  Interpolation Theory gives techniques for constructing spaces from two
initial Banach spaces. We provide several conditions under which the
restriction of a holomorphic map $f:X_0+X_1 \rightarrow Y_0+Y_1$ to the
interpolated spaces (using some specific interpolation methods), where
$f_{|X_0}:X_0 \rightarrow Y_0$ is compact, is also compact and holomorphic.
