  An operator $T$ from a Banach lattice $E$ into a Banach space is disjointly
non-singular ($DN$-$S$, for short) if no restriction of $T$ to a subspace
generated by a disjoint sequence is strictly singular. We obtain several
results for $DN$-$S$ operators, including a perturbative characterization. For
$E=L_p$ ($1< p<\infty$) we improve the results, and we show that the $DN$-$S$
operators have a different behavior in the cases $p=2$ and $p\neq 2$. As an
application we prove that the strongly embedded subspaces of $L_p$ form an open
subset in the set of all closed subspaces.
