Details
| Original language | English |
|---|---|
| Pages (from-to) | 145-185 |
| Number of pages | 41 |
| Journal | Mathematische Nachrichten |
| Volume | 199 |
| Issue number | 1 |
| Publication status | Published - 1999 |
| Externally published | Yes |
Abstract
A Boutet de Monvel type calculus is developed for boundary value problems on (possibly) noncompact manifolds. It is based on a class of weighted symbols and Sobolev spaces. If the underlying manifold is compact, one recovers the standard calculus. The following is proven: (1) The algebra script G sign of Green operators of order and type zero is a spectrally invariant Fréchet subalgebra of ℒ(H), H a suitable Hubert space, i. e., script G sign ∩ ℒ(H)-1 = script G sign-1. (2) Focusing on the elements of order and type zero is no restriction since there are order reducing operators within the calculus. (3) There is a necessary and sufficient criterion for the Fredholm property of boundary value problems, based on the invertibility of symbols modulo lower order symbols, and (4) There is a holomorphic functional calculus for the elements of script G sign in several complex variables.
Keywords
- Boundary value problems, Boutet de Monvel's calculus, Fréchet algebras, Noncompact manifolds, Spectral invariance
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Mathematische Nachrichten, Vol. 199, No. 1, 1999, p. 145-185.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Fréchet algebra techniques for boundary value problems on noncompact manifolds
T2 - Fredholm criteria and functional calculus via spectral invariance
AU - Schrohe, Elmar
N1 - Copyright: Copyright 2018 Elsevier B.V., All rights reserved.
PY - 1999
Y1 - 1999
N2 - A Boutet de Monvel type calculus is developed for boundary value problems on (possibly) noncompact manifolds. It is based on a class of weighted symbols and Sobolev spaces. If the underlying manifold is compact, one recovers the standard calculus. The following is proven: (1) The algebra script G sign of Green operators of order and type zero is a spectrally invariant Fréchet subalgebra of ℒ(H), H a suitable Hubert space, i. e., script G sign ∩ ℒ(H)-1 = script G sign-1. (2) Focusing on the elements of order and type zero is no restriction since there are order reducing operators within the calculus. (3) There is a necessary and sufficient criterion for the Fredholm property of boundary value problems, based on the invertibility of symbols modulo lower order symbols, and (4) There is a holomorphic functional calculus for the elements of script G sign in several complex variables.
AB - A Boutet de Monvel type calculus is developed for boundary value problems on (possibly) noncompact manifolds. It is based on a class of weighted symbols and Sobolev spaces. If the underlying manifold is compact, one recovers the standard calculus. The following is proven: (1) The algebra script G sign of Green operators of order and type zero is a spectrally invariant Fréchet subalgebra of ℒ(H), H a suitable Hubert space, i. e., script G sign ∩ ℒ(H)-1 = script G sign-1. (2) Focusing on the elements of order and type zero is no restriction since there are order reducing operators within the calculus. (3) There is a necessary and sufficient criterion for the Fredholm property of boundary value problems, based on the invertibility of symbols modulo lower order symbols, and (4) There is a holomorphic functional calculus for the elements of script G sign in several complex variables.
KW - Boundary value problems
KW - Boutet de Monvel's calculus
KW - Fréchet algebras
KW - Noncompact manifolds
KW - Spectral invariance
UR - http://www.scopus.com/inward/record.url?scp=0001115649&partnerID=8YFLogxK
U2 - 10.1002/mana.19991990108
DO - 10.1002/mana.19991990108
M3 - Article
AN - SCOPUS:0001115649
VL - 199
SP - 145
EP - 185
JO - Mathematische Nachrichten
JF - Mathematische Nachrichten
SN - 0025-584X
IS - 1
ER -