A sharpened Schwarz-Pick operatorial inequality for nilpotent operators Haykel Gaaya 47A1247B35operator theorynumerical radiusnumerical rangeeigenvaluesvon Neumann inequalitiescompression shiftToeplitz matricesunitary dilation$\rho$-dilationsPoncelet property Let $S(\phi)$ denote the extremal operator defined by the compression of the unilateral shift $S$ to the model subspace $H(\phi) = \mathbb{H}^2\ominus\phi\mathbb{H}^2 $ as the following: $S(\phi)f(z) = P(zf(z))$, where $P$ denotes the orthogonal projection from $\mathbb{H}^2$ onto $ H(\phi)$ and $\phi$ is an inner function on the unit disc. In this mathematical note, we give an explicit formula of the numerical radius of $S(\phi)$ in the particular case where $\phi$ is a finite Blaschke product with unique zero and give an estimate on the general case. We establish also a sharpened Schwarz-Pick operatorial inequality generalizing a U. Haagerup and P. de la Harpe result for nilpotent operators [U. Haagerup and P. de la Harpe, \textit{The numerical radius of a nilpotent operator on a Hilbert space}, Proc. Amer. Math. Soc. \textbf{115} (1992), no. 2, 371--379]. Indiana University Mathematics Journal 2012 text pdf 10.1512/iumj.2012.61.4946 10.1512/iumj.2012.61.4946 en Indiana Univ. Math. J. 61 (2012) 223 - 248 state-of-the-art mathematics http://iumj.org/access/