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Positive p-summing operators, vector measures and tensor products

  • Oscar Blasco (a1)

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In this paper we shall introduce a certain class of operators from a Banach lattice X into a Banach space B (see Definition 1) which is closely related to p-absolutely summing operators defined by Pietsch [8].

These operators, called positive p-summing, have already been considered in [9] in the case p = 1 (there they are called cone absolutely summing, c.a.s.) and in [1] by the author who found this space to be the space of boundary values of harmonic B-valued functions in .

Here we shall use these spaces and the space of majorizing operators to characterize the space of bounded p-variation measures and to endow the tensor product with a norm in order to get as its completion in this norm.

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References

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1.Blasco, O., Boundary values of vector valued harmonic functions considered as operators, Studio Math. 86 (1987), 1933.
2.Bochner, S., Additive set functions on groups, Ann. of Math. 40 (1939), 769799.
3.Diestel, J. and Uhl, J. J., Vector Measures (Amer. Math. Soc. Mathematical Surveys 15, (1977)).
4.Dinculeanu, N., Vector Measures (Pergamon Press, New York, 1967).
5.Heinrich, S., Nielsen, M. J. and Olsen, G., Order bounded operators and tensor products of Banach lattices, Math. Scand. 49 (1981), 99127.
6.Leader, S., The theory of -spaces for finitely additives set functions, Ann. of Math. (2) 58 (1953), 528543.
7.Lindenstrauss, J. and Tzafiri, L., Classical Banach Spaces, Vols. I and II (Springer-Verlag, Berlin, 1979).
8.Pietsch, A., Absolut p-summierende Abbildungen in normmierten Rieumen, Studia Math. 28 (1967), 333353.
9.Schaeffer, H. H., Banach Lattices and Positive Operators (Springer-Verlag, Berlin, 1974).
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Positive p-summing operators, vector measures and tensor products

  • Oscar Blasco (a1)

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