Abstract view
The DunfordPettis Property for Symmetric Spaces


Published:20000801
Printed: Aug 2000
Anna Kamińska
Mieczysław Mastyło
Abstract
A complete description of symmetric spaces on a separable measure
space with the DunfordPettis property is given. It is shown that
$\ell^1$, $c_0$ and $\ell^{\infty}$ are the only symmetric sequence
spaces with the DunfordPettis property, and that in the class of
symmetric spaces on $(0, \alpha)$, $0 < \alpha \leq \infty$, the only
spaces with the DunfordPettis property are $L^1$, $L^{\infty}$, $L^1
\cap L^{\infty}$, $L^1 + L^{\infty}$, $(L^{\infty})^\circ$ and $(L^1 +
L^{\infty})^\circ$, where $X^\circ$ denotes the norm closure of $L^1
\cap L^{\infty}$ in $X$. It is also proved that all Banach dual
spaces of $L^1 \cap L^{\infty}$ and $L^1 + L^{\infty}$ have the
DunfordPettis property. New examples of Banach spaces showing that
the DunfordPettis property is not a threespace property are also
presented. As applications we obtain that the spaces $(L^1 +
L^{\infty})^\circ$ and $(L^{\infty})^\circ$ have a unique symmetric
structure, and we get a characterization of the DunfordPettis
property of some K\"otheBochner spaces.
MSC Classifications: 
46E30, 46B42 show english descriptions
Spaces of measurable functions ($L^p$spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Banach lattices [See also 46A40, 46B40]
46E30  Spaces of measurable functions ($L^p$spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B42  Banach lattices [See also 46A40, 46B40]
