A description of Banach space-valued Orlicz hearts

作者：Labuschagne Coenraad C A^*; Offwood Theresa M

来源：Central European Journal of Mathematics, 2010, 8(6): 1109-1119.

DOI：10.2478/s11533-010-0065-9

• Bochner norm
• l-norm
• Banach lattice
• Banach space
• Martingale
• Orlicz space
• Orlicz heart
• Tensor product
• Radon-Nikodym property

Let Y be a Banach space, (Omega, Sigma, mu) a probability space and phi a finite Young function. It is shown that the Y-valued Orlicz heart H-phi(mu, Y) is isometrically isomorphic to the l-completed tensor product H-phi(mu)(circle times) over tilde Y-r of the scalar-valued Orlicz heart H-phi(mu) and Y, in the sense of Chaney and Schaefer. As an application, a characterization is given of the equality of (H-phi(mu)(circle times) over tilde tY)* and H phi(mu)*(circle times) over tilde tY* in terms of the Radon-Nikodym property on Y*. Convergence of norm-bounded martingales in H phi(mu, Y) is characterized in terms of the Radon-Nikodym 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-Nikodym property if and only if E (circle times) over tilde Y-t has the Radon-Nikodym property. As a corollary, the Radon-Nikodym property in H-phi(mu, Y) is described in terms of the Radon-Nikodym property on H phi(mu) and Y.

• 出版日期2010-12