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Zur Spektralinvarianz von Algebren von Pseudodifferentialoperatoren in der \(L^ p\)-Theorie. (The spectral invariance of algebras of pseudodifferential operators in the \(L^ p\)-theory). (German) Zbl 0674.47033
Let \(\Psi^ 0_{1,\delta}(0\leq \delta <1)\) be the Hörmander class of pseudodifferential operators. It is shown that if an operator P from \(\Psi^ 0_{1,\delta}\) is invertible in \({\mathfrak L}(L^ p({\mathbb{R}}^ n))\) then \(P^{-1}\in \Psi^ 0_{1,\delta}\). This statement means, in particular, that \(\Psi^ 0_{1,\delta}\) is a so-called \(\Psi\)-algebra in \({\mathfrak L}(L^ p({\mathbb{R}}^ n))\). The proof is based on methods developed earlier by R. Beals, R. R. Coifman, Y. Meyer and H. O. Cordes.
Reviewer: J.Banas

47Gxx Integral, integro-differential, and pseudodifferential operators
35S05 Pseudodifferential operators as generalizations of partial differential operators
46H05 General theory of topological algebras
46K05 General theory of topological algebras with involution
Full Text: DOI EuDML
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