Commutators of small rank and reducibility of operator semigroups
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- by Ali Jafarian, Alexey I. Popov, Mehdi Radjabalipour and Heydar Radjavi PDF
- Proc. Amer. Math. Soc. 142 (2014), 4277-4289 Request permission
Abstract:
It is easy to see that if $\mathcal {G}$ is a non-abelian group of unitary matrices, then for no members $A$ and $B$ of $\mathcal {G}$ can the rank of $AB-BA$ be one. We examine the consequences of the assumption that this rank is at most two for a general semigroup $\mathcal {S}$ of linear operators. Our conclusion is that under obviously necessary, but trivial, size conditions, $\mathcal {S}$ is reducible. In the case of a unitary group satisfying the hypothesis, we show that it is contained in the direct sum $\mathcal {G}_1\oplus \mathcal {G}_2$, where $\mathcal {G}_1$ is at most $3\times 3$ and $\mathcal {G}_2$ is abelian.References
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Additional Information
- Ali Jafarian
- Affiliation: University of New Haven, 300 Boston Post Road, West Haven, Connecticut 06516
- Email: ajafarian@newhaven.edu
- Alexey I. Popov
- Affiliation: Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada
- MR Author ID: 775644
- Email: a4popov@uwaterloo.ca
- Mehdi Radjabalipour
- Affiliation: Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada (on sabbatical from the Iranian Academy of Sciences, Tehran, Iran)
- Email: radjabalipour@ias.ac.ir
- Heydar Radjavi
- Affiliation: Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada
- MR Author ID: 143615
- Email: hradjavi@uwaterloo.ca
- Received by editor(s): January 16, 2013
- Published electronically: August 13, 2014
- Additional Notes: The second and fourth authors’ research was supported in part by NSERC (Canada)
The third author’s research was supported in part by the Iranian National Science Foundation - Communicated by: Pamela B. Gorkin
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 4277-4289
- MSC (2010): Primary 47D03, 20M20; Secondary 47B47, 51F25
- DOI: https://doi.org/10.1090/S0002-9939-2014-12217-9
- MathSciNet review: 3266995