标题： 半希尔伯特空间上有界线性算子空间中的光滑性 显示英文标题

标题： Smoothness in the space of bounded linear operators on semi-Hilbert space

摘要： 给定一个Hilbert空间$\mathbb{H}$上的非零正算子$A$，自然地在$\mathbb{H}$上诱导出一个半内积。 在这项工作中，我们引入了在所得半Hilbert空间上有界线性算子的\emph{$A$光滑性}概念，并研究了其各种性质。 我们提供了对$A$-光滑性的全面表征，针对$A$-有界算子，并进一步分析了$A$-光滑性，根据其$A$-范数取得集对$A$-紧算子进行分析。 利用这些特征，我们证明了在$A$有界算子处半范数$\|\cdot\|_A$的 Gateaux 可微性等价于其$A$光滑性。 此外，我们表征了$2\times 2$块对角矩阵的$A$光滑性。 显示英文摘要

摘要： Given a nonzero positive operator $A$ on a Hilbert space $\mathbb{H}$, a semi-inner product is naturally induced on $\mathbb{H}$. In this work, we introduce the notion of \emph{$A$-smoothness} for bounded linear operators on the resulting semi-Hilbert space and investigate its various properties. We provide a comprehensive characterization of the $A$-smoothness for $A$-bounded operators and further analyze the $A$-smoothness of $A$-compact operators in terms of their $A$-norm attainment sets. Utilizing these characterizations, we establish that G\^{a}teaux differentiability of the semi-norm $\|\cdot\|_A$ at an $A$-bounded operator is equivalent to its $A$-smoothness. Furthermore, we characterize the $A$-smoothness of $2\times 2$ block diagonal matrices.