Abel’s Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions

We obtain full description of eigenvalues and eigenvectors of composition operators $${C_{\varphi}:\fancyscript{A}\mathbb{R}\to \fancyscript{A}\mathbb{R}}$$Cφ:AR→AR for a real analytic self map$${\varphi:\mathbb{R} \rightarrow \mathbb{R} }$$φ:R→R as well as an isomorphic description of corresponding eigenspaces. We completely characterize those$${\varphi}$$φ for which Abel’s equation$${f\circ \varphi=f+1}$$f∘φ=f+1 has a real analytic solution on the real line. We find cases when the operator$${C_{\varphi}}$$Cφ has roots using a constructed embedding of$${\varphi}$$φ into the so-called real analytic iteration semigroups.