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

Abstract

In this paper we consider a class of evolution operators with coefficients depending on time and space variables $$(t,x) \in {\mathbb {T}}\times {\mathbb {R}}^n$$ ( t , x ) ∈ T × R n , where $${\mathbb {T}}$$ 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 $${\mathbb {R}}^n$$ R n .