THE NUMERICAL RADIUS OF A NILPOTENT OPERATOR ON A HILBERT-SPACE

Let T be a bounded linear operator of norm 1 on a Hilbert space H such that T(n) = 0 for some n greater-than-or-equal-to 2. Then its numerical radius satisfies w(T) less-than-or-equal-to cos-pi/(n + 1) and this bound is sharp. Moreover, if there exists a unit vector xi is-an-element-of H such that [[T-xi[xi][ = cos-pi/(n + 1), 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(theta) = SIGMA(k = -n + 1)n - 1 f(k)e(ik-theta) is positive, one has [f1[ less-than-or-equal-to f0 cos-pi/(n + 1); moreover, there is essentially one polynomial for which equality holds.