A \(\psi ^ *\) algebra of pseudodifferential operators on noncompact manifolds
A class of pseudodifferential operators (pdo) of order zero is shown to have the spectral invariance property: for a pdo A with symbol in \(SG_ 1^{(0,0)}(X)\) on an SG-compatible manifold X, invertibility of A as an operator on \(L^ 2(X)\) implies that the inverse also is a pdo with symbol in \(SG_ 1^{(0,0)}(X)\). Moreover, this suggests a topology on the space of zero order pdo which makes it a \(\Psi ^ *\) subalgebra of \({\mathcal L}(L^ 2(X))\) in the sense of \textit{B.Gramsch}. Following an idea of \textit{R. Beals} and \textit{H. O. Cordes}, the proof is based on a characterization of pdo by commutators.
scientific article; zbMATH DE number 4026192
scientific article