 2052, Australia current address Math.-Geogr. Fakutät Katholitsche Universität Eichstätt D-85071 Eichstätt, Germany
³ Departamento de Anáisis Matemático Facultad de Matemáticas Universidad de Sevilla Aptdo. 1160 41080 Sevilla, Spain

Studia Mathematica, Tome 150 (2002) no. 2, pp. 133-149

Résumé

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.

Détail

DOI : 10.4064/sm150-2-3

Keywords: characterization given those banach space valued vector measures finite variation whose associated integration operator mapsto int kern compact linear map banach space moreover every infinite dimensional banach space there exist nontrivial vector measures finite variation compact other still finite variation compact has infinite variation never compact