A description of Banach space-valued Orlicz hearts

Coenraad Labuschagne; Theresa Offwood

Open Mathematics, Tome 8 (2010), p. 1109-1119 / Harvested from The Polish Digital Mathematics Library

Résumé

Let Y be a Banach space, (Ω, Σ; μ) a probability space and φ a finite Young function. It is shown that the Y-valued Orlicz heart H φ(μ, Y) is isometrically isomorphic to the l-completed tensor product $H_{ϕ} (μ) {\tilde{\otimes}}_{l} Y$ of the scalar-valued Orlicz heart Hφ(μ) and Y, in the sense of Chaney and Schaefer. As an application, a characterization is given of the equality of $(H_{ϕ} (μ) {\tilde{\otimes}}_{l} Y) *$ and $H_{ϕ} (μ) * {\tilde{\otimes}}_{l} Y *$ in terms of the Radon-Nikodým property on Y. Convergence of norm-bounded martingales in H φ(μ, Y) is characterized in terms of the Radon-Nikodým property on Y. Using the associativity of the l-norm, an alternative proof is given of the known fact that for any separable Banach lattice E and any Banach space Y, E and Y have the Radon-Nikodým property if and only if $E {\tilde{\otimes}}_{l} Y$ has the Radon-Nikodým property. As a corollary, the Radon-Nikodým property in H φ(μ, Y) is described in terms of the Radon-Nikodým property on H φ(μ) and Y.

Publié le : 2010-01-01

Zbl 1218.46010

EUDML-ID : urn:eudml:doc:268954

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     author = {Coenraad Labuschagne and Theresa Offwood},
     title = {A description of Banach space-valued Orlicz hearts},
     journal = {Open Mathematics},
     volume = {8},
     year = {2010},
     pages = {1109-1119},
     zbl = {1218.46010},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0065-9}
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Coenraad Labuschagne; Theresa Offwood. A description of Banach space-valued Orlicz hearts. Open Mathematics, Tome 8 (2010) pp. 1109-1119. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0065-9/

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