Yosida-Hewitt type decompositions for weakly compact operators and operator-valued measures (Q2644002)

Let \(X,Y\) be Banach spaces and \((\Omega, \Sigma, \mu)\) be a measure space. For a finitely-additive operator-valued measure \(m: \Sigma \to L(X,Y)\), some sufficient conditions are given for the existence of a representation \(m = m_c + m_s\), where \(m_c\) is countably additive in he uniform operator topology and absolutely continuous with respect to \(\mu\), and \(m_s\) is weakly singular with respect to \(\mu\). To obtain this kind of results, the author studies decompositions of weakly compact operators on Köthe-Bochner function spaces and applies them to the natural integration operator \(T_m : L_\infty(X) \to Y\) generated by \(m\).