Preprint No.
A-01-12
Dmitriy Bilik, Vladimir Kadets, Roman Shvidkoy, Gleb Sirotkin,
Dirk Werner
Narrow operators on vector-valued sup-normed spaces
Abstract:
We characterise narrow and strong Daugavet operators on
$C(K,E)$-spaces; these are in a way the largest sensible classes of
operators for which the norm equation $\|\Id+T\| = 1+\|T\|$ is valid.
For certain separable range spaces $E$ including all finite-dimensional ones
and locally uniformly convex ones we show that an unconditionally
pointwise convergent sum of narrow operators
on $C(K,E)$ is narrow, which implies
for instance the known result that these spaces do not have
unconditional FDDs. In a different vein, we construct two narrow
operators on $C([0,1],\ell_1)$ whose sum is not narrow.
Keywords: Daugavet property, narrow operator, strong Daugavet
operator, USD-nonfriendly spaces, $C(K,E)$-spaces
Mathematics Subject Classification (MSC2000): 46B20; 46B04, 46B28,
46E40, 47B38
Language: ENG
Available: Pr-A-01-12.ps
Contact: Dirk Werner, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (werner@math.fu-berlin.de)
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