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

Abstract

LetXbe a completely regular Hausdorff space and letE,·Eand(F,·F)be Banach spaces. LetCb(X,E)be the space of allE-valued bounded, continuous functions onX, equipped with the strict topologyβσ. We study the relationship between important classes of(βσ,·F)-continuous linear operatorsT:Cb(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.

A Riesz representation theory for completely regular Hausdorff spaces and its applications

“…Next, in [37] in case C b .X /˝E is assumed to beˇ -dense in C b .X; E/ (for example, if X has a -compact dense subset; resp. X is a D-space; resp.…”

“…In the present paper we study the Riemann-Stieltjes integration of bounded continuous functions with respect to Borel operator measures. This approach is very natural and has been used by many authors (see [9-11, 14, 19, 33]) and is different from the concept of integration developed by the present author in [36] and [37], where the integral of bounded continuous functions is defined as aˇz-continuous extension of the so-called immediate integral of [1], [2, Sections G-H] (see [36,Lemma 6 and Theorem 9]). …”