投影理论的后续:定理与群体作用

Jean-Francois Niglio
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引用次数: 1

摘要

在本文中,我们希望扩展从第一篇题为“投影理论”的文章中获得的一些结果。我们已经证明了单参数投影算子可以由单位圆构造。正如在前一篇文章中所讨论的,这些算子形成一个李群,称为投影群。在第一部分中,我们将展示我第一篇文章中的概念与现有理论[1][2]是一致的。在第二节中,我们将证明这两个算子不仅是互同的,而且我们还可以利用旋转群[3][4]来定义一个群作用。证明了群以一种非忠实但∞可传递的方式作用于的元素,这种方式与两种群运算一致。最后,在最后一节中,我们利用算子和Hadamard积在矩阵运算中定义了群运算;这种构造与第一篇文章中定义的组操作一致。
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A Follow-Up on Projection Theory: Theorems and Group Action
In this article, we wish to expand on some of the results obtained from the first article entitled Projection Theory. We have already established that one-parameter projection operators can be constructed from the unit circle . As discussed in the previous article these operators form a Lie group known as the Projection Group. In the first section, we will show that the concepts from my first article are consistent with existing theory [1] [2]. In the second section, it will be demonstrated that not only such operators are mutually congruent but also we can define a group action on  by using the rotation group [3] [4]. It will be proved that the group acts on elements of  in a non-faithful but ∞-transitive way consistent with both group operations. Finally, in the last section we define the group operation  in terms of matrix operations using the operator and the Hadamard Product; this construction is consistent with the group operation defined in the first article.
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