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

Abstract

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 Abel's 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.Mathematics Subject Classification. Primary 47B33, 46E10, 39B12; Secondary 47A15, 47A75, 39B22, 30D05.

A hypercyclicity criterion for non-metrizable topological vector spaces

“…The idea of the previous example can be extended to certain composition operators C ϕ : A (I) → A (I) for bounded open intervals I in R. These results will appear elsewhere. We should also note the following alternative argument provided by José Bonet: A classical result of Belitskii and Lyubich [2] (see also [4]) shows that any real analytic diffeomorphism without fixed points ϕ : R → R is real analytic conjugate to the shift x → x + 1. As a consequence, for any real analytic diffeomorphism without fixed points ϕ : I → I on an open interval I ⊂ R, the composition operator C ϕ : A (I) → A (I) is sequentially hypercyclic.…”

Applications of Banach Limit in Ulam Stability

“…As for the Abel equation, its complex solutions were studied in [40,41], whereas in [42], Equation ( 9) was investigated from functional analytic and operator theoretical point of view.…”