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Multiplicative Commutators of Operators

Published online by Cambridge University Press:  20 November 2018

Arlen Brown
Affiliation:
University of Michigan
Carl Pearcy
Affiliation:
University of Michigan
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An invertible operator T on a Hilbert space is a multiplicative commutator if there exist invertible operators A and B on such that T = ABA–1B–1. In this paper we discuss the question of which operators are, and which are not, multiplicative commutators. The analogous question for additive commutators (operators of the form ABBA) has received considerable attention and has, in fact, been completely settled (2). The present results represent the information we have been able to obtain by carrying over to the multiplicative problem the techniques that proved efficacious in the additive situation. While these results remain incomplete, they suffice, for example, to enable us to determine precisely which normal operators are multiplicative commutators.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Brown, A., Halmos, P. R., and Pearcy, C., Commutators of operators on Hilbert space, Can. J. Math., 17 (1965), 695708.CrossRefGoogle Scholar
2. Brown, A. and Pearcy, C., Structure of commutators of operators, Ann. Math., 82 (1965), 112127.CrossRefGoogle Scholar
3. Deckard, D. and Pearcy, C., Another class of invertible operators without square roots, Proc. Amer. Math. Soc., 14(1963), 445449.CrossRefGoogle Scholar
4. Lumer, G. and Rosenblum, M., Linear operator equations, Proc. Amer. Math. Soc., 10 (1959), 3241.CrossRefGoogle Scholar
5. Pearcy, C., On commutators of operators on Hilbert space, Proc. Amer. Math. Soc, 16 (1965), 5359.CrossRefGoogle Scholar
6. Shoda, K., Über den Kommutator der Matrizen, J. Math. Soc. Japan, 3 (1951), 7881.CrossRefGoogle Scholar
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