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Commutator characterization of periodic pseudodifferential operators. (English) Zbl 0976.47029
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.

MSC:
 47G30 Pseudodifferential operators 58J40 Pseudodifferential and Fourier integral operators on manifolds 47B47 Commutators, derivations, elementary operators, etc. 47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
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References:
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