论文标题
之前的 - 希尔伯特$ c^*$ - 模块的正交性的操作员平等和表征
Operator equalities and Characterizations of Orthogonality in Pre-Hilbert $C^*$-Modules
论文作者
论文摘要
在本文的第一部分中，我们在$ c^*$ - 代数上使用状态，以建立一些与三角形不平等相等的等效陈述，以及用于hilbert $ c^*$模块元素的平行四边形身份。我们还表征了三角形不平等中的平等情况，用于Hilbert $ c^*$ - 模块上的可相邻操作员。然后，我们为毕达哥拉斯的身份提供了一些必要和充分的条件，用于在hilbert $ c^*$ - 模块中的两个向量，假设其内部产物具有负实际部分。我们介绍了毕达哥拉斯正交性的概念，并讨论其特性。我们在平行四边形定律和某些极限条件下为希尔伯特太空运营商描述了这一概念。我们提出了几个示例，以说明Birkhoff - James，Roberts和Pythagoras正交之间的关系，以及Hilbert $ C^*$模块框架中通常的正交性。
In the first part of the paper, we use states on $C^*$-algebras in order to establish some equivalent statements to equality in the triangle inequality, as well as to the parallelogram identity for elements of a pre-Hilbert $C^*$-module. We also characterize the equality case in the triangle inequality for adjointable operators on a Hilbert $C^*$-module. Then we give certain necessary and sufficient conditions to the Pythagoras identity for two vectors in a pre-Hilbert $C^*$-module under the assumption that their inner product has negative real part. We introduce the concept of Pythagoras orthogonality and discuss its properties. We describe this notion for Hilbert space operators in terms of the parallelogram law and some limit conditions. We present several examples in order to illustrate the relationship between the Birkhoff--James, Roberts, and Pythagoras orthogonalities, and the usual orthogonality in the framework of Hilbert $C^*$-modules.
