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Lushness, Numerical Index 1 and the Daugavet Property in Rearrangement Invariant Spaces

Published online by Cambridge University Press:  20 November 2018

Vladimir Kadets ,
Miguel Martín ,
Javier Merí  and
Dirk Werner
Vladimir Kadets
Affiliation:
Department of Mechanics and Mathematics, Kharkov National University, pl. Svobody 4, 61022 Kharkov, Ukraine, e-mail: vova1kadets@yahoo.com
Miguel Martín
Affiliation:
Departamento de Análisis Matematico, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain, e-mail: mmartins@ugr.es, jmeri@ugr.es
Javier Merí
Affiliation:
Departamento de Análisis Matematico, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain, e-mail: mmartins@ugr.es, jmeri@ugr.es
Dirk Werner
Affiliation:
Department of Mathematics, Freie Universitöt Berlin, Arnimallee 6, D-14 195 Berlin, Germany, e-mail: werner@math.fu-berlin.de
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Abstract

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We show that for spaces with 1–unconditional bases lushness, the alternative Daugavet property and numerical index 1 are equivalent. In the class of rearrangement invariant (r.i.) sequence spaces the only examples of spaces with these properties are ${{c}_{0,}}{{\ell }_{1}}$ and ${{\ell }_{\infty }}$. The only lush r.i. separable function space on $\left[ 0,1 \right]$ is ${{L}_{1}}\left[ 0,1 \right]$; the same space is the only r.i. separable function space on $\left[ 0,1 \right]$ with the Daugavet property over the reals.

Keywords

46B0446E30lush spacenumerical indexDaugavet propertyKöthe spacerearrangement invariant space
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 2 , 01 April 2013 , pp. 331 - 348
Copyright
Copyright © Canadian Mathematical Society 2013

References

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