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 $\|\mathrm{Id}+T\| = 1+\|T\|$ 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.