Interplay between complex symmetry and Koenigs eigenfunctions

Abstract

We investigate the relationship between the complex symmetry of composition operators $C_{\varphi }f=f\circ \varphi$ induced on the classical Hardy space $H^2(\mathbb{D})$ by an analytic self-map $\varphi$ 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 $\varphi$ is a Schröder map (fixes a point $a\in \mathbb{D}$ with $0<|{\varphi }^{\prime }(a)|<1$) and $\sigma$ is its Koenigs eigenfunction, then $C_{\varphi }$ 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 Schröder symbols consist entirely of complex symmetric operators.

10 pages