The numerical radius of a nilpotent operator on a Hilbert space
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- by Uffe Haagerup and Pierre de la Harpe PDF
- Proc. Amer. Math. Soc. 115 (1992), 371-379 Request permission
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 $\left | {\left \langle {T\xi |\xi } \right \rangle } \right | = \cos \frac {\pi }{{\left ( {n + 1} \right )}}$, 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 $|{f_1}| \leq {f_0}\cos \frac {\pi }{{\left ( {n + 1} \right )}}$; moroever, there is essentially one polynomial for which equality holds.References
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Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 115 (1992), 371-379
- MSC: Primary 47A12; Secondary 47A10
- DOI: https://doi.org/10.1090/S0002-9939-1992-1072339-6
- MathSciNet review: 1072339