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English

The numerical radius of a nilpotent operator on a Hilbert space

Published inProceedings of the American Mathematical Society, vol. 115, no. 2, p. 371-379
Publication date1992
Abstract

ABSTRACTL.e t T be a bounded linear operatoro f norm 1 on a Hilbert space H such that T" = 0 for some n > 2. Then its numerical radius satisfies w (T) < cos (n'n+ 1) and this bound is sharp. Moreover, if there exists a unit vector s E H such that I( TIg) l= cos (n+'+1 ) then T has a reducings ubspace of dimension n on which T is the usual n-shift. The proofs show that these facts are related to the following result of Fejer: if a trigonometric polynomial f(6) = Zk-!n+i fkeikO is positive, one has If, I < fo cos (n+); moroever, there is essentially one polynomial for which equality holds.

Citation (ISO format)
HAAGERUP, Uffe, DE LA HARPE, Pierre. The numerical radius of a nilpotent operator on a Hilbert space. In: Proceedings of the American Mathematical Society, 1992, vol. 115, n° 2, p. 371–379.
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  • PID : unige:12076
ISSN of the journal0002-9939
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