Real analyticity of composition is shy

Abstract

Dahmen and Schmeding have obtained the result that although the smooth Lie group $G$ of real analytic diffeomorphisms $\mathbb S^{\,1.}\to\mathbb S^{\,1.}$ has a compatible analytic manifold structure, it does not make $G$ a real analytic Lie group since the group multiplication is not real analytic. The authors considered this result "surprising" for the applied concept of infinite-dimensional real analyticity for maps $E\to F$, defined by the property that locally a holomorphic extension $E_{\mathbb C}\to F_{\mathbb C}$ exist. In this note we show that this type of real analyticity is quite rare for composition maps ${\rm f\,}\varphi:x\mapsto\varphi\circ x$ when $\varphi$ is real analytic. Specifically, we show that the smooth Fr\'echet space map ${\rm f\,}\varphi:C\,(\mathbb R)\to C\,(\mathbb R)$ for real analytic $\varphi:\mathbb R\to\mathbb R$ is real analytic in the above sense only if $\varphi$ is the restriction to $\mathbb R$ of some entire function $\mathbb C\to\mathbb C$. We also discuss the possibility of proving that the set of these "admissible" functions $\varphi$ be "small" in the space $A\,(\mathbb R)$ of real analytic functions either in the Baire categorical sense, or in the measure theoretic sense of shyness.