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Counterexample to a Question on Commutators

Alan McIntosh
Proceedings of the American Mathematical Society
Vol. 29, No. 2 (Jul., 1971), pp. 337-340
DOI: 10.2307/2038137
Stable URL: http://www.jstor.org/stable/2038137
Page Count: 4
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Counterexample to a Question on Commutators
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Abstract

We show that it is possible for two selfadjoint operators $A$ and $B$ in a Hilbert space $H$ with bounded commutator $AB - BA$ to have the property that $|A|B - B|A|$ is unbounded (where $|A|$ denotes the positive square root of $A^2$). The proof reduces to showing that for all natural numbers $n$, there exist a bounded positive operator $U$ and a bounded operator $V$ satisfying $\|UV - VU\| \geqq n\| UV + VU\|$.

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