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 a 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.

中文翻译：

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Pre-Hilbert C*-Modules 中的算子等式和正交性表征

在论文的第一部分，我们使用状态$C^{*}$-代数，以建立一些等价的三角不等式，以及前希尔伯特元素的平行四边形恒等式$C^{*}$-模块。我们还为希尔伯特上的可并算子刻画了三角形不等式中的等式情况$C^{*}$-模块。然后我们对前希尔伯特中的两个向量的毕达哥拉斯恒等式给出一定的充要条件$C^{*}$-module 假设它们的内积具有负实部。我们介绍毕达哥拉斯正交性的概念并讨论其性质。我们根据平行四边形定律和一些极限条件来描述希尔伯特空间算子的这个概念。我们举几个例子来说明 Birkhoff-James、Roberts 和 Pythagoras 正交性之间的关系，以及 Hilbert 框架中通常的正交性$C^{*}$-模块。