Abstract
We obtain full description of eigenvalues and eigenvectors
of composition operators Cϕ : A (R) → A (R) for a real analytic self
map ϕ : R → R as well as an isomorphic description of corresponding
eigenspaces. We completely characterize those ϕ for which Abels equation
f ◦ ϕ = f + 1 has a real analytic solution on the real line. We find
cases when the operator Cϕ has roots using a constructed embedding of
ϕ into the so-called real analytic iteration semigroups.