Order-preserving isomorphisms of algebras of pseudo-differential operators

07.07.2022, 16:00  –  2.09.0.14
Forschungsseminar Differentialgeometrie

Alessandro Contini (Leibniz University Hanover)

In 1971 Egorov proved his famous Theorem, stating that conjugating a pseudo-differential operator with an invertible Fourier Integral Operator produces a new \(\Psi\)DO with the same principal symbol. This implies in particular that the map \(\varphi(A)\colon=FAF'\) for \(F'\) a parametrix of the elliptic FIO \(F\) is an order-preserving automorphism of the algebra \(\Psi(M)\) for \(M\) a manifold. In the subsequent years the following question (a kind of converse to Egorov's result) was addressed: can we characterize all the order-preserving algebra isomorphisms between \(\Psi(X)\) and \(\Psi(Y)\) for any manifolds \(X,Y\)? Positive answers were given first by Duistermaat and Singer (with refinements by Mathai and Melrose) in the Hörmander calculus, then by Christianson for semi-classical operators, and finally by Battisti-Coriasco-Schrohe for the Boutet de Monvel algebra. In this talk I shall recall and give an overview of the above mentioned concepts and results. Then I shall dedicate some time to present the so-called SG-calculus of pseudo-differential operators, and finally I will outline some ideas and problems encountered up until now. This is based on work-in-progress during my current time as a PhD student with Elmar Schrohe.