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\(p\)-variations of vector measures with respect to vector measures and integral representation of operators. (English) Zbl 1337.46034

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.

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
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