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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