Composition operators on Gelfand-Shilov classes

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 $\psi'.$ We find the optimal index $d'$ for which $C_\psi(\Sigma_d({\mathbb R}))\subset \Sigma_{d'}({\mathbb R})$ holds for any non-constant polynomial $\psi.$