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 \({\mathbb {R}}^n\) .
