COMPLEX INTERPOLATION AND REGULAR OPERATORS BETWEEN BANACH-LATTICES
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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 $R_p = (R_infty,R_1)^ heta$ if $ heta = 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 $Ssubset L_p$ is a subspace, every regular operator $u : S o L_p$ admits a regular extension $ ilde u : L_p o 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.