Weakly compact operators and strict topologies

Let X be a completely regular Hausdorff space and C b ( X ) be the Banach space of all real-valued bounded continuous functions on X, endowed with the uniform norm. It is shown that every weakly compact operator T from C b ( X ) to a quasicomplete locally convex Hausdorff space E can be uniquely decomposed as T = T 1 + T 2 + T 3 + T 4 , where T k : C b ( X ) → E ( k = 1 , 2 , 3 , 4 ) are weakly compact operators, and T 1 is tight, T 2 is purely τ-additive, T 3 is purely σ-additive and T 4 is purely finitely additive. Moreover, we derive a generalized Yosida–Hewitt decomposition for E-valued strongly bounded regular Baire measures.