Narrow operators on vector-valued sup-normed spaces

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 $\mathrm{\parallel }Id+T\mathrm{\parallel }=1+\mathrm{\parallel }T\mathrm{\parallel }$ 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 $C([0,1],{\mathrm{\ell }}_1)$ whose sum is not narrow.