Weakly compact, unconditionally converging, and Dunford-Pettis operators on spaces of vector-valued continuous functions

Abstract

Given a compact Hausdorff spaceX, EandFtwo Banach spaces, letT: C(X, E) → Fdenote a bounded linear operator (hereC(X, E)stands for the Banach space of all continuousE-valued functions defined onXunder supremum norm). It is well known [4] that any such operatorThas a finitely additive representing measureGthat is defined on the σ–field of Borel subsets ofXand thatGtakes its values in the space of all bounded linear operators fromEinto the second dual of F. The representing measure G enjoys a host of many important properties; we refer the reader to [4] and [5] for more on these properties. The question of whether properties of the operator T can be characterized in terms of properties of the representing measure has been considered by many authors, see for instance[1], [2], [3] and [6]. Most characterizations presented (see [3] concerning weakly compact operators or [3] and [6] concerning unconditionally converging operators) were given under additional assumptions on the Banach space E. The aim of this paper is to show that one cannot drop the assumptions on E, indeed as we shall soon show many of the operator characterizations characterize the Banach space E itself. More specifically, it is known [3] that ifE*andE**have the Radon-Nikodym property then a bounded linear operatorT: C(X, E) → Fis weakly compact if and only if the measureGis continuous at Ø (also called strongly bounded), i.e. limn||G|| (Bn) = 0 for every decreasing sequenceBn↘ Ø of Borel subsets ofX(here ||G|| (B) denotes the semivariation ofGatB), and if for every Borel setBthe operatorG(B)is a weakly compact operator fromEtoF. In this paper we shall show that if one wants to characterize weakly compact operators as those operators with the above mentioned properties thenE*andE**must both have the Radon-Nikodym property. This will constitute the first part of this paper and answers in the negative a question of [2]. In the second part we consider unconditionally converging operators onC(X, E). It is known [6] that ifT: C(X, E) → Fis an unconditionally converging operator, then its representing measureGis continuous at 0 and, for every Borel setB, G(B)is an unconditionally converging operator fromEtoF. The converse of the above result was shown to be untrue by a nice example (see [2]). Here again we show that if one wants to characterize unconditionally converging operators as above, then the Banach space E cannot contain a copy of c0. Finally, in the last section we characterize Banach spacesEwith the Schur property in terms of properties of Dunford-Pettis operators onC(X, E)spaces.