Interplay between complex symmetry and Koenigs eigenfunctions

We investigate the relationship between the complex symmetry of composition operators $C_ϕf=f\circ ϕ$ induced on the classical Hardy space $H^2(\mathbb{D})$ by an analytic self-map $ϕ$ 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 $ϕ$ is a Schröder map (fixes a point $a\in \mathbb{D}$ with $0<|ϕ'(a)|<1$) and $σ$ is its Koenigs eigenfunction, then $C_ϕ$ is complex symmetric if and only if $(σ^n)_{n\in \mathbb{N}}$ 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.

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