Proper improvement of well-known numerical radius inequalities and their applications

TL;DR

This paper introduces improved inequalities for the numerical radius of bounded linear operators on Hilbert spaces, providing tighter bounds and applications to estimating zeros of complex polynomials.

Contribution

It establishes new, non-trivial inequalities for the numerical radius that improve upon classical bounds, with applications to polynomial zero estimation.

Findings

Derived a new inequality: $w^2(T) \,\leq\, \min_{0\leq \alpha \leq 1} \| \alpha T^*T + (1-\alpha) TT^* \|$.

Showed the inequalities are non-trivial improvements over existing bounds.

Applied the results to estimate bounds for zeros of complex monic polynomials.

Abstract

New inequalities for the numerical radius of bounded linear operators defined on a complex Hilbert space $\mathcal{H}$ are given. In particular, it is established that if $T$ is a bounded linear operator on a Hilbert space $\mathcal{H}$ then \[ w^2(T)\leq \min_{0\leq \alpha \leq 1} \left \| \alpha T^*T +(1-\alpha)TT^* \right \|,\] where $w(T)$ is the numerical radius of $T.$ The inequalities obtained here are non-trivial improvement of the well-known numerical radius inequalities. As an application we estimate bounds for the zeros of a complex monic polynomial.