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arXiv · 1202.3962

A sharpened Schwarz-Pick operatorial inequality for nilpotent operators

Haykel Gaaya
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Abstract

Let denote by $S(ϕ)$ the extremal operator defined by the compression of the unilateral shift $S$ to the model subspace $ H(ϕ)=H^{2} \ominus ϕH^{2} $ as the following $S(ϕ)f(z)=P(zf(z)),$ where $P$ denotes the orthogonal projection from the Hardy space $H^{2}$ onto $ H(ϕ)$ and $ϕ$ is an inner function on the unit disc. In this mathematical notes, we give an explicit formula of the numerical radius of the truncated shift $S(ϕ)$ in the particular case where $ϕ$ is a finite Blaschke product with unique zero and 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

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Haykel Gaaya. 2012-02-17. A sharpened Schwarz-Pick operatorial inequality for nilpotent operators. https://arxiv.org/abs/1202.3962

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