Operators on Spaces of Bounded Vector-Valued Continuous Functions with Strict Topologies

Abstract EN

Let X be a completely regular Hausdorff space, and let (E,‖·‖E) and (F,‖·‖F) be Banach spaces.

Let Cb(X,E) be the space of all E-valued bounded, continuous functions defined on X, equipped with the strict topologies βz, where z=σ,∞,p,τ,t.

General integral representation theorems of (βz,‖·‖F)-continuous linear operators T:Cb(X,E)→F with respect to the corresponding operator-valued measures are established.

Strongly bounded and (βz,‖·‖F)-continuous operators T:Cb(X,E)→F are studied.

We extend to “the completely regular setting” some classical results concerning operators on the spaces C(X,E) and Co(X,E), where X is a compact or a locally compact space.