Globally solvable time-periodic evolution equations in Gelfand-Shilov classes

In this paper we consider a class of evolution operators with coefficients depending on time and space variables $(t,x)\in \mathbb{T}\times {\mathbb{R}}^n$, where $\mathbb{T}$ is the one-dimensional torus and prove necessary and sufficient conditions for their global solvability in (time-periodic) Gelfand-Shilov spaces. The argument of the proof is based on a characterization of these spaces in terms of the eigenfunction expansions given by a fixed self-adjoint, globally elliptic differential operator on ${\mathbb{R}}^n$.