Composition operators on Gelfand-Shilov classes

Abstract

We study composition operators on global classes of ultradifferentiable functions of Beurling type invariant under Fourier transform. In particular, for the classical Gelfand-Shilov classes ${\Sigma }_d,d>1,$ we prove that a necessary condition for the composition operator $f\mapsto f\circ \psi$ to be well defined is the boundedness of ${\psi }^{\prime }.$ We find the optimal index $d^{\prime }$ for which $C_{\psi }({\Sigma }_d(\mathbb{R}))\subset {\Sigma }_{d^{\prime }}(\mathbb{R})$ holds for any non-constant polynomial $\psi .$