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Let X be a completely regular Hausdorff space and let E , · E and ( F , · F ) be Banach spaces. Let C b ( X , E ) be the space of all E -valued bounded, continuous functions on X , equipped with the strict topology β σ . We study the relationship between important classes of ( β σ , · F ) -continuous linear operators T : C b ( X , E ) →F (strongly bounded, unconditionally converging, weakly completely continuous, completely continuous, weakly compact, nuclear, and strictly singular) and the corresponding operator measures given by Riesz representing theorems. Some applications concerning the coincidence among these classes of operators are derived.

Some Classes of Continuous Operators on Spaces of Bounded Vector-Valued Continuous Functions with the Strict Topology