Composition operators between de Branges-Rovnyak and Hardy spaces

Let $d\ge 1$ and $\phi :B_d\to \mathbb{D}$ 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 $C_{\phi }$ 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 ${\phi }_{\zeta }$, $\zeta \in \mathrm{\partial }{B}_d$.