The numerical radius of a nilpotent operator on a Hilbert space

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.