Topologizability and Related Properties of the Iterates of Composition Operators in Gelfand-Shilov Classes

We analyse the behaviour of the iterates of composition operators defined by polynomials acting on global classes of ultradifferentiable functions of Beurling type which are invariant under the Fourier transform. In particular, we determine the polynomials ψ for which the sequence of iterates of the composition operator Cψ is topologizable (m-topologizable) acting on certain Gelfand-Shilov spaces defined by mean of Braun-Meise-Taylor weights. We prove that the composition operators Cψ with ψ a polynomial of degree greater than one are always topologizable in certain settings involving Gelfand-Shilov spaces, just like in the Schwartz space. Unlike in the Schwartz space setting, composition operators Cψ associated with polynomials ψ are not always m−topologizable. We also deal with the composition operators Cψ with ψ being an affine function acting on Sω(R) and find a complete characterization of topologizability and m−topologizability.