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

Abstract

We consider the Banach space $H_{\mathrm{ap}}^{\infty }({\mathbb{C}}_0)$ 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 $H_{\mathrm{ap}}^{\infty }({\mathbb{C}}_0)$ that are uniformly continuous on ${\mathbb{C}}_0$. We prove a strong version of Montel's theorem for $H_{\mathrm{ap}}^{\infty }({\mathbb{C}}_0)$ and characterize the bounded composition operators on $H_{\mathrm{ap}}^{\infty }({\mathbb{C}}_0)$ and $A_{\mathrm{ap}}({\mathbb{C}}_0)$, as well as the compact composition operators on $H_{\mathrm{ap}}^{\infty }({\mathbb{C}}_0)$ and certain subspaces of it.

19 pages