Dunford–Pettis sets

Abstract

Bibasic sequences are used to study relative weak compactness and relative norm compactness of Dunford-Pettis sets. A Banach space X has the Dunford-Pettis property provided that every weakly compact operator with domain X and range an arbitrary Banach space Y maps weakly compact sets in X into norm compact sets in Y. Localizing this notion, a bounded subset A of X is said to be a Dunford-Pettis subset of X if T(A) is relatively norm compact in Y whenever T: X -Y is a weakly compact operator. Consequently, a Banach space X has the Dunford-Pettis property if and only if each of its weakly compact sets is a Dunford-Pettis set. The survey article by Diestel [5] is an excellent source of information about classical results in Banach spaces which relate to the Dunford-Pettis property. Kevin Andrews utilized Dunford-Pettis sets in a study of the Bochner integral in [1]. In Theorem 1 of [1], Andrews showed that a subset K of X is a Dunford-Pettis subset of X if and only if limn (sup{ jx* (x) I x C K}) = 0 whenever (x*) is a weakly null sequence in X* ( = the continuous linear dual of X). In Corollary 4 of [1], Andrews used this characterization to show that if X has the Dunford-Pettis property and ?1 does not embed in X, then the space Ll(pLi,X) has the Dunford-Pettis property. E. Bator showed in [2] that a dual space has the weak Radon-Nikodym property if and only if each Dunford-Pettis subset of X is relatively compact. In addition to reproducing Bator's result, Emmanuele [8] established several other structure properties for Banach spaces in which all Dunford-Pettis sets are relatively compact. Since every bounded subset of a Banach space X whose dual space X* has the Schur property is a Dunford-Pettis subset of X, it is clear that there are Dunford-Pettis sets which are not relatively weakly compact. However, we note that Odell [13, p. 377] showed that every sequence in a Dunford-Pettis set has a weakly Cauchy subsequence. In this paper we study Dunford-Pettis sets which fail to be relatively norm or weakly compact. The following definitions and notation will be helpful. A sequence (Xnv, f2) in XxX* is called bibasic [14, p. 85], [4] if (xn) is a basic sequence in X, (fn) is a basic sequence in X*, and fi*(xj) = ?ij. If (Xn, fn) is a bibasic sequence, Xo = [Xn], Received by the editors April 14, 1998 and, in revised form, March 15, 2000. 2000 Mathematics Subject Classification. Primary 46B20; Secondary 46B15, 46B45.