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 ${\mathrm{\Sigma }}_d,d>1,$ we prove that a necessary condition for the composition operator $f\mapsto f\circ \psi$ to be well defined is the boundedness of ${\psi }^{\mathrm{\prime }}\mathrm{.}$ We find the optimal index $d^{\mathrm{\prime }}$ for which $C_{\psi }({\mathrm{\Sigma }}_d(\mathbb{R}))\subset {\mathrm{\Sigma }}_{d^{\mathrm{\prime }}}(\mathbb{R})$ holds for any non-constant polynomial $\psi \mathrm{.}$