Montel's theorem and composition operators for analytic almost periodic functions

We consider the Banach space $H^\infty_{\mathrm{ap}}(\mathbb{C}_0)$ of bounded analytic functions on the open right half-plane $\mathbb{C}_0$ that are almost periodic on some smaller half-plane, as well as the subspace $A_{\mathrm{ap}}(\mathbb{C}_0)$ of those functions in $H^\infty_{\mathrm{ap}}(\mathbb{C}_0)$ that are uniformly continuous on $\mathbb{C}_0$. We prove a strong version of Montel's theorem for $H^\infty_{\mathrm{ap}}(\mathbb{C}_0)$ and characterize the bounded composition operators on $H^\infty_{\mathrm{ap}}(\mathbb{C}_0)$ and $A_{\mathrm{ap}}(\mathbb{C}_0)$, as well as the compact composition operators on $H^\infty_{\mathrm{ap}}(\mathbb{C}_0)$ and certain subspaces of it.