Almost commuting unitaries with spectral gap are near commuting unitaries
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- by Tobias J. Osborne
- Proc. Amer. Math. Soc. 137 (2009), 4043-4048
- DOI: https://doi.org/10.1090/S0002-9939-09-10026-6
- Published electronically: August 7, 2009
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Abstract:
Let $\mathcal {M}_n$ be the collection of $n\times n$ complex matrices equipped with operator norm. Suppose $U, V \in \mathcal {M}_n$ are two unitary matrices, each possessing a gap larger than $\Delta$ in their spectrum, which satisfy $\|UV-VU\| \le \epsilon$. Then it is shown that there are two unitary operators $X$ and $Y$ satisfying $XY-YX = 0$ and $\|U-X\| + \|V-Y\| \le E(\Delta ^2/\epsilon ) \left (\frac {\epsilon }{\Delta ^2}\right )^{\frac 16}$, where $E(x)$ is a function growing slower than $x^{\frac {1}{k}}$ for any positive integer $k$.References
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Bibliographic Information
- Tobias J. Osborne
- Affiliation: Department of Mathematics, Royal Holloway, University of London, Egham, TW20 0EX, United Kingdom
- Email: tobias.osborne@rhul.ac.uk
- Received by editor(s): September 15, 2008
- Received by editor(s) in revised form: April 18, 2009
- Published electronically: August 7, 2009
- Communicated by: Marius Junge
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 4043-4048
- MSC (2000): Primary 15A15, 15A27, 47A55; Secondary 47B47
- DOI: https://doi.org/10.1090/S0002-9939-09-10026-6
- MathSciNet review: 2538565