Complex interpolation of Hardy-type subspaces

Abstract

We consider the problem of complex interpolation of certain Hardy-type subspaces of Köthe function spaces. For example, suppose $X_0$ and $X_1$ are Köthe function spaces on the unit circle $\mathbf{T},$ 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.