Extension of operators defined on reflexive subspaces of $L^1$ and $L^1/H^1$

Inperpolation theory is used to develop a general pattern for proving extension theorems mentioned in the title. In the case where the range space $G$ is a $w^*$-closed subspace of $L^\infty$ or $H^\infty$ with reflexive annihilator $F$, a necessary and sufficient condition on $G$ is found for such an extension to be always possible. Specifically, $F$ must be Hilbertian and become complemented in $L^p$ $(1<p\le2)$ after a suitable change of density.