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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Counterexample to a question on commutators


Author: Alan McIntosh
Journal: Proc. Amer. Math. Soc. 29 (1971), 337-340
MSC: Primary 47.10
MathSciNet review: 0276798
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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 $ \left\vert A \right\vert B - B\left\vert A \right\vert$ is unbounded (where $ \left\vert A \right\vert$ 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 $ \left\Vert {UV - VU} \right\Vert \geqq n\left\Vert {UV + VU} \right\Vert$.


References [Enhancements On Off] (What's this?)

  • [1] A.-P. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092–1099. MR 0177312
  • [2] C. R. Putnam, Commutation properties of Hilbert space operators and related topics, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 36, Springer-Verlag New York, Inc., New York, 1967. MR 0217618

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0276798-4
Keywords: Hilbert space, selfadjoint operator, commutator
Article copyright: © Copyright 1971 American Mathematical Society