Integral Representations for a Class of Operators on \(L_E^1\)

Abstract

Let \((X, \mathcal{A}, \mu)\) be a finite measure space, \(E\) a locally convex Hausdorff space, \(L_{E}^{1}\) the space of functions \(f: X \to E\) which are \(\mu\)-integrable by semi-norms, \(P(\mu, E)\) the space of Pettis integrable functions and \(P_{1}(\mu, E)\) those elements of \(P(\mu, E)\) which are measurable by semi-norms. We prove that a linear continuous mapping \( T: L_{E}^{1} \to E\) is of the form \(T(f)= \int g f d \mu\) (\(g \in L^{\infty}\)) if and only if \( h( T(f))=0\) whenever \( h\circ f=0 \) for any \( f \in L_{E}^{1}, h \in E'\). Similar results are proved for \(P(\mu, E)\) and \(P_{1}(\mu, E)\).