Composition operators on Hardy-Smirnov spaces

TL;DR

This paper studies composition operators on Hardy-Smirnov spaces, providing new characterizations of adjoints, Hermitian, and unitary operators, and exploring complex symmetry properties under various conditions.

Contribution

It offers new proofs and characterizations of adjoint formulas, Hermitian, and unitary composition operators on Hardy-Smirnov spaces, including concrete examples and symmetry analysis.

Findings

Characterization of composition operators with adjoint as composition operators

New proof of Gallardo-Gutiérrez and Montes-Rodríguez's adjoint formula

Identification of conditions for complex symmetry and examples of conjugations

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 $\tau:\mathbb{U}\rightarrow\Omega$ 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 $H^{2}(\Omega).$