Bochner Representable Operators on Spaces of Continuous Functions

Abstract

Let C_b(X) be the Banach lattice of all bounded continuous real-valued functions on a completely regular Hausdorff space X and beta denote the natural strict topology on C_b(X). For a Banach space (E,Vert cdot Vert _E), a linear operator T:C_b(X)rightarrow E is said to be tight if Vert T(u_alpha )Vert _Erightarrow 0 whenever (u_alpha ) is a uniformly bounded net in C_b(X) such that u_alpha rightarrow 0 uniformly on all compact sets in X. It is shown that a linear operator T:C_b(X)rightarrow E is nuclear tight if and only if T is a nuclear operator between the locally convex space (C_b(X),beta ) and a Banach space E and if and only if T is Bochner representable, that is, there exist a positive Radon measure mu on X and a E-valued mu -Bochner integrable function g on X so that T(u)=int _X u(x)g(x)dmu for all uin C_b(X).