标题： 某些半希尔伯特空间中算子的$A$-数值半径的上界及其应用 显示英文标题

标题： Certain Upper Bounds on the $A$-Numerical Radius of Operators in Semi-Hilbertian Spaces and Their Applications

摘要： 考虑一个复数希尔伯特空间$\left(\mathcal{H}, \langle \cdot, \cdot \rangle\right)$，其上配备了一个正的有界线性算子$A$在$\mathcal{H}$上。 这通过半内积$\langle x, y \rangle_A = \langle Ax, y \rangle$诱导出一个半范数$\|\cdot\|_A$，对于$x, y \in \mathcal{H}$。 在这一半希尔伯特空间的设定中，我们研究了有界算子$T$的$A$-数值半径$w_A(T)$和$A$-算子半范数$\|T\|_A$。 本文对半希尔伯特空间中算子的$A$-数值半径现有的上界进行了显著改进。 我们建立了几个新的不等式，这些不等式提供的估计比目前文献中的更精确。 值得注意的是，我们改进了$A$-算子半范数的三角不等式，提供了对算子行为更精确的描述。 为了验证我们的理论进展，我们提供了具体的例子，展示了我们结果的有效性。 这些例子说明了我们的界限相比之前的结果提供了更紧的估计，证实了我们发现的实际相关性。 我们的工作有助于更广泛地理解数值半径不等式及其在算子理论中的应用，对泛函分析及相关领域具有潜在的影响。 显示英文摘要

摘要： Consider a complex Hilbert space $\left(\mathcal{H}, \langle \cdot, \cdot \rangle\right)$ equipped with a positive bounded linear operator $A$ on $\mathcal{H}$. This induces a semi-norm $\|\cdot\|_A$ through the semi-inner product $\langle x, y \rangle_A = \langle Ax, y \rangle$ for $x, y \in \mathcal{H}$. In this semi-Hilbertian space setting, we investigate the $A$-numerical radius $w_A(T)$ and $A$-operator semi-norm $\|T\|_A$ of bounded operators $T$. This paper presents significant improvements to existing upper bounds for the $A$-numerical radius of operators in semi-Hilbertian spaces. We establish several new inequalities that provide sharper estimates than those currently available in the literature. Notably, we refine the triangle inequality for the $A$-operator semi-norm, offering more precise characterizations of operator behavior. To validate our theoretical advancements, we provide concrete examples that demonstrate the effectiveness of our results. These examples illustrate how our bounds offer tighter estimates compared to previous ones, confirming the practical relevance of our findings. Our work contributes to the broader understanding of numerical radius inequalities and their applications in operator theory, with potential implications for functional analysis and related fields.