Characterization of some classes of operators on spaces of vector-valued continuous functions. (English) Zbl 0564.47013

Let \({\mathcal C}\) denote either the class of weakly compact, Dunford-Pettis, Dieudonné or unconditionally converging operators between Banach spaces. Let C(K,E) be the Banach space of all continuous functions on a compact Hausdorff space K, with values in a Banach space E, under the supremum norm. It is well known that if T is a linear bounded operator from C(K,E) into a Banach space F, belonging to \({\mathcal C}\), then its representing measure m satisfies: (i) m has semi-variation continuous at \(\emptyset\) and (ii) for every Borel set \(A\subset K\), m(A) is an operator from E into F which belongs to \({\mathcal C}\). These two conditions are by no means sufficient to characterize the operators between C(K,E) and F belonging to \({\mathcal C}\). One of the main results of this paper is that conditions (i) and (ii) above characterize the operators from C(K,E) into F belonging to \({\mathcal C}\), for every pair of Banach spaces E and F, if and only if K is a compact dispersed space. This result is sharpened in different senses, depending on the concrete class of operators considered.


47B38 Linear operators on function spaces (general)
46E40 Spaces of vector- and operator-valued functions
46G10 Vector-valued measures and integration
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