On regular Fréchet-Lie groups. VII: The group generated by pseudo- differential operators of negative order
[For part VI, see ibid. 6, 217-246 (1983; Zbl 0537.58005).] The Lie group structure of pseudo-differential operators of nonpositive order is investigated. Let \({\mathfrak P}^ m\) be the algebra of pseudo- differential operators of order m (m\(\leq 0)\) on an n-dimensional compact manifold without boundary. Various topologies \(T_{m'}\) (m'\(\leq m)\) are introduced into \({\mathfrak P}^ m\), where \(T_{m'-1}>T_{m'}\). Under each \(T_{m'}\) and the inverse limit topology \(T_{-\infty}\) of the system \(\{T_{m'}\), m'\(\leq m\}\), \({\mathfrak P}^ m\) is a Fréchet algebra. Let G\({\mathfrak P}^ m\) be the group generated by the invertible elements of \(1+{\mathfrak P}^ m\). It is proved that G\({\mathfrak P}^ m\) is a regular Fréchet-Lie group under each \(T_{m'}\) (m'\(\leq \min (m,-n-1))\) and \(T_{-\infty}\). Moreover, the *-operation is continuous with respect to the above topologies and G\({\mathfrak P}^ m\) is *-closed, where the *- operation is a mapping which assigns to an operator its adjoint operator.
Lie group structure of pseudo-differential operators
scientific article; zbMATH DE number 3901702