Composition operators between model and Hardy spaces

Abstract

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