Fundamental aspects of vector-valued Banach limits
Izvestiya. Mathematics , Tome 80 (2016) no. 2, pp. 316-328
Voir la notice de l'article provenant de la source Math-Net.Ru
This paper is divided into four parts. In the first we study the
existence of vector-valued Banach limits and show that a real Banach space
with a monotone Schauder basis admits vector-valued Banach limits if and
only if it is $1$-complemented in its bidual. In the second we prove
two vector-valued versions of Lorentz' intrinsic characterization of almost
convergence. In the third we show that the unit sphere in the space
of all continuous linear operators from $\ell_\infty(X)$ to $X$ which are
invariant under the shift operator on $\ell_\infty(X)$ cannot be obtained
via compositions of surjective linear isometries with vector-valued Banach
limits. In the final part we show that if $X$ enjoys the Krein–Milman
property, then the set of vector-valued Banach limits is a face of the unit
ball in the space of all continuous linear operators from $\ell_\infty(X)$
to $X$ which are invariant under the shift operator on $\ell_\infty(X)$.
Keywords:
Banach limit, almost convergence, group of isometries, extremal structure.
@article{IM2_2016_80_2_a2,
author = {F. J. Garcia-Pacheco and F. J. Perez-Fernandez},
title = {Fundamental aspects of vector-valued {Banach} limits},
journal = {Izvestiya. Mathematics },
pages = {316--328},
publisher = {mathdoc},
volume = {80},
number = {2},
year = {2016},
language = {en},
url = {http://geodesic.mathdoc.fr/item/IM2_2016_80_2_a2/}
}
F. J. Garcia-Pacheco; F. J. Perez-Fernandez. Fundamental aspects of vector-valued Banach limits. Izvestiya. Mathematics , Tome 80 (2016) no. 2, pp. 316-328. http://geodesic.mathdoc.fr/item/IM2_2016_80_2_a2/