Marian Nowak

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

Let be a completely regular Hausdorff space, and let and be Banach spaces. Let be the space of all -valued bounded, continuous functions defined on , equipped with the strict topologies , where . General integral representation theorems of -continuous linear operators with respect to the corresponding operator-valued measures are established. Strongly bounded and -continuous operators are studied. We extend to “the completely regular setting” some classical results concerning operators on the spaces and , where is a compact or a locally compact space.

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

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.

Integral representation of linear operators on Orlicz-Bochner spaces

AbstractLet (Ω, Σ, μ) be a σ-finite measure space and let

Strong topologies on vector-valued function spaces

\mathcal{L}(X,Y)

Topological Properties of the Complex Vector Lattice of Bounded Measurable Functions

stand for the space of all bounded linear operators between Banach spaces (X; ‖ • ‖X) and (Y; ‖ • ‖Y). We study the problem of integral representation of linear operators from an Orlicz-Bochner spaceLϕ(μ,X) toY with respect to operator measures

Mackey Topologies on Vector-Valued Function Spaces

m : \sum \to \mathcal{L}(X,Y)