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Almost commuting matrices need not be nearly commuting


Author: Man Duen Choi
Journal: Proc. Amer. Math. Soc. 102 (1988), 529-533
MSC: Primary 47A55; Secondary 15A27, 47B47
DOI: https://doi.org/10.1090/S0002-9939-1988-0928973-3
MathSciNet review: 928973
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Abstract: Let $ {\mathcal{M}_n}$ be the collection of $ n \times n$ complex matrices with the Hilbert-space-operator norm. There exist two concrete matrices $ A,B \in {\mathcal{M}_n}$ with $ \vert\vert A\vert\vert = 1 - 1/n,\vert\vert B\vert\vert \leq 1,\vert\vert AB - BA\vert\vert \leq 2/n$, but $ \vert\vert A - R\vert\vert + \vert\vert B - S\vert\vert \geq 1 - 1/n$ for all commuting pairs $ R,S \in {\mathcal{M}_n}$. It is shown explicitly that there is a natural obstruction which prevents almost commuting matrices to get close to any commuting pairs.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0928973-3
Keywords: Almost commuting matrices, Hilbert-space-operator norm, signature of a matrix, commuting approximant, pertubation
Article copyright: © Copyright 1988 American Mathematical Society

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