Counterexample to a question on commutators
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- by Alan McIntosh PDF
- Proc. Amer. Math. Soc. 29 (1971), 337-340 Request permission
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 $\left | A \right |B - B\left | A \right |$ is unbounded (where $\left | A \right |$ 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 $\left \| {UV - VU} \right \| \geqq n\left \| {UV + VU} \right \|$.References
- A.-P. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092–1099. MR 177312, DOI 10.1073/pnas.53.5.1092
- 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
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 29 (1971), 337-340
- MSC: Primary 47.10
- DOI: https://doi.org/10.1090/S0002-9939-1971-0276798-4
- MathSciNet review: 0276798