On Dunford-Pettis operators that are Pettis-representable

Abstract

Let E be a Banach 8pace. It i8 shown that if every DunfordPettis operator T: L' [0, 1] -_ E* is Pettis-repre8entable, then every operator T: L' [0, 1] -_ E* is Pettis-representable. It is well known that a bounded linear operator T from L1 [0, 1] into a Banach space E that is Bochner or Pettis-representable is a Dunford-Pettis operator, i.e., T maps weakly convergent sequences into norm convergent sequences. It was suspected for a while that any Dunford-Pettis operator T: L1[0, 1] L1[0, 1] is Bochner representable, but Coste (see [3, p. 90]) gave an exainple of a convolution type operator T: L1[0,11] L1[0,1] that is a Dunford-Pettis operator but is not Bochner representable. Coste's example suggested the following problem: If every Dunford-Pettis operator from L1 [0, 1] into a Banach space E is Bochner representable, is every bounded linear operator T: Ll [0,1] -E Bochner representable? In [2] Bourgain solved the above problem affirmatively. A parallel problem to the one solved by Bourgain can now be asked as follows: If every DunfordPettis operator T from L1 [0, 1] to a Banach space E is Pettis-representable, is every bounded linear operator T: L1 [0, 1] E Pettis-representable? In [10] we showed that the answer to the above problem is positive when the Banach space E is complemented in a Banach lattice. In fact under this hypothesis one can conclude that every bounded linear operator T: L1 [0, 1] E is Bochner representable (see [5]). In this paper, we will show that the answer is also positive for dual Banach spaces, indeed we shall show that if every Dunford-Pettis operator T from L1 into a dual Banach space E* is Pettis-representable then every bounded linear operator is Pettis-representable. By an operator between two Banach spaces, we always mean a bounded linear operator. All the notions used in this paper and not defined can be found in [3, 5]. Let E be a Banach space and let T be an operator from L1 [0, 1] to E. The operator T is said to be Bochner (resp. Pettis) representable if there exists g: [0,1] -E Bochner-integrable and essentially bounded (resp. Pettis-integrable and scalarly essentially bounded) such that for every f in Ll [0, 1], T(f) Bochner-fJ f * g dX (resp., Pettis-f1 f * g dX). A Banach space E is said to have the Radon-Nikodym property (RNP) (resp., the weak-Radon-Nikodym property (WRNP)), if every operator T: L1 [0, 1] -E is Bochner-representable (resp., Pettis-representable). For more about the RNP and the WRNP see [3, 5, 7, 8, 9]. Received by the editors August 15, 1981. 1980 Mathematics Subject Classification. Primary 46B22, 46G10.