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Journal of Operator Theory

Volume 82, Issue 2, Fall 2019  pp. 369-382.

Commutators close to the identity

Authors:  Terence Tao
Author institution:UCLA Department of Math., Los Angeles, CA 90095-1555, U.S.A.

Summary: Let $D,X \in B(H)$ be bounded operators on an infinite dimensional Hilbert space $H$. If the commutator $[D,X] = DX-XD$ lies within $\varepsilon $ in operator norm of the identity operator $1_{B(H)}$, then it was observed by Popa that one has the lower bound $\| D \| \|X\| \geqslant \frac{1}{2} \log \frac{1}{\varepsilon }$ on the product of the operator norms of $D,X$; this is a quantitative version of the Wintner--Wielandt theorem that $1_{B(H)}$ cannot be expressed as the commutator of bounded operators. On the other hand, it follows easily from the work of Brown and Pearcy that one can construct examples in which $\|D\| \|X\| = O(\varepsilon ^{-2})$. In this note, we improve the Brown--Pearcy construction to obtain examples of $D,X$ with $\| [D,X] - 1_{B(H)} \| \leqslant \varepsilon $ and $\| D\| \|X\| = O( \log^{5} \frac{1}{\varepsilon } )$.


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