Compact composition operators on model spaces

Abstract

Let $\varphi: B_d\to\mathbb{D}$, $d\ge 1$, be a holomorphic function, where $B_d$ denotes the open unit ball of $\mathbb{C}^d$ and $\mathbb{D}= B_1$. Let $\Theta: \mathbb{D} \to \mathbb{D}$ be an inner function and let $K^p_\Theta$ denote the corresponding model space. For $p>1$, we characterize the compact composition operators $C_\varphi: K^p_\Theta \to H^p(B_d)$, where $H^p(B_d)$ denotes the Hardy space.