Inequalities for the $A$-Norm and $A$-Numerical Radius of Operator Sums in Semi-Hilbertian Spaces with Applications

Abstract

This paper establishes several new inequalities for the $A$-norm and $A$-numerical radius of operator sums in semi-Hilbertian spaces, significantly advancing the existing theory. We present two fundamental refinements of the generalized triangle inequality for operator norms, providing sharper estimates than previously known results. Our investigation yields novel bounds for the $A$-numerical radius of products and commutators of operators, with particular attention to their Cartesian decompositions. The developed framework enables applications to quantum mechanics, where we derive improved uncertainty relations and perturbation bounds, and to partial differential equations, where we obtain stability estimates for nonlocal elliptic operators. Through concrete examples, we demonstrate the optimality of our inequalities and their advantages over classical results. The theoretical contributions are complemented by potential applications in functional analysis, operator theory, and mathematical physics, suggesting directions for future research in semi-Hilbertian operator theory.