Blasco, O.; Calabuig, J. M.; Sánchez-Pérez, E. A. \(p\)-variations of vector measures with respect to vector measures and integral representation of operators. (English) Zbl 1337.46034 Banach J. Math. Anal. 9, No. 1, 273-285 (2015). Summary: In this paper we provide two representation theorems for two relevant classes of operators from any \(p\)-convex order continuous Banach lattice with weak unit into a Banach space: the class of continuous operators and the class of cone absolutely summing operators. We prove that they can be characterized as spaces of vector measures with finite \(p\)-semivariation and \(p\)-variation, respectively, with respect to a fixed vector measure. We give in this way a technique for representing operators as integrals with respect to vector measures. Cited in 3 Documents MSC: 46G10 Vector-valued measures and integration 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) 47B60 Linear operators on ordered spaces Keywords:vector measures; operator; \(p\)-variation; \(p\)-semivariation; vector valued integration; cone absolutely summing operators PDFBibTeX XMLCite \textit{O. Blasco} et al., Banach J. Math. Anal. 9, No. 1, 273--285 (2015; Zbl 1337.46034) Full Text: DOI Euclid