In this article, we completely characterize the complex symmetry, cyclicity and hypercyclicity of composition operators C φ f = f • φ induced by affine self-maps φ of the right half-plane C + on the Hardy-Hilbert space H 2 (C + ). The interplay between complex symmetry and cyclicity plays a key role in the analysis. We also provide new proofs for the normal, self-adjoint and unitary cases and for an adjoint formula discovered by Gallardo-Gutiérrez and Montes-Rodríguez.2010 Mathematics Subject Classification. Primary 47B33, 47A16, 47B32.

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We investigate composition operators CΦ on the Hardy-Smirnov space H 2 (Ω) induced by analytic self-maps Φ of an open simply connected proper subset Ω of the complex plane. When the Riemann map τ : U → Ω used to define the norm of H 2 (Ω) is a linear fractional transformation, we characterize the composition operators whose adjoints are composition operators. As applications of this fact, we provide a new proof for the adjoint formula discovered by Gallardo-Gutiérrez and Montes-Rodríguez and we give a new approach to describe all Hermitian and unitary composition operators on H 2 (Ω). Additionally, if the coefficients of τ are real, we exhibit concrete examples of conjugations and describe the Hermitian and unitary composition operators which are complex symmetric with respect to specific conjugations on H 2 (Ω). We finish this paper showing that if Ω is unbounded and Φ is a non-automorphic self-map of Ω with a fixed point, then CΦ is never complex symmetric on H 2 (Ω).

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