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Commutation Properties of Operator Polynomials

Published online by Cambridge University Press:  09 April 2009

S. R. Caradus
S. R. Caradus
Affiliation:
Australian National University, CanberraQueen's University at, Kingston
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Suppose A and B are continuous linear operators mapping a complex Banach space X into itself. For any polynomial pC, it is obvious that when A commutes with B, then p(A) commutes with B. To see that the reverse implication is false, let A be nilpotent of order n. Then An commutes with all B but A cannot do so. Sufficient conditions for the implication: p(A) commutes with B implies A commutes with B: were given by Embry [2] for the case p(λ) = λn and Finkelstein and Lebow [3] in the general case. The latter authors proved in fact that if f is a function holomorphic on σ(A) and if f is univalent with non-vanishing derivative on σ(A), then A can be expressed as a function of f(A).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 1 , February 1971 , pp. 98 - 100
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Caradus, S. R., ‘A Note on a paper by J. T. Marti’, Comment. Math. Helv. 44 (3) (1969), 282–283.CrossRefGoogle Scholar
[2]Embry, M. R., ‘Nth Roots of Operators’, Proc. Amer. Math. Soc. 19 (1) (1968), 63–68.Google Scholar
[3]Finkelstein, M. and Lebow, A., ‘A Note on Nth Roots of Operators’, Proc. Amer. Math. Soc. 21 (1969), 250.Google Scholar
[4]Marti, J. T., ‘Operational Calculus for Two Commuting Closed Operators’, Comment. Math. Helv. 43 (1968), 87–97.CrossRefGoogle Scholar
[5]Stone, M. H., ‘On Unbounded Operators in Hilbert Space’, J. Indian Math. Soc. (N. S.) 15 (1951), 155–192.Google Scholar