The complete continuity property in Bochner function spaces

Abstract

Let X be a Banach space, (fi, X, X) a finite measure space, and 1 < p < oo . It is shown that LP (I, X) has the complete continuity property if and only if X has it. A similar result about LlA(G, X) is also given. I. Introduction Let X be a Banach space, let (Q, X, X) be a finite measure space, and let 1 < p < oo. We denote by LP(X, X) the Banach space of all (class of) X- valued p-Bochner A-integrable functions (class of) with its usual norm. If X is the scalar field then LP(X, X) will be denoted by LP(X). A Banach space X is said to have the complete continuity property if for ev- ery finite measure space (K, 3r, p), every bounded operator T: LX(K, 3r, p) —► X is a Dunford-Pettis operator. Any Banach space with the Radon-Nikodym property (RNP) has the complete continuity property. In particular, any LP(X), 1 < p < oo , has the complete continuity property. It is well known (see (DU)) that if X has the (RNP) then LP(X, X) has the same property. Recently, Saab and Saab (SS) observed that if X is a dual Banach space that has the complete continuity property then LP(X, X) enjoys the same property. They also asked (SS, Question 13) whether IP(X, X) has the complete continuity property whenever X does. In this paper we will show that the answer is always affirmative. The question of when a property passes from the Banach space X to LP(X, X) was exten- sively studied by several authors in the past. Let us recall that Kwapien (Kw) showed that LP(X, X) (1 < p < oo) contains a copy of en if and only if X contains a copy of Co . Talagrand (T) showed that if X is weakly sequentially complete then LP(X, X) (1 <p < oo) is weakly sequentially complete. Kalton, Saab, and Saab (KSS) were able to prove that the property (u) also passes from X to LP(X, X) (1 < p < oo). Mendoza (M) succeeded in showing that X contains a complemented copy of lx if and only if LP(X, X) (1 < p < oo)