\((r,p)\)-absolutely summing operators on the space \(C(T,X)\) and applications.
In the paper under review, the author gives necessary and sufficient conditions for an operator \(U:C(T,X)\to Y\) to be \((r,p)\)-absolutely summing \((1\leq p\leq r< \infty)\), \(T\) being a compact Hausdorff space, \(X,Y\) Banach spaces, and \(C(T,X)\) the Banach space of all continuous \(X\)-valued functions defined on \(T\) endowed with the supremum norm. More precisely, it is proved that if \(U:C(T,X)\to Y\) is an \((r,p)\)-absolutely summing operator and \(G\) is its representing measure, then \(G(E)\) is also an \((r,p)\)-absolutely summing operator from \(X\) into \(Y\) for each \(E\in\Sigma\) \((\Sigma\) denotes the \(\sigma\)-field of Borel subsets of \(T)\) and \(\| G(E)\|_{r,p} (T)=\sup \{(\sum^n_{i=1} \| G(E_i)\|^r_{r, p})^{1/r}; \{E_1, \dots,E_n\}\subset \Sigma\) a finite partition of \(T\}\leq \| U\|_{r,p}\) with \(\|\cdot \|_{r,p}\) being the \((r,p)\)-absolutely summing norm. Vice versa, it is shown that if \(G(E)\) is an \((r,p) \)-absolutely summing operator from \(X\) into \(Y\) for each \(E\in\Sigma\) and \(G:\Sigma\to As_{p,r}(X,Y)\) has finite variation with respect to the \((r,p)\)-absolutely summing norm on \(As_{p,r} (X,Y)\) \((As_{p,r}(X,Y)\) is the Banach space of all \((r,p)\)-absolutely summing operators from \(X\) into \(Y\) endowed with the \((r,p)\)-absolutely summing norm), then \(U\) is an \((r,p)\)-absolutely summing operator. The author also proves that the injective tensor product of an integral operator and an \((r,p) \)-absolutely summing operator is an \((r,p)\)-absolutely summing operator.
scientific article; zbMATH DE number 1750396