The Representation of Linear Operators

A satisfactory representation theory in terms of integrals for linear operators from an Lp space to an arbitrary Banach space X has not as yet been developed for p>l. For the case p = l, Dunford and Pettis [3] have obtained results provided X satisfies certain rather general restrictions. In particular, the representations for the cases in which X is an L space or an Lp space are well known. The case in which the domain is L and X is an arbitrary space has been treated by Rickart [6] who obtained a representation in terms of the Phillips-Birkhoff integral [5; 2]. If the range space X is a function space in which the values of the functions at a fixed point in their common domain of definition yield a linear functional over the space, the problem can frequently be treated if the form of the linear functional over the domain space is known by considering the operator as a family of linear functionals over the space. In the present paper we use essentially this technique to obtain a representation for operators from Lp to L in terms of the Lebesgue integral. We denote by (R, ), (S, y) measure spaces where R, S are sets, , 7 completely additive measures defined over rr-rings J(R), J(S) of subsets in R, S respectively, and we assume that R, S are unions of countable families of sets of finite measure. The set of all realvalued functions {x(/)} defined over R which are measurable and such that I x(t) \ p is integrable over R with respect to ) is the space of essentially bounded functions with norm ||x|| =ess sup( |x(/)|. By a theorem of F. Riesz [7], the linear functional over LP(R, l, p' defined by 1/p + l/p' = 1 and hGL°>(R, <p) if p = l. We shall make extensive use of the theorem of Radon and Nikodym [8] which states that L(R, <p) is isometrically isomorphic to the space of completely additive and absolutely continuous set functions A(R, cp) defined over J(R) where F£.A(R, <p) implies that ||e||