The Bartle-Dunford-Schwartz and the Dinculeanu-Singer theorems revisited

Abstract

Let $X$ and $Y$ be Banach spaces and let $\Omega$ be a compact Hausdorff space. Denote by $\mathcal{C}_{p}(\Omega,X)$ the space of $p$-continous $X$-valued functions, $1\leq p\leq \infty$. For operators $S\in\mathcal{L}(\mathcal{C}(\Omega),\mathcal{L}(X,Y))$ and $U\in\mathcal{L}(\mathcal{C}_{p}(\Omega,X),Y)$, we establish integral representation theorems with respect to a vector measure $m:\Sigma\rightarrow \mathcal{L}(X,Y^{**})$, where $\Sigma$ denotes the $\sigma$-algebra of Borel subsets of $\Omega$. The first theorem extends the classical Bartle-Dunford-Schwartz representation theorem. It is used to prove the second theorem, which extends the classical Dinculeanu-Singer representation theorem, also providing to it an alternative simpler proof. For the latter (and the main) result, we build the needed integration theory, relying on a new concept of the $q$-semivariation, $1\leq q\leq \infty$, of a vector measure $m:\Sigma\rightarrow \mathcal{L}(X,Y^{**})$.