Banach spaces in which Dunford-Pettis sets are relatively compact

Abstract

Introduction. Let E be a Banach space and X a bounded subset of E. X is called a Dunford-Pettis set if for any weak null sequence (x*) c E* one has lim sup ix* (x)] = 0. n X This note is devoted to a study of the family of Banach spaces with the property that their Dunford-Pettis subsets are relatively compact; we shall say that such a space has the (DPrcP). Our interest in this class of Banach spaces is motivated by the following fact: in the paper [4] we proved that a dual Banach space with the Weak Radon-Nikodym Property ([8], in short (WRNP)) has the (DPrcP); since any Banach space with the (DPrcP) has the so called Compact Range Property ([8], in short (CRP)), it turns out that the result from [4] can be reversed, so obtaining a new characterization of the (WRNP) in dual spaces; this result makes the (WRNP) in dual spaces easier to be handled: for instance, we are able to answer a question by Ruess, [12], when defining a research program concerning projective tensor products of Banach spaces. There is another reason that could as well motivate our study: we state that dominated operators from special C(K, E) spaces taking values in a Banach space with the (DPrcP) are Dunford-Pettis; if E is finite dimensional it is well-known that this is always verified, but when the dimension of E is infinite the above result is no longer true. When looking for hypotheses on E and F making dominated operators Dunford-Pettis we realize that this happens if E has the Dunford-Pettis property ([2]) and F the (DPrcP); and these are in a sense the best hypotheses one can consider. All of these facts are contained in Section 1. Section 2 contains some more examples of Banach spaces with the (DPrcP) as well as some permanence results.