Bochner Representable Operators on Spaces of Continuous Functions

Abstract

AbstractLet $$C_b(X)$$ C b ( X ) be the Banach lattice of all bounded continuous real-valued functions on a completely regular Hausdorff space X and $$\beta $$ β denote the natural strict topology on $$C_b(X)$$ C b ( X ) . For a Banach space $$(E,\Vert \cdot \Vert _E)$$ ( E , ‖ · ‖ E ) , a linear operator $$T:C_b(X)\rightarrow E$$ T : C b ( X ) → E is said to be tight if $$\Vert T(u_\alpha )\Vert _E\rightarrow 0$$ ‖ T ( u α ) ‖ E → 0 whenever $$(u_\alpha )$$ ( u α ) is a uniformly bounded net in $$C_b(X)$$ C b ( X ) such that $$u_\alpha \rightarrow 0$$ u α → 0 uniformly on all compact sets in X. It is shown that a linear operator $$T:C_b(X)\rightarrow E$$ T : C b ( X ) → E is nuclear tight if and only if T is a nuclear operator between the locally convex space $$(C_b(X),\beta )$$ ( C b ( X ) , β ) and a Banach space E and if and only if T is Bochner representable, that is, there exist a positive Radon measure $$\mu $$ μ on X and a E-valued $$\mu $$ μ -Bochner integrable function g on X so that $$T(u)=\int _X u(x)g(x)d\mu $$ T ( u ) = ∫ X u ( x ) g ( x ) d μ for all $$u\in C_b(X)$$ u ∈ C b ( X ) .