This is an announcement for the paper “The Bartle-Dunford-Schwartz and the Dinculeanu-Singer theorems revisited” by Fernando Muñozhttps://arxiv.org/find/math/1/au:+Munoz_F/0/1/0/all/0/1, Eve Ojahttps://arxiv.org/find/math/1/au:+Oja_E/0/1/0/all/0/1, Cándido Piñeirohttps://arxiv.org/find/math/1/au:+Pineiro_C/0/1/0/all/0/1.
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 $m:\Sigma\rightarrow\mathcal{L}(X, Y_{**})$.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.07312