  We show that if $T$ is a narrow operator on $X=X_{1}\oplus_{1} X_{2}$ or
$X=X_{1}\oplus_{\infty} X_{2}$, then the restrictions to $X_{1}$ and $X_{2}$
are narrow and conversely. We also characterise by a version of the Daugavet
property for positive operators on Banach lattices which unconditional sums of
Banach spaces inherit the Daugavet property, and we study the Daugavet property
for ultraproducts.
