Representation of Operators Defined on the Space of Bochner Integrable Functions

The representation of linear operators, on the Banach space of Bochner integrable functions, has been the object of much study for the past fifty years. Dunford and Pettis began this investigation in 1940 with the representation of weakly compact and norm compact operators on L1(R) by a Bochner integral, see [6,8]. Andrews has extended their study to the case of the space L1(E), of E-valued, Bochner integrable functions, see [1,2]. The theory of liftings has also been used by Dinculeanu [7] and others to obtain a representation for the general linear operator on L1(E). It is worth noting that the representation is often related to the Radon-Nikodym property of a Banach space, see [2,11]. In this paper, we investigate the representation of operators on a space of Banach-valued Bochner integrable functions which are defined on a perfect measure space. A summary of the paper follows. In the first section, we recall some definitions and notations that we need in the sequel. The second section points out three known results. The most important is the fact that a function defined on a compact Hausdorff space, with values in a Banach space E, is Bochner integrable for every regular Borel measure if it is continuous with respect to the weak topology σ(E, E′). In the third section, we give two representation theorems for an operator T : L1(E) → D, defined on the space of E-valued Bochner integrable functions on a perfect measure space, and with values in a Banach space D. In fact, we prove that, for such an operator T : L1(E) → D, there is a bounded and strongly integrable function g, which is continuous with