BANACH SPACES \MITH THE (CRP) AND DOMTNATED OPERATORS ON C(K)

Abstract. We characterize Banach spaces with the (CRP) by means of dominated operators on C(K), improving an old result by J.J. Uhl; hence we apply our main theorem to the study of Pietsch integral operators and to Dunford-Pettis operators. 1. In [8] J.J. Uhl showed the following result: "Let K be a compact Hausdorff space and.E a Banach space with the Radon-Nikodym property. If. T: C(K)---+ E is a dominated operator, then 7 is compact". Now we improve his result to the case of Ba^nach spaces with the (CRP), also showing a converse of this result; i.e. we characterize the Banach spaces with the (CRP) by means of compact dominated operators on C(K). Hence, we apply our theorem to the study of compactness of Pietsch integral operators and to Dunford-Pettis operators. 2. This section is devoted to state the definitions we need as well as the main theorem of the paper. Throughout.I ( will denote a compact Hausdorff space and E aBanach space. Deffnition 1..E is said to possess the compact range propertS in symbols (CRP), if any E-valued countably additive measure p, defined on a o-field of subsets of an arbitrary set.9, with finite variation lpl(S), has relatively compact range. Note that if E has the Radon-Nikodym property then it has the (CRP). Since any Banach space with the weak Radon-Nikodym property has the (CRP), it is easy to see that there exist Banach spaces with the (CRP) but without the Radon-Nikodym property. In passing we observe that if.E has the pr-(CRP) (see [6] for this definition) for a non-atomic measure pr then it has the (CRP). A dominated operator f: CQ{)--r E will be defined according to Definition 1 in [3], p. 379. Work performed under th