Topologizability and related properties of the iterates of composition operators in Gelfand-Shilov classes
Angela A. Albanese, Héctor Ariza
TL;DR
The paper analyzes topologizability and $m$-topologizability of iterates of composition operators $C_\psi$ induced by polynomials on Gelfand-Shilov spaces $\mathcal{S}_\omega(\mathbb{R})$ defined via Braun-Meise-Taylor weights, with invariance under the Fourier transform. It proves that for polynomials of degree $>1$ the iterates form a topologizable family from $\mathcal{S}_\omega(\mathbb{R})$ to $\mathcal{S}_{\omega(\cdot^{1/a})}(\mathbb{R})$ when $a>2$, and that this topology may extend to $\mathcal{S}_\omega(\mathbb{R})$ under additional log-type conditions; however, $m$-topologizability can fail in general, as shown by explicit counterexamples (e.g., $\psi(x)=x^2$ with $\omega(t)=|t|^{1/s}$, $s>1$) and repelling fixed-point arguments. In the degree-one case, translations yield $m$-topologizability on $\mathcal{S}_\omega(\mathbb{R})$ (and on $\Sigma_s(\mathbb{R})$), while nontrivial dilations do not, though topologizability can be recovered in extended target scales $\mathcal{S}_{\omega(\bullet^{1/(1+\delta)})}(\mathbb{R})$ for any $\delta>0$. The work thus delineates when iterates of polynomial-induced composition operators behave topologically in ultradifferentiable Gelfand-Shilov contexts, highlighting a sharp contrast with the Schwartz space and identifying open problems for fixed-point configurations.
Abstract
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 $C_ψ$ is topologizable (m-topologizable) acting on certain Gelfand-Shilov spaces defined by mean of Braun-Meise-Taylor weights. We prove that the composition operators $C_ψ$ 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 $C_ψ$ associated with polynomials $ψ$ are not always $m-$topologizable. We also deal with the composition operators $C_ψ$ with $ψ$ being an affine function acting on $\mathcal{S}_ω(\mathbb{R})$ and find a complete characterization of topologizability and m-topologizability