Complex symmetry and cyclicity of composition operators on $H^2(\mathbb{C}_+)$

TL;DR

This paper characterizes the complex symmetry, cyclicity, and hypercyclicity of affine-induced composition operators on the Hardy space of the right half-plane, providing new proofs for special cases and an adjoint formula.

Contribution

It offers a complete characterization of these properties for composition operators induced by affine maps on $H^2( ext{half-plane})$, including new proofs and formulas.

Findings

Characterization of complex symmetry, cyclicity, and hypercyclicity for these operators.

New proofs for normal, self-adjoint, and unitary cases.

An explicit adjoint formula for the operators.

Abstract

In this article, we completely characterize the complex symmetry, cyclicity and hypercyclicity of composition operators $C_\phi f=f\circ\phi$ induced by affine self-maps $\phi$ of the right half-plane $\mathbb{C}_+$ on the Hardy-Hilbert space $H^2(\mathbb{C}_+)$. We also provide new proofs for the normal, self-adjoint and unitary cases and for an adjoint formula discovered by Gallardo-Guti\'{e}rrez and Montes-Rodr\'{i}gues.