Absolutely (r, q)-Summing Operators on Vector-Valued Function Spaces

Let X and Y be Banach spaces and let be a compact Hausdorff space. In 1973, Swartz, in his by now classical theorem, characterized the absolute summability of an operator U from to Y in terms of its associated operator and of its representing measure m. We study the interplay between U, , and m in the context of absolutely (r, q)-summing operators, considering the spaces of p-continuous functions on , , instead of . This encompasses the Swartz theorem together with its existing extensions on absolutely (r, q)-summing operators, providing, among others, an improvement even to the Swartz theorem. Counterexamples are exhibited to indicate sharpness of our results.