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

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.