Tighter Inequalities for $A$-Numerical Radii of Operator Matrices and Their Applications

TL;DR

This paper develops sharper bounds for the $A$-numerical radius of operator matrices in semi-Hilbertian spaces, enhancing theoretical tools with applications in quantum mechanics, PDEs, and control theory.

Contribution

It introduces novel inequalities for $A$-numerical radius, generalizing classical results and providing a unified framework for semi-Hilbertian space operator analysis.

Findings

New upper bounds for $A$-numerical radius of operator matrices.

Refined inequalities involving $A$-absolute value operators and Schwarz-type bounds.

Applications demonstrated in quantum mechanics, PDE stability, and control systems.

Abstract

This paper establishes new upper bounds for the $A$-numerical radius of operator matrices in semi-Hilbertian spaces by leveraging the $A$-Buzano inequality and developing refined techniques for operator matrices. We present several sharp inequalities that generalize and improve existing results, including novel bounds for $2 \times 2$ operator matrices involving $A$-absolute value operators and mixed Schwarz-type inequalities, refined power inequalities relating $A$-numerical radius to operator norms with optimal parameter selection, and a unified framework extending classical numerical radius inequalities to semi-Hilbertian spaces. The results are supported by detailed examples demonstrating their sharpness, including cases of equality, and we investigate their relationship to classical numerical radius inequalities, showing how our framework provides tighter estimates through $A$-operator seminorms and $A$-adjoint techniques. These theoretical advances have applications in quantum mechanics (operator bounds for quantum channels), partial differential equations (stability analysis of discretized operators), and control theory (hybrid system energy management). Our work contributes to operator theory in semi-Hilbertian spaces by providing new tools for analyzing operator matrices through $A$-numerical radius inequalities, with particular emphasis on the interplay between operator structure and the semi-inner product induced by positive operators.