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Narrow operators on vector-valued sup-normed spaces

Dmitriy Bilik·Vladimir Kadets·Roman Shvidkoy·Gleb Sirotkin·Dirk Werner
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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.

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Dmitriy Bilik, Vladimir Kadets, Roman Shvidkoy, Gleb Sirotkin, Dirk Werner. 2001-06-27. Narrow operators on vector-valued sup-normed spaces. https://arxiv.org/abs/math/0106227

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