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 $\|\mathrm{Id}+T\| = 1+\|T\|$ is valid. For certain separable range spaces $E$, including all finite-dimensional spaces and all locally uniformly convex spaces, we show that an unconditionally pointwise convergent sum of narrow operators on $C(K,E)$ is narrow. This 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.
Citation
Dmitriy Bilik. Vladimir Kadets. Roman Shvidkoy. Gleb Sirotkin. Dirk Werner. "Narrow operators on vector-valued sup-normed spaces." Illinois J. Math. 46 (2) 421 - 441, Summer 2002. https://doi.org/10.1215/ijm/1258136201
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