Composition operators on Hardy-Smirnov spaces

Abstract

We investigate composition operators CΦ on the Hardy-Smirnov space H2(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 H2(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 H2(Ω). These descriptions are particular cases of the results obtained by Gunatillake-Jovovic-Smith in 2015 and Gunatillake in 2017. 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 H2(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 H2(Ω).