Elevating Precision in Inequalities for Numerical Radii and Operator Matrices

TL;DR

This paper develops more precise inequalities for the numerical radius of operators and matrices, utilizing the generalized Aluthge transform to improve bounds and extend previous results in operator theory.

Contribution

It introduces enhanced numerical radius inequalities and bounds for operator matrices, advancing the accuracy of existing theoretical estimates.

Findings

Established a range of improved numerical radius inequalities.

Derived bounds for 2x2 operator matrices that refine previous estimates.

Utilized the generalized Aluthge transform to obtain new inequalities.

Abstract

In this paper, we aim to establish a range of numerical radius inequalities. These discoveries will bring us to a recently validated numerical radius inequality and will present numerical radius inequalities that exhibit enhanced precision when compared to those recently established for particular cases. Additionally, we employ the generalized Aluthge transform for operators to deduce a set of inequalities pertaining to the numerical radius. Moreover, we set forth various upper and lower bounds for the numerical radius of $2\times 2$ operator matrices, refining and expanding upon the bounds determined previously.