Composition operators between de Branges-Rovnyak and Hardy spaces

Abstract

Let $d\ge 1$ and $\varphi: B_d\to\mathbb{D}$ be a holomorphic function, where $B_d$ denotes the open unit ball of $\mathbb{C}^d$ and $\mathbb{D} = B_1$. Let $b: \mathbb{D} \to \mathbb{D}$ be a holomorphic function and $\mathcal {H}(b)$ denote the corresponding de Branges-Rovnyak space. We show that compactness of the composition operator $C_\varphi$ from $\mathcal{H}(b)$ to the Hardy space $H^2(B_d)$ is related to natural restrictions on the Nevanlinna counting functions of the slice-functions $\varphi_\zeta$, $\zeta\in \partial B_d$.