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Operator norms determined by their numerical ranges

Published online by Cambridge University Press:  20 January 2009

Colin M. McGregor
Colin M. McGregor
Affiliation:
Department of Mathematics, King's College, Aberdeen
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This paper owes its origin to the following question posed by A. M. Sinclair, “If a linear algebra with identity has two equivalent unital algebra norms, |.|1 and |.|2, whose corresponding numerical radii, v1 and v2, are equal on the whole algebra, are |.|1 and |.|2 related? Are they, for example, necessarily equal?” We do not give a complete answer to this question but are able to give sufficient conditions on algebras of operators for v1 = v2 to imply |.|1 = |.|2 That this implication does not hold for an arbitrary algebra with identity is demonstrated by means of a counter-example. The result for operator algebras is used to deduce some essentially non numerical range results for equivalent operator norms.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 17 , Issue 3 , June 1971 , pp. 249 - 255
Copyright
Copyright © Edinburgh Mathematical Society 1971

References

REFERENCES

(1)Bonsall, F. F. and Duncan, J., Numerical ranges of operators on normed spaces and of elements of normed algebras, London Math. Soc. Lecture Note Series, No. 2 (1971).Google Scholar
(2)Duncan, J., Mcgregor, C. M., Pryce, J. D. and White, A. J., The numerical index of a normed space, J. London Math. Soc. (2) 2 (1970), 481–488.CrossRefGoogle Scholar