Spectral invariance, ellipticity, and the Fredholm property for pseudodifferential operators on weighted Sobolev spaces

✍ Scribed by Elmar Schrohe

• Publisher: Springer
• Year: 1992
• Tongue: English
• Weight: 840 KB
• Volume: 10
• Category: Article
• ISSN: 0232-704X
• DOI: 10.1007/bf00136867

✦ Synopsis

The pseudodifferential operators with symbols in the Grushin classes S,~, 0 < < p < 1, of slowly varying symbols are shown to form spectrally invariant unital Fr6chet-*-algebras (*-algebras) in £(L 2 (Rn)) and in £(Hz t ) for weighted Sobolev spaces Hit defined via a weight function y. In all cases, the Fredholm property of an operator can be characterized by uniform ellipticity of the symbol. This gives a converse to theorems of Grushin and Kumano-go-Taniguchi. Both, the spectrum and the Fredholm spectrum of an operator turn out to be independent of the choices of s, t and .

The characterization of the Fredholm property by uniform ellipticity leads to an index theorem for the Fredholm operators in these classes, extending results of Fedosov and Hormander.