Weakly $${p}$$-Dunford Pettis sets in $$ {L_1(\mu ,X)}$$

Abstract

Sets in Banach spaces that are mapped into norm compact sets by operators $$T:X\rightarrow \ell _p$$ (called weakly p-Dunford Pettis sets), for $$1< p< \infty $$ , are studied in arbitrary Banach spaces X and in the space $$L_1(\mu , X)$$ of Bochner integrable functions. Sufficient conditions for a subset of $$L_1(\mu , X)$$ to be a weakly p-Dunford Pettis set are given. It is shown that if $$X^*\in C_{p}$$ , and K is a bounded and uniformly $$L_p$$ -integrable subset of $$L_p(\mu , X)$$ , then K is a weakly p-DP set in $$L_1(\mu , X)$$ .