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

Marian Nowak. "A Riesz representation theory for completely regular Hausdorff spaces and its applications." Open Mathematics 14.1 (2016): 474-496. <http://eudml.org/doc/285710>.

@article{MarianNowak2016,
abstract = {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.},
author = {Marian Nowak},
journal = {Open Mathematics},
keywords = {Spaces of vector-valued continuous functions; Strict topologies; Operator measures; Strongly bounded operators; Unconditionally converging operators; Weakly compact operators; spaces of vector-valued continuous functions; strict topologies; operator measures; strongly bounded operators; unconditionally converging operators; weakly compact operators},
language = {eng},
number = {1},
pages = {474-496},
title = {A Riesz representation theory for completely regular Hausdorff spaces and its applications},
url = {http://eudml.org/doc/285710},
volume = {14},
year = {2016},
}

TY - JOUR
AU - Marian Nowak
TI - A Riesz representation theory for completely regular Hausdorff spaces and its applications
JO - Open Mathematics
PY - 2016
VL - 14
IS - 1
SP - 474
EP - 496
AB - 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.
LA - eng
KW - Spaces of vector-valued continuous functions; Strict topologies; Operator measures; Strongly bounded operators; Unconditionally converging operators; Weakly compact operators; spaces of vector-valued continuous functions; strict topologies; operator measures; strongly bounded operators; unconditionally converging operators; weakly compact operators
UR - http://eudml.org/doc/285710
ER -