Interplay between complex symmetry and Koenigs eigenfunctions

TL;DR

This paper explores the connection between complex symmetry of composition operators on Hardy space and their Koenigs eigenfunctions, revealing conditions for symmetry and analyzing the structure of their commutants.

Contribution

It establishes a characterization of complex symmetric composition operators via Koenigs eigenfunctions and conjugate-orthogonality, and examines the structure of their commutants.

Findings

Complex symmetry of $C_{}$ is characterized by the conjugate-orthogonality of Koenigs sequences.

Koenigs eigenfunctions' conjugate-orthogonality is linked to the operator's symmetry.

Commutants of complex symmetric composition operators with Schr"{o}der symbols are entirely composed of complex symmetric operators.

Abstract

We investigate the relationship between the complex symmetry of composition operators $C_{\phi}f=f\circ \phi$ induced on the classical Hardy space $H^2(\mathbb{D})$ by an analytic self-map $\phi$ of the open unit disk $\mathbb{D}$ and its Koenigs eigenfunction. A generalization of orthogonality known as conjugate-orthogonality will play a key role in this work. We show that if $\phi$ is a Schr\"{o}der map (fixes a point $a\in \mathbb{D}$ with $0<|\phi'(a)|<1$) and $\sigma$ is its Koenigs eigenfunction, then $C_{\phi}$ is complex symmetric if and only if $(\sigma^n)_{n\in \mathbb{N}}$ is complete and conjugate-orthogonal in $H^2(\mathbb{D})$. We study the conjugate-orthogonality of Koenigs sequences with some concrete examples. We use these results to show that commutants of complex symmetric composition operators with Schr\"{o}der symbols consist entirely of complex symmetric operators.