On the numerical radius of the truncated adjoint Shift

TL;DR

This paper explores the relationship between Taylor coefficients of positive rational functions on the torus and the numerical radius of a specific extremal operator, extending previous inequalities and connecting classical results with operator theory.

Contribution

It establishes a connection between Taylor coefficients of rational functions and the numerical radius of a truncated shift operator, completing a line of research initiated in 2002.

Findings

Derived bounds for the numerical radius of the extremal operator

Connected classical inequalities with operator theory concepts

Extended previous results to finite Blashke products with a unique zero

Abstract

A celebrated thorem of Fejer (1915) asserts that for a given positive trigonometric polynomial $\sum_{j=-n+1}^{n-1}c_{j}e^{ijt}$, we have $\lvert c_{1}\lvert\leqslant c_{0}\cos\frac{\pi}{n+1}$. A more recent inequality due to U. Haagerup and P. de la Harpe asserts that, for any contraction $T$ such that $T^{n}=0$, for some $n\geq2$, the inequality $\omega_{2}(T)\leqslant\cos\frac{\pi}{n+1}$ holds, and $\omega_{2}(T)=\cos\frac{\pi}{n+1}$ when T is unitarily equivalent to the extremal operator ${S}^{\ast}_{n}={\bbs}_{\lvert{\C}^{n}}={\bbs}_{\lvert Ker (u_{n}(\bbs))}$ where $u_{n}(z)=z^{n}$ and $\bbs$ is the adjoint of the shift operator on the Hilbert space of all square summable sequences. Apparently there is no relationship between them. In this mathematical note, we show that there is a connection between Taylor coefficients of positive rational functions on the torus and numerical radius of the extremal operator $\bbs(\phi)=\bbs_{\lvert Ker(\phi(\bbs))}$ for a precise inner function $\phi$. This result completes a line of investigation begun in 2002 by C. Badea and G. Cassier \cite{Cassier}. An upper and lower bound of the numerical radius of $\bbs(\phi)$ are given where $\phi$ is a finite Blashke product with unique zero.