On estimate for numerical radius of some contractions

Abstract

For the numerical radius of an arbitrary nilpotent operator T on a Hilbert space H, Haagerup and de la Harpe proved the inequality \(w(T) \leqslant \left\| T \right\|cos\frac{\pi }{{n + 1}}\), where $n \geq 2$ is the nilpotency order of the operator T. In the present paper, we prove a Haagerup-de la Harpe-type inequality for the numerical radius of contractions from more general classes.