Details
| Original language | English |
|---|---|
| Pages (from-to) | 237-254 |
| Number of pages | 18 |
| Journal | Annals of Global Analysis and Geometry |
| Volume | 10 |
| Issue number | 3 |
| Publication status | Published - Jan 1992 |
| Externally published | Yes |
Abstract
The pseudodifferential operators with symbols in the Grushin classes \~Sinf0supρ,δ, 0 ≤ δ < ρ ≤ 1, of slowly varying symbols are shown to form spectrally invariant unital Frécher-*-algebras (Ψ*-algebras) in L(L2(Rn)) and in L(Hγst) for weighted Sobolev spaces Hinfγsup stdefined via a weight d function γ. 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-Ta-Taniguchi. Both, the spectrum and the Fredholm spectrum of an operator turn out to be independent of the choices of s, t and γ. 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 Hörmander.
Keywords
- ellipticity, Fredholm operators, MSC 1991: 47G30, 47A53, 47D25, 46H35, Pseudodifferential operators, spectral invariance
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Social Sciences(all)
- Political Science and International Relations
- Mathematics(all)
- Geometry and Topology
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In: Annals of Global Analysis and Geometry, Vol. 10, No. 3, 01.1992, p. 237-254.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Spectral invariance, ellipticity, and the Fredholm property for pseudodifferential operators on weighted Sobolev spaces
AU - Schrohe, Elmar
N1 - Copyright: Copyright 2007 Elsevier B.V., All rights reserved.
PY - 1992/1
Y1 - 1992/1
N2 - The pseudodifferential operators with symbols in the Grushin classes \~Sinf0supρ,δ, 0 ≤ δ < ρ ≤ 1, of slowly varying symbols are shown to form spectrally invariant unital Frécher-*-algebras (Ψ*-algebras) in L(L2(Rn)) and in L(Hγst) for weighted Sobolev spaces Hinfγsup stdefined via a weight d function γ. 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-Ta-Taniguchi. Both, the spectrum and the Fredholm spectrum of an operator turn out to be independent of the choices of s, t and γ. 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 Hörmander.
AB - The pseudodifferential operators with symbols in the Grushin classes \~Sinf0supρ,δ, 0 ≤ δ < ρ ≤ 1, of slowly varying symbols are shown to form spectrally invariant unital Frécher-*-algebras (Ψ*-algebras) in L(L2(Rn)) and in L(Hγst) for weighted Sobolev spaces Hinfγsup stdefined via a weight d function γ. 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-Ta-Taniguchi. Both, the spectrum and the Fredholm spectrum of an operator turn out to be independent of the choices of s, t and γ. 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 Hörmander.
KW - ellipticity
KW - Fredholm operators
KW - MSC 1991: 47G30, 47A53, 47D25, 46H35
KW - Pseudodifferential operators
KW - spectral invariance
UR - http://www.scopus.com/inward/record.url?scp=0041095295&partnerID=8YFLogxK
U2 - 10.1007/BF00136867
DO - 10.1007/BF00136867
M3 - Article
AN - SCOPUS:0041095295
VL - 10
SP - 237
EP - 254
JO - Annals of Global Analysis and Geometry
JF - Annals of Global Analysis and Geometry
SN - 0232-704X
IS - 3
ER -