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

TL;DR

This paper develops a comprehensive operator-theoretic and function-theoretic framework for analytic almost periodic functions on the right half-plane. It introduces strong Montel-type results for the space $H^ abla_ ext{ap}( abla_0)$ via the concept of joint almost periodicity, and proves a density result $A_ ext{ap}( abla_0)=\overline{\mathscr{P}_D}$, connecting to Dirichlet polynomials. It then characterizes bounded and compact composition operators on $H^ abla_ ext{ap}( abla_0)$ and $A_ ext{ap}( abla_0)$, including their subspaces, by describing symbols of the form $\varphi(s)=a s+\psi(s)$ with $a\ge0$ and $\psi$ almost periodic. The results generalize and unify Bayart–type Montel and composition-operator theorems without relying on Bohr's theorem, highlighting the role of almost periodicity in the analytic structure of these spaces and their Dirichlet-series connections.

Abstract

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.

Theorems & Definitions (63)

• Definition
• Definition
• Theorem 1.1
• Definition
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Theorem 2.2
• Remark