标题： 关于$A$-正交性保持和半希尔伯特空间中的布兰科-科尔多夫斯基-图恩谢克定理 显示英文标题

标题： On $A$-orthogonality preservation and Blanco-Koldobsky-Turnšek theorem in semi-Hilbert spaces

摘要： 我们研究在由希尔伯特空间$\mathbb{H}.$上的正算子$A$引入的半希尔伯特框架内，$A$有界算子在一点处对$A$正交性的局部保持。我们提供了这种保持的完整特征。此外，我们根据$A$正交性保持探讨了$A$有界算子的$A$范数达到集的性质。 我们还研究了 $A$-有界算子的最小 $A$-范数达到集的类似性质。 然后，我们将 $A$-等距算子表征为保持 $A$-正交性的 $A$-范数一算子。 最后，我们表征了希尔伯特空间中的那些子集，使得由一个 $A$-范数一算子的这种保持性可以推出该算子是一个 $A$-等距算子。 显示英文摘要

摘要： We investigate the local preservation of $A$-orthogonality at a point by $A$-bounded operators within the semi-Hilbertian framework induced by a positive operator $A$ on a Hilbert space $\mathbb{H}.$ We provide complete characterizations of such preservation. Additionally, we explore properties of the $A$-norm attainment set of an $A$-bounded operator in light of $A$-orthogonality preservation. We also study analogous properties for the minimum $A$-norm attainment set of an $A$-bounded operator. We then characterize the $A$-isometries as the $A$-norm one operators preserving $A$-orthogonality. Finally, we characterize those subsets of Hilbert spaces for which such preservation by an $A$-norm one operator implies that the operator is an $A$-isometry.