Commutator characterization of periodic pseudodifferential operators (Q1978085)

Let \(M\) be a closed smooth orientable manifold and \(\mathbb{T}^n\) be the torus \(\mathbb{R}^n/\mathbb{Z}^n\). The set of pseudo-differential operators of order \(m\in\mathbb{R}\) on \(M\), in the Hörmander sense, is denoted by \(\Psi^m(M)\). Let \(\text{Op }S^m(\mathbb{T}^n)\) be the set of periodic pseudodifferential operators, in the Agranovich sense. The aim of the paper is a proof of the equality \(\Psi^m(\mathbb{T}^n)= \text{Op }S^m(\mathbb{T}^n)\), by means of commutators. The main step in the proof is a global commutator characterization of \(\Psi^m(M)\). The paper is self-contained: at the beginning of each section, the author clearly reviews definitions and calculus of the pseudodifferential operators which are involved in the section.