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Sums of SCD sets and their applications to SCD operators and narrow operators

100%

Kadets V., Shepelska V.

Open Mathematics

2010

tom 8

nr 1

129-134

We answer two open questions concerning the recently introduced notions of slicely countably determined (SCD) sets and SCD operators in Banach spaces. An application to narrow operators in spaces with the Daugavet property is given.

A description of Banach space-valued Orlicz hearts

63%

Labuschagne C., Offwood T.

Open Mathematics

2010

tom 8

nr 6

1109-1119

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.

Ograniczanie wyników

2 Open Mathematics

1 Kadets V.

1 Labuschagne C.

1 Offwood T.

1 Shepelska V.

2 2010

Wyniki wyszukiwania

Sums of SCD sets and their applications to SCD operators and narrow operators

A description of Banach space-valued Orlicz hearts