A \(\psi ^ *\) algebra of pseudodifferential operators on noncompact manifolds
type: Item
instance of: scholarly article
title: A \(\psi ^ *\) algebra of pseudodifferential operators on noncompact manifolds
zbMATH Open document ID: 0631.47035
DOI: 10.1007/BF01194158
author: Elmar Schrohe
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 47L30
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 47Gxx
Mathematics Subject Classification ID: 58J40
zbMATH DE Number: 4026192
zbMATH Keywords: pseudodifferential operators
zbMATH Keywords: spectral invariance property
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4026192