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On Lie group representations and operator ranges.
- Source :
-
Proceedings of the American Mathematical Society . Oct2021, Vol. 149 Issue 10, p4317-4329. 13p. - Publication Year :
- 2021
-
Abstract
- In this paper, Lie group representations on Hilbert spaces are studied in relation with operator ranges. Let R be an operator range of a Hilbert space H. Given the set Λ of R-invariant operators, and given a Lie group representation ρ : G → GL(H), we discuss the induced semigroup homomorphism ρ : ρ−1(Λ) → B(R) for the operator range topology on R. In one direction, we work under the assumption ρ−1 (Λ) = G, so ρ : G →B(R) is in fact a group representation. In this setting, we prove that ρ is continuous (and smooth) if and only if the tangent map dρ is R-invariant. In another direction, we prove that for the tautological representations of unitary or invertible operators of an arbitrary infinite-dimensional Hilbert space H, the set ρ−1(Λ) is neither a group for a large set of nonclosed operator ranges R nor closed for all nonclosed operator ranges R. Both results are proved by means of explicit counterexamples. [ABSTRACT FROM AUTHOR]
- Subjects :
- *HILBERT space
*UNITARY operators
*TOPOLOGY
*HOMOMORPHISMS
*LIE groups
Subjects
Details
- Language :
- English
- ISSN :
- 00029939
- Volume :
- 149
- Issue :
- 10
- Database :
- Academic Search Index
- Journal :
- Proceedings of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 152160847
- Full Text :
- https://doi.org/10.1090/proc/15554