Let \(n\ge 1\) and \(\varphi : \mathbb {D}^n\rightarrow \mathbb {D}\) be a holomorphic function, where \(\mathbb {D}\) denotes the open unit disk of \(\mathbb {C}\) . Let \(\Theta : \mathbb {D} \rightarrow \mathbb {D}\) be an inner function and \(K^p_\Theta \) , \(p>0\) , denote the corresponding model space. We obtain characterizations of the compact composition operators \(C_\varphi : K^2_\Theta \rightarrow H^2(\mathbb {D}^n)\) , \(n\ge 1\) , and \(C_\varphi : K^p_\Theta \rightarrow H^p(\mathbb {D}^n)\) , \(1<p<\infty \) , where \(H^p(\mathbb {D}^n)\) denotes the Hardy space.
