Composition operators between model and Hardy spaces

Abstract

Let $n\ge 1$ and $\ddot{I}†:{\mathbb{D}}^n\to \mathbb{D}$ be a holomorphic function, where $\mathbb{D}$ denotes the open unit disk of $\mathbb{C}$. Let $\hat{I}˜:\mathbb{D}\to \mathbb{D}$ be an inner function and $K_{\hat{I}}^p˜$, $p>0$, denote the corresponding model space. We obtain characterizations of the compact composition operators $C_{\ddot{I}}†:K_{\hat{I}}^p˜\to H^p({\mathbb{D}}^n)$, $1<p<\infty$, where $H^p({\mathbb{D}}^n)$ denotes the Hardy space.

8 pages