Let
T
T
be a bounded linear operator of norm 1 on a Hilbert space
H
H
such that
T
n
=
0
{T^n} = 0
for some
n
≥
2
n \geq 2
. Then its numerical radius satisfies
w
(
T
)
≤
cos
π
(
n
+
1
)
w\left ( T \right ) \leq \cos \frac {\pi }{{\left ( {n + 1} \right )}}
and this bound is sharp. Moreover, if there exists a unit vector
ξ
∈
H
\xi \in H
such that
|
⟨
T
ξ
|
ξ
⟩
|
=
cos
π
(
n
+
1
)
\left | {\left \langle {T\xi |\xi } \right \rangle } \right | = \cos \frac {\pi }{{\left ( {n + 1} \right )}}
, then
T
T
has a reducing subspace of dimension
n
n
on which
T
T
is the usual
n
n
-shift. The proofs show that these facts are related to the following result of Fejer: if a trigonometric polynomial
f
(
θ
)
=
∑
k
=
−
n
+
1
n
−
1
f
k
e
i
k
θ
f\left ( \theta \right ) = \sum \nolimits _{k = - n + 1}^{n - 1} {{f_k}{e^{ik\theta }}}
is positive, one has
|
f
1
|
≤
f
0
cos
π
(
n
+
1
)
|{f_1}| \leq {f_0}\cos \frac {\pi }{{\left ( {n + 1} \right )}}
; moroever, there is essentially one polynomial for which equality holds.