标题： 更紧的不等式用于$A$-数值半径的算子矩阵及其应用 显示英文标题

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

摘要： 本文通过利用$A$-Buzano不等式并发展针对算子矩阵的精炼技术，建立了半希尔伯特空间中$A$-数值半径的新上界。我们提出了几个精确的不等式，这些不等式推广并改进了现有结果，包括涉及$A$-绝对值算子和混合Schwarz型不等式的$2 \times 2$算子矩阵的新界限，将$A$-数值半径与算子范数相关联的精炼幂不等式，以及通过最优参数选择得到的更紧致的估计，还提供了一个统一框架，将经典的数值半径不等式扩展到半希尔伯特空间。结果通过详细示例得到支持，展示了它们的紧致性，包括相等的情况，并我们研究了它们与经典数值半径不等式的关系，表明我们的框架如何通过$A$-算子半范数和$A$-伴随技术提供更紧致的估计。这些理论进展在量子力学（量子信道的算子界）、偏微分方程（离散化算子的稳定性分析）和控制理论（混合系统能量管理）中有应用。 我们的工作通过提供新的工具来分析通过$A$-数值半径不等式进行的算子矩阵，从而为半希尔伯特空间中的算子理论做出贡献，特别强调算子结构与由正算子诱导的半内积之间的相互作用。 显示英文摘要

摘要： 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.