Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let Cb(X, E) be the space of all E-valued bounded, continuous functions on X, equipped with the strict topology β. We develop the Riemman-Stieltjes-type Integral representation theory of (β, || · ||F) -continuous operators T : Cb(X, E) → F with respect to the representing Borel operator measures. For X being a k-space, we characterize strongly bounded (β, || · ||F)-continuous operators T : Cb(X, E) → F. As an application, we study (β, || · ||F)-continuous weakly compact and unconditionally converging operators T : Cb(X, E) → F. In particular, we establish the relationship between these operators and the corresponding Borel operator measures given by the Riesz representation theorem. We obtain that if X is a k-spaceand E is reflexive, then (Cb(X, E), β) has the V property of Pełczynski.
@article{bwmeta1.element.doi-10_1515_math-2016-0043,
author = {Marian Nowak},
title = {A Riesz representation theory for completely regular Hausdorff spaces and its applications},
journal = {Open Mathematics},
volume = {14},
year = {2016},
pages = {474-496},
zbl = {1360.46035},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_math-2016-0043}
}
Marian Nowak. A Riesz representation theory for completely regular Hausdorff spaces and its applications. Open Mathematics, Tome 14 (2016) pp. 474-496. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_math-2016-0043/