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

Let $X$ and $Y$ be Banach spaces and let $\mathrm{\Omega }$ be a compact Hausdorff space. Denote by ${\mathcal{C}}_p(\mathrm{\Omega },X)$ the space of $p$-continous $X$-valued functions, $1\le p\le \mathrm{\infty }$. For operators $S\in \mathcal{L}(\mathcal{C}(\mathrm{\Omega }),\mathcal{L}(X,Y))$ and $U\in \mathcal{L}({\mathcal{C}}_p(\mathrm{\Omega },X),Y)$, we establish integral representation theorems with respect to a vector measure $m:\Sigma\rightarrow $\mathcal{L}$(X,Y^{**})$,where$\Sigma$denotesthe$\sigma$-algebra of Borel subsets of $\mathrm{\Omega }$. 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, $1\le q\le \mathrm{\infty }$, of a vector measure $m:\mathrm{\Sigma }\to \mathcal{L}(X,Y^{\ast \ast })$.