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The numerical radius of a nilpotent operator on a Hilbert space 

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Published in  Proceedings of the American Mathematical Society. 1992, vol. 115, no. 2, p. 371379  
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 nshift. 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.  
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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. 371379. https://archiveouverte.unige.ch/unige:12076 