Decompositions for weakly compact operators on the space of totally measurable functions

Let ΣΣ be a σσ-algebra of subsets of a non-empty set ΩΩ and let XX be a Banach space. Let B(Σ,X)B(Σ,X) denote the Banach space of totally ΣΣ-measurable XX-valued functions defined on ΩΩ. We derive a Yosida–Hewitt type decomposition for weakly compact operators from B(Σ,X)B(Σ,X) to a Banach space YY. As an application, we obtain a Yosida–Hewitt type decomposition for operator-valued measures.