Details
| Original language | English |
|---|---|
| Pages (from-to) | 141-167 |
| Number of pages | 27 |
| Journal | Bulletin des Sciences Mathematiques |
| Volume | 155 |
| Early online date | 24 Jan 2019 |
| Publication status | Published - Sept 2019 |
Abstract
Given a Lie group G of quantized canonical transformations acting on the space L2(M) over a closed manifold M, we define an algebra of so-called G-operators on L2(M). We show that to G-operators we can associate symbols in appropriate crossed products with G, introduce a notion of ellipticity and prove the Fredholm property for elliptic elements. This framework encompasses many known elliptic theories, for instance, shift operators associated with group actions on M, transversal elliptic theory, transversally elliptic pseudodifferential operators on foliations, and Fourier integral operators associated with coisotropic submanifolds.
Keywords
- Crossed product, Elliptic operator, Fourier integral operator, Fredholm operator, Quantized canonical transformation
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Bulletin des Sciences Mathematiques, Vol. 155, 09.2019, p. 141-167.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Elliptic operators associated with groups of quantized canonical transformations
AU - Savin, A.
AU - Schrohe, E.
AU - Sternin, B.
N1 - Funding Information: The authors are grateful to S. Coriasco, Yu.A. Kordyukov, V.E. Nazaikinskii and R. Nest for useful remarks. The authors were supported by DFG (grant SCHR 319/8-1 ). Boris Sternin was also funded by RFBR (grants 15-01-08392 and 16-01-00373 ) and Anton Savin by “ RUDN University Program 5-100 ”. The main results of this paper were announced at the Conference on Analysis in Hannover, October 2016. The authors are grateful to the referee for useful remarks. Finally, note that recently algebraic aspects of index problems for G -operators associated with discrete groups of quantized canonical transformations were studied in [12] .
PY - 2019/9
Y1 - 2019/9
N2 - Given a Lie group G of quantized canonical transformations acting on the space L2(M) over a closed manifold M, we define an algebra of so-called G-operators on L2(M). We show that to G-operators we can associate symbols in appropriate crossed products with G, introduce a notion of ellipticity and prove the Fredholm property for elliptic elements. This framework encompasses many known elliptic theories, for instance, shift operators associated with group actions on M, transversal elliptic theory, transversally elliptic pseudodifferential operators on foliations, and Fourier integral operators associated with coisotropic submanifolds.
AB - Given a Lie group G of quantized canonical transformations acting on the space L2(M) over a closed manifold M, we define an algebra of so-called G-operators on L2(M). We show that to G-operators we can associate symbols in appropriate crossed products with G, introduce a notion of ellipticity and prove the Fredholm property for elliptic elements. This framework encompasses many known elliptic theories, for instance, shift operators associated with group actions on M, transversal elliptic theory, transversally elliptic pseudodifferential operators on foliations, and Fourier integral operators associated with coisotropic submanifolds.
KW - Crossed product
KW - Elliptic operator
KW - Fourier integral operator
KW - Fredholm operator
KW - Quantized canonical transformation
UR - http://www.scopus.com/inward/record.url?scp=85063593566&partnerID=8YFLogxK
U2 - 10.1016/j.bulsci.2019.01.010
DO - 10.1016/j.bulsci.2019.01.010
M3 - Article
AN - SCOPUS:85063593566
VL - 155
SP - 141
EP - 167
JO - Bulletin des Sciences Mathematiques
JF - Bulletin des Sciences Mathematiques
SN - 0007-4497
ER -