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Derivability, variation and range of a vector measure

100%
Rodríguez-Piazza L.
Studia Mathematica
|
1994-1995
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tom 112
|
nr 2
165-187
EN
We prove that the range of a vector measure determines the σ-finiteness of its variation and the derivability of the measure. Let F and G be two countably additive measures with values in a Banach space such that the closed convex hull of the range of F is a translate of the closed convex hull of the range of G; then F has a σ-finite variation if and only if G does, and F has a Bochner derivative with respect to its variation if and only if G does. This complements a result of [Ro] where we proved that the range of a measure determines its total variation. We also give a new proof of this fact.
2
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Conical measures and properties of a vector measure determined by its range

63%
Rodríguez-Piazza L., Romero-Moreno M. C.
Studia Mathematica
|
1997
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tom 125
|
nr 3
255-270
EN
We characterize some properties of a vector measure in terms of its associated Kluvánek conical measure. These characterizations are used to prove that the range of a vector measure determines these properties. So we give new proofs of the fact that the range determines the total variation, the σ-finiteness of the variation and the Bochner derivability, and we show that it also determines the (p,q)-summing and p-nuclear norm of the integration operator. Finally, we show that Pettis derivability is not determined by the range and study when every measure having the same range of a given measure has a Pettis derivative.
3
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Compactness of the integration operator associated with a vector measure

51%
Okada S., Ricker W., Rodríguez-Piazza L.
Studia Mathematica
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2002
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tom 150
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nr 2
133-149
EN
A characterization is given of those Banach-space-valued vector measures m with finite variation whose associated integration operator Iₘ: f ↦ ∫fdm is compact as a linear map from L¹(m) into the Banach space. Moreover, in every infinite-dimensional Banach space there exist nontrivial vector measures m (with finite variation) such that Iₘ is compact, and other m (still with finite variation) such that Iₘ is not compact. If m has infinite variation, then Iₘ is never compact.
4
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Some translation-invariant Banach function spaces which contain c₀

51%
Lefèvre P., Li D., Queffélec H., Rodríguez-Piazza L.
Studia Mathematica
|
2004
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tom 163
|
nr 2
137-155
EN
We produce several situations where some natural subspaces of classical Banach spaces of functions over a compact abelian group contain the space c₀.
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