Generalized numerical radius inequalities for certain operator matrices

TL;DR

This paper introduces new inequalities for the $q$-numerical radius of various structured operator matrices, providing bounds and computational methods, and suggests future research directions.

Contribution

It develops novel $q$-numerical radius inequalities for specific structured operator matrices, including tridiagonal, circulant, and skew circulant types.

Findings

Established bounds for $q$-numerical radius of operator matrices.

Derived inequalities for circulant and skew circulant matrices.

Provided algorithms and examples for computing $q$-numerical radii.

Abstract

In this article, a series of new inequalities involving the $q$-numerical radius for $n\times n$ tridiagonal, and anti-tridiagonal operator matrices has been established. These inequalities serve to establish both lower and upper bounds for the $q$-numerical radius of operator matrices. Additionally, we developed $q$-numerical radius inequalities for $n\times n$ circulant, skew circulant, imaginary circulant, imaginary skew circulant operator matrices. Important examples have been used to illustrate the developed inequalities. In this regard, analytical expressions and a numerical algorithm have also been employed to obtain the $q$-numerical radii. We also provide a concluding section, which may lead to several new problems in this area.