标题： 算子和在半希尔伯特空间中的$A$-范数和$A$-数值半径的不等式及其应用 显示英文标题

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

摘要： 本文建立了半希尔伯特空间中算子和的$A$-范数和$A$-数值半径的几个新不等式，显著推进了现有理论。 我们提出了算子范数广义三角不等式的两个基本改进，提供了比之前已知结果更精确的估计。 我们的研究得出了算子乘积和对易子的$A$-数值半径的新界，特别关注它们的笛卡尔分解。 所发展的框架可用于量子力学，其中我们推导出改进的不确定性关系和扰动界，并用于偏微分方程，其中我们得到非局部椭圆算子的稳定性估计。 通过具体例子，我们展示了不等式的最优性及其相对于经典结果的优势。 理论贡献通过在泛函分析、算子理论和数学物理中的潜在应用得到补充，表明了半希尔伯特算子理论未来研究的方向。 显示英文摘要

摘要： 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.