We consider the problem of complex interpolation of certain Hardy-type subspaces of K\"othe function spaces. For example, suppose $X_0$ and $X_1$ are K\"othe function spaces on the unit circle $\bold T,$ and let $H_{ X_0}$ and $H_{ X_1}$ be the corresponding Hardy spaces. Under mild conditions on $X_0,X_1$ we give a necessary and sufficient condition for the complex interpolation space $[H_{ X_0},H_{ X_1}]_{ \theta}$ to coincide with $H_{ X_{ \theta}}$ where $X_{ \theta}=[X_0,X_1]_{ \theta}.$ We develop a very general framework for such results and our methods apply to many more general sitauations including the vector-valued case.
Categories: math.FA