Complex symmetric composition operators

TL;DR

This paper characterizes which linear fractional composition operators on the Hardy space are complex symmetric, providing a clear criterion for their symmetry based on the properties of the symbol map.

Contribution

It offers a complete characterization of complex symmetric linear fractional composition operators on the Hardy space, advancing understanding of their structural properties.

Findings

Identifies conditions under which $C_$ is complex symmetric

Provides a classification of linear fractional symbols leading to symmetry

Enhances understanding of symmetry in composition operators

Abstract

Let $\varphi$ be a linear fractional self-map of the open unit disk $\mathbb{D}$ and $H^2$ the Hardy space of analytic functions on $\mathbb{D}$. The goal of this article is to characterize the linear fractional composition operators $C_\varphi f=f\circ\varphi$ on $H^2$ that are complex symmetric.