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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Almost commuting unitaries with spectral gap are near commuting unitaries


Author: Tobias J. Osborne
Journal: Proc. Amer. Math. Soc. 137 (2009), 4043-4048
MSC (2000): Primary 15A15, 15A27, 47A55; Secondary 47B47
Published electronically: August 7, 2009
MathSciNet review: 2538565
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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 $ \Vert UV-VU\Vert \le \epsilon$. Then it is shown that there are two unitary operators $ X$ and $ Y$ satisfying $ XY-YX = 0$ and $ \Vert U-X\Vert + \Vert V-Y\Vert \le E(\Delta^2/\epsilon) \left(\frac{\epsilon}{\Delta^2}\right)^{\frac16}$, where $ E(x)$ is a function growing slower than $ x^{\frac{1}{k}}$ for any positive integer $ k$.


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Additional Information

Tobias J. Osborne
Affiliation: Department of Mathematics, Royal Holloway, University of London, Egham, TW20 0EX, United Kingdom
Email: tobias.osborne@rhul.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10026-6
PII: S 0002-9939(09)10026-6
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
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.