Interpolation of Holomorphic functions

Interpolation Theory gives techniques for constructing spaces from two initial Banach spaces. We provide several conditions under which the restriction of a holomorphic map $f:X_0+X_1\to Y_0+Y_1$ to the interpolated spaces (using some specific interpolation methods), where $f_{\mathrm{\mid }{X}_0}:X_0\to Y_0$ is compact, is also compact and holomorphic.