Almost commuting unitaries with spectral gap are near commuting unitaries

Osborne, Tobias

doi:10.1090/S0002-9939-09-10026-6

Almost commuting unitaries with spectral gap are near commuting unitaries
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by Tobias J. Osborne PDF

Proc. Amer. Math. Soc. 137 (2009), 4043-4048 Request permission

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 $k$.

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