Interplay between complex symmetry and Koenigs eigenfunctions

We investigate the relationship between complex symmetry of composition operators C ϕ f = ∘ induced on classical Hardy space H 2 ( D ) by an analytic self-map open unit disk and its Koenigs eigenfunction. A generalization orthogonality known as conjugate-orthogonality will play a key role in this work. show that if is Schröder map (fixes point ∈ with 0 < | ′ 1 σ eigenfunction, then symmetric only n N complete conjugate-orthogonal . study sequences some concrete examples. use these results to commutants symbols consist entirely operators.