Composition operators on Hardy-Smirnov spaces

We investigate composition operators $C_{\Phi}$ on the Hardy-Smirnov space $H^{2}(\Omega)$ induced by analytic self-maps $\Phi$ of an open simply connected proper subset $\Omega$ of the complex plane. When the Riemann map $\tau:\mathbb{U}\rightarrow\Omega$ used to define the norm of $H^{2}(\Omega)$ 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-Guti\'{e}rrez and Montes-Rodr\'{i}guez and we give a new approach to describe all Hermitian and unitary composition operators on $H^{2}(\Omega).$ Additionally, if the coefficients of $\tau$ 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 $H^{2}(\Omega).$ We finish this paper showing that if $\Omega$ is unbounded and $\Phi$ is a non-automorphic self-map of $\Omega$ with a fixed point, then $C_{\Phi}$ is never complex symmetric on $H^{2}(\Omega).$