Composition operators on Hardy-Smirnov spaces
We investigate composition operators $C_Φ$ 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 $τ:\mathbb{U}\rightarrowΩ$ 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 $C_Φ$ is never complex symmetric on $H^{2}(Ω).$