A note on composition operators on spaces of real analytic functions

Abstract

We characterize composition operators on spaces of real analytic functions which are open onto their images. We give an example of a semiproper map ϕ such that the associated composition operator is not open onto its image.1. ϕ is semiproper (i.e., for every compact set K in N there is a compact set L in M such that ϕ(L) = ϕ(M ) ∩ K); 2. ϕ(M ) has the "global extension property"; 3. ϕ(M ) has the "semiglobal extension property".

Composition operators with closed range for one-dimensional smooth symbols

“…It is worth mentioning that for semiproper real analytic symbols ψ : R n → R m we know the full characterization of composition operators with closed range not only for the space of smooth functions [7] but also for the space of real analytic functions (for details see the works of Domański and Langenbruch [9][10][11] and Domański et al [12]). …”

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

“…Since f 0 (R) is a halfline unbounded from above (or the whole line) we get q•f 0 (R) = T. Then by [21,Th. 3.2], C q•f0 is open onto its image.…”

“…The composition operator is definitely one of the most natural linear operators of analysis and there is an extensive literature on that subject: see the monographs in case of spaces of holomorphic functions [19,39] and the papers on a real analytic case [13,14,[20][21][22][23][24]. For a literature on the space of real analytic functions see a recent survey [20].…”

“…We conclude this section recalling when composition operators on A (R) are open, see [22]. • ϕ is surjective;…”

A note on dynamics in functional spaces

“…In [4,5], composition operators on spaces of real analytic functions were studied. Dynamical properties of topological transitivity and hypercyclicity of a composition operator C ϕ on certain subspaces of C(X) have also been extensively studied in connection with the topological properties of the underlying map ϕ ; see [2,9].…”