Composition operators on Hardy-Smirnov spaces

Abstract

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 $\tau :\mathbb{U}\to \Omega$ 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-Gutiérrez and Montes-Rodríguez 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 $H^2(\Omega ).$

References in corpus (1)

• Complex symmetric partial isometries