  We study certain interpolation and extension properties of the space of
regular operators between two Banach lattices. Let $R_p$ be the space of all
the regular (or equivalently order bounded) operators on $L_p$ equipped with
the regular norm. We prove the isometric identity $R_p = (R_\infty,R_1)^\theta$
if $\theta = 1/p$, which shows that the spaces $(R_p)$ form an interpolation
scale relative to Calder\'on's interpolation method. We also prove that if
$S\subset L_p$ is a subspace, every regular operator $u : S \to L_p$ admits a
regular extension $\tilde u : L_p \to L_p$ with the same regular norm. This
extends a result due to Mireille L\'evy in the case $p = 1$. Finally, we apply
these ideas to the Hardy space $H^p$ viewed as a subspace of $L_p$ on the
circle. We show that the space of regular operators from $H^p$ to $L_p$
possesses a similar interpolation property as the spaces $R_p$ defined above.
