article

Title: A sharpened Schwarz-Pick operatorial inequality for nilpotent operators

Authors: Haykel Gaaya

Issue: Volume 61 (2012), Issue 1, 223-248

Abstract:

$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].$