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

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.