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The numerical radius of a nilpotent operator on a Hilbert space

Authors: Uffe Haagerup and Pierre de la Harpe
Journal: Proc. Amer. Math. Soc. 115 (1992), 371-379
MSC: Primary 47A12; Secondary 47A10
MathSciNet review: 1072339
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Abstract: Let $ T$ be a bounded linear operator of norm 1 on a Hilbert space $ H$ such that $ {T^n} = 0$ for some $ n \geq 2$. Then its numerical radius satisfies $ w\left( T \right) \leq \cos \frac{\pi }{{\left( {n + 1} \right)}}$ and this bound is sharp. Moreover, if there exists a unit vector $ \xi \in H$ such that $ \left\vert {\left\langle {T\xi \vert\xi } \right\rangle } \right\vert = \cos \frac{\pi }{{\left( {n + 1} \right)}}$, then $ T$ has a reducing subspace 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\left( \theta \right) = \sum\nolimits_{k = - n + 1}^{n - 1} {{f_k}{e^{ik\theta }}} $ is positive, one has $ \vert{f_1}\vert \leq {f_0}\cos \frac{\pi }{{\left( {n + 1} \right)}}$; moroever, there is essentially one polynomial for which equality holds.

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Keywords: Numerical radius, nilpotent operator, positive trigonometric polynomial
Article copyright: © Copyright 1992 American Mathematical Society

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