  Let $E$ and $F$ be Banach lattices. Let $G$ be a vector sublattice of $E$ and
$T: G\rightarrow F$ be an order continuous positive compact (resp. weakly
compact) operators. We show that if $G$ is an ideal or an order dense
sublattice of $E$, then $T$ has a norm preserving compact (resp. weakly
compact) positive extension to $E$ which is likewise order continuous on $E$.
In particular, we prove that every compact positive orthomorphism on an order
dense sublattice of $E$ extends uniquely to a compact positive orthomorphism on
$E$.
