Weakly compact operators and strict topologies

Let X   be a completely regular Hausdorff space and Cb(X)Cb(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 Cb(X)Cb(X) to a quasicomplete locally convex Hausdorff space E   can be uniquely decomposed as T=T1+T2+T3+T4T=T1+T2+T3+T4, where Tk:Cb(X)→ETk:Cb(X)→E(k=1,2,3,4)(k=1,2,3,4) are weakly compact operators, and T1T1 is tight, T2T2 is purely τ  -additive, T3T3 is purely σ  -additive and T4T4 is purely finitely additive. Moreover, we derive a generalized Yosida–Hewitt decomposition for E-valued strongly bounded regular Baire measures.