Linear operators on the space of bounded continuous functions with strict topologies (Q618824)

Let \(X\) be a completely regular Hausdorff space and let \((C_b(X),\beta_{\sigma})\) stand for the space of real-valued bounded continuous functions on \(X\) equipped with the strict topology \(\beta_{\sigma}\). The author deals with linear operators acting on \((C_b(X),\beta_{\sigma})\) with values in an arbitrary Banach space \((E,||\cdot||_E)\). The paper is divided into three parts, the first one of which is an expository section. In the second one, the author shows in Proposition 2.2 that a linear map \(T: C_b(X)\to E\) is \((\beta_{\sigma},||\cdot||_E)\)-weakly compact if and only if it is weakly compact and \((\beta_{\sigma},||\cdot||_E)\)-continuous. This leads to Corollary 2.3, saying that, for \((\beta_{\sigma},||\cdot||_E)\)-continuous maps \(T: C_b(X)\to E\), both properties of weak compactness are equivalent. The last part is devoted to a Yosida-Hewitt type decomposition for weakly compact operators on \(C_b(X)\). The main result of this section (Theorem 3.1) says that every weakly compact \(T: C_b(X)\to E\) can be uniquely decomposed as \(T=T_1+T_2\) with \(T_1,T_2\) both weakly compact, \(T_1\) \(\sigma\)-additive and \(T_2\) purely finitely additive. The second part of this theorem describes a unique decomposition of the representing measure \(m:\mathcal{B}\to E\) into a sum of two strongly bounded measures \(m=m_c+m_p\) which satisfy \(e'\circ m_c\in M_{\sigma}(X)\) and \(e'\circ m_p\in M_{pfa}(X)\) for every functional \(e'\in E'\).