Compactness in L 1 , D - P Operators , Geometry of Banach Spaces

A type of oscillation modeled on BMO is introduced to characterize norm compactness in L 1. This result is used to characterize the bounded linear operators from L 1 into a Banach space X that map weakly convergent sequences onto norm convergent sequences (i.e. are Dunford-Pettis). This characterization is used to study the geometry of Banach spaces X with the property that all bounded linear operators from L 1 into X are Dunford-Pettis. The main result of this paper is a BMO-style oscillation characterization of L 1-norm compactness. From this result we obtain a characterization of the bounded linear operators from L 1 into a Banach space X that map weakly convergent sequences onto norm convergent sequences (i.e. are Dunford-Pettis). With such characterizations in hand, we study the geometry of Banach spaces X with the property that all bounded linear operators from L 1 into X are Dunford-Pettis. One way to insure that a relatively weakly compact set K in L 1 is relatively norm compact is to be able to nd, for each > 0, a nite measurable partition of 0; 1] such that for each f in K and each A in , osc f A ess sup !2A f(!) ? ess inf !2A f(!) : This condition is too strong to characterize norm compactness.