Weakly compact operators and strict topologies

摘要

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 T-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.