A description of Banach space-valued Orlicz hearts

Coenraad Labuschagne; Theresa Offwood

Open Mathematics (2010)

  • Volume: 8, Issue: 6, page 1109-1119
  • ISSN: 2391-5455

Abstract

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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 ⊗ ˜ 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 ⊗ ˜ l Y * and H ϕ μ * ⊗ ˜ 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 ⊗ ˜ 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.

How to cite

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Coenraad Labuschagne, and Theresa Offwood. "A description of Banach space-valued Orlicz hearts." Open Mathematics 8.6 (2010): 1109-1119. <http://eudml.org/doc/268954>.

@article{CoenraadLabuschagne2010,
abstract = {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 \[ H\_\varphi \left( \mu \right)\tilde\{\otimes \}\_l Y \] 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 \[ \left( \{H\_\varphi \left( \mu \right)\tilde\{\otimes \}\_l Y\} \right)* \] and \[ H\_\varphi \left( \mu \right)*\tilde\{\otimes \}\_l Y* \] 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.},
author = {Coenraad Labuschagne, Theresa Offwood},
journal = {Open Mathematics},
keywords = {Bochner norm; l-norm; Banach lattice; Banach space; Martingale; Orlicz space; Orlicz heart; Tensor product; Radon-Nikodým property; -norm; martingale; tensor product},
language = {eng},
number = {6},
pages = {1109-1119},
title = {A description of Banach space-valued Orlicz hearts},
url = {http://eudml.org/doc/268954},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Coenraad Labuschagne
AU - Theresa Offwood
TI - A description of Banach space-valued Orlicz hearts
JO - Open Mathematics
PY - 2010
VL - 8
IS - 6
SP - 1109
EP - 1119
AB - 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 \[ H_\varphi \left( \mu \right)\tilde{\otimes }_l Y \] 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 \[ \left( {H_\varphi \left( \mu \right)\tilde{\otimes }_l Y} \right)* \] and \[ H_\varphi \left( \mu \right)*\tilde{\otimes }_l Y* \] 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.
LA - eng
KW - Bochner norm; l-norm; Banach lattice; Banach space; Martingale; Orlicz space; Orlicz heart; Tensor product; Radon-Nikodým property; -norm; martingale; tensor product
UR - http://eudml.org/doc/268954
ER -

References

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