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

Elizabeth M. Bator¹ ; Paul W. Lewis¹ ; Dawn R. Slavens²

¹ Department of Mathematics University of North Texas Denton, TX 76203-1430, U.S.A.
² Department of Mathematics Midwestern State University Wichita Falls, TX 76308, U.S.A.

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 54 (2006) no. 1, pp. 63-73.

Résumé

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.

Détail

DOI : 10.4064/ba54-1-6

Keywords: emmanuele showed mit sigma sigma algebra sets banach space mit sigma countably additive finite variation mit sigma dunford pettis set extension theorem setting bounded finitely additive vector measures established characterization strongly bounded operators abstract continuous function spaces given characterization motivates study set operators class maps extend results saab dealing unconditionally converging operators characterization existence countably additive non strongly bounded representing measure terms presented characterization resolves question posed