Numerical radius inequalities and estimation of zeros of polynomials

TL;DR

This paper presents new inequalities relating the numerical radius and spectral radius of operators, and applies these results to derive bounds for the zeros of complex polynomials.

Contribution

It introduces refined inequalities for the numerical radius of operators and develops bounds for polynomial zeros, advancing operator theory and polynomial root estimation.

Findings

Refined numerical radius inequalities for bounded linear operators.

New bounds for zeros of complex polynomials.

Inequalities involving spectral radius and numerical radius for operator sums.

Abstract

Let $A$ be a bounded linear operator defined on a complex Hilbert space and let $|A|=(A^*A)^{1/2}$ be the positive square root of $A$. Among other refinements of the well known numerical radius inequality $w^2(A)\leq \frac12 \|A^*A+AA^*\|$, we show that \begin{eqnarray*} w^2(A)&\leq&\frac{1}{4} w^2 \left(|A|+i|A^*|\right)+\frac{1}{8}\left\||A|^2+|A^*|^2\right \|+\frac{1}{4}w\left(|A||A^*|\right) &\leq& \frac12 \|A^*A+AA^*\|. \end{eqnarray*} Also, we develop inequalities involving numerical radius and spectral radius for the sum of the product operators, from which we derive the following inequalities $$ w^p(A) \leq \frac{1}{\sqrt{2} } w(|A|^p+i|A^*|^p )\leq \|A\|^p$$ for all $p\geq 1.$ Further, we derive new bounds for the zeros of complex polynomials.