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...

Let E be a Banach function space over a finite and atomless measure space (Ω,Σ,μ) and let $(X,|\left|·\right|{|}_X)$ and $(Y,|\left|·\right|{|}_Y)$ be real Banach spaces. A linear operator T acting from the Köthe-Bochner space E(X) to Y is said to be absolutely continuous if $\left|\right|T\left(1_{Aₙ}f\right){\left|\right|}_Y\to 0$ whenever μ(Aₙ) → 0, (Aₙ) ⊂ Σ. In this paper we examine absolutely continuous operators from E(X) to Y. Moreover, we establish relationships between different classes of linear operators from E(X) to Y.

In this note we review some results about:1. Representation of Absolutely (∞,p) summing operators (∏∞,p) in C(K,E)2. Dunford-Pettis properties.

In the context of spaces of homogeneous type, we develop a method to deterministically construct dyadic grids, specifically adapted to a given combinatorial situation. This method is used to estimate vector-valued operators rearranging martingale difference sequences such as the Haar system.

With the help of recent adjacent dyadic constructions by Hytönen and the author, we give an alternative proof of results of Lechner, Müller and Passenbrunner about the $L^p$-boundedness of shift operators acting on functions $f\in L^p(X;E)$ where 1 < p < ∞, X is a metric space and E is a UMD space.

The algebraic and topological reflexivity of C(X) and C(X,E) are investigated by using representations for the into isometries due to Holsztyński and Cambern.

Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinder Ω = Σ × ℝ with $\Sigma \subset {ℝ}^{n-1}$, a bounded domain of class $C^{1,1}$, are obtained in the space $L^q(ℝ;L²\left(\Sigma \right))$, q ∈ (1,∞). As a preparation, spectral decompositions of vector-valued homogeneous Sobolev spaces are studied. The main theorem is proved using the techniques of Schauder decompositions, operator-valued multiplier functions and R-boundedness of operator families.

For 1 ≤ q < ∞, let ${}_q\left(\right)$ denote the Banach algebra consisting of the bounded complex-valued functions on the unit circle having uniformly bounded q-variation on the dyadic arcs. We describe a broad class ℐ of UMD spaces such that whenever X ∈ ℐ, the sequence space ℓ²(ℤ,X) admits the classes ${}_q\left(\right)$ as Fourier multipliers, for an appropriate range of values of q > 1 (the range of q depending on X). This multiplier result expands the vector-valued Marcinkiewicz Multiplier Theorem in the direction q >...

Soit $K$ un compact de $C^n$ de la forme $K={\Pi }_{i=1}^r{K}_i$ où chaque $K_i$ est soit l’adhérence d’un domaine strictement pseudoconvexe dans $C^{n_i}$, soit l’adhérence d’un polyèdre de Weil régulier, ou encore un compact de $C$. $E$ étant un espace de Fréchet, on montre que lorsque $f$ appartient à ${\mathbf{C}}^1(K,E)$ avec $\bar{\partial }f\cong 0$ alors $f$ est approchable uniformément sur $K$ par des fonctions holomorphes au voisinage de $K$ et à valeurs dans $E$. On donne également des résultats de localisation pour l’espace $H(K,E)$.