SPACES OF INTEGRABLE FUNCTIONS WITH RESPECT TO VECTOR MEASURES OF CONVEX RANGE AND FACTORIZATION OF OPERATORS FROM Lp-SPACES

Let (Ω,Σ) be a measurable space and let X be a Banach space. Throughout this paper G will be a countably additive vector measure G : Σ → X. Consider the space L1(G) of (classes of real) G-integrable functions, following the definition of Bartle, Dunford and Schwartz [1] and Lewis [9]. The properties of this space have been studied by Kluvánek and Knowles [7], Okada [10] and Curbera [2]. In the first part of this paper (Section 1) we investigate the relation between the convexity of the range of G and the structure of the space L1(G). We use an alternate definition of the norm of L1(G) that is obtained by identifying each function f ∈ L1(G) with an operator from an L∞ space. In the second part (Section 2), we apply these ideas to obtain several properties of operators from an Lp space to a Banach space. In particular, we use the results of Dinculeanu that relates vector measures and operators from Lp spaces (see [6]) to obtain a factorization theorem. As an application, we also show a description of the range of p′-summing operators from Lp spaces. We use well-known results about general Vector Measure Theory (see [5]). The notation is standard. If x′ ∈ X ′, |x′G| is the variation of the scalar measure x′G defined by x′G(A) := 〈G(A), x′〉. The semivariation of G in a set A ∈ Σ is given by ‖G‖(A) = sup{|x′G|(A) : x′ ∈ BX′}. We write χA for the characteristic function of A ∈ Σ and Ac for Ω \ A. If 1 ≤ p < ∞, p′ is the (extended) real number that satisfies 1/p+1/p′ = 1. The space of linear and continuous operators between the Banach spaces Y and X is denoted by L(Y,X). A measurable real function defined on Ω is G-integrable if it is x′Gintegrable for each x′ ∈ X ′, and for every A ∈ Σ there is an element ∫A fdG