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Commutators of Operators on Hilbert Space

Published online by Cambridge University Press:  20 November 2018

Arlen Brown
Affiliation:
University of Michigan
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The purpose of this paper is to record some progress on the problem of determining which (bounded, linear) operators A on a separable Hilbert space H are commutators, in the sense that there exist bounded operators B and C on H satisfying A = BCCB. It is thus natural to consider this paper as a continuation of the sequence (2; 3; 5). In §2 we show that many infinite diagonal matrices (with scalar entries) are commutators and that every weighted unilateral and bilateral shift is a commutator.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Dixmier, J., Les algèbres d﹜ opérateurs dans l'espace Hilbertien (Paris, 1957).Google Scholar
2. Halmos, P. R., Commutators of operators, Amer. J. Math., 74 (1952), 237240.CrossRefGoogle Scholar
3. Halmos, P. R., Commutators of operators II, Amer. J. Math., 76 (1954), 191198.CrossRefGoogle Scholar
4. Halmos, P. R., A glimpse into Hilbert space] Chap. I, Lectures on Mathematics (New York, 1963).Google Scholar
5. Pearcy, C., On commutators of operators on Hilbert space, Proc. Amer. Math. Soc, 16 (1965), 5359.CrossRefGoogle Scholar
6. Wielandt, H., Über die Unbeschränktheit der Operatoren der Quantenmechanik, Math. Ann., 121 (1949), 21.CrossRefGoogle Scholar
7. Wintner, A., The unboundedness of quantum-mechanical matrices, Phys. Rev., 71 (1947), 738739.CrossRefGoogle Scholar
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