Complex Interpolation and Regular Operators between Banach Lattices

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 ∞ , R 1) θ if θ = 1/p, which shows that the spaces (R p) form an interpolation scale relative to Calderón's interpolation method. We also prove that if S ⊂ L p is a subspace, every regular operator u : S → L p admits a regular extension ˜ u : L p → L p with the same regular norm. This extends a result due to Mireille Lévy 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.