Noncommutative ball maps
Noncommutative ball maps
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非交换球映射
DOI：
10.1016/j.jfa.2009.03.008
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发表时间：
2008
影响因子：
1.7
通讯作者：
Helton
中科院分区：
文献类型：
--
作者：
J. Helton;I. Klep;S. McCullough;Nicholas Slinglend;Helton
在本文中，我们分析的问题，涉及矩阵变量，我们使用非交换代数设置。更具体地说，我们使用一类函数（称为NC解析函数）定义的幂级数在noncommuting变量和评估这些功能的所有维度的矩阵集，我们称这种情况为无量纲。这些类型的函数最近被用于无量纲线性系统工程问题的研究。在本文中，我们刻画了NC解析映射，发送无量纲矩阵球的无量纲矩阵球和进行边界的边界，这样的地图，我们称之为“NC球映射”。我们发现，到正规化，NC球映射是直接和的单位映射与NC解析映射的球到球。也就是说，“NC球映射”是非常简单的，与D 'Angelo在C.另一类数学上自然的映射带有一个非对易可分辨边界的变体，但我们的结果是有限的。我们将感兴趣的几种类型的非交换球，传统的，但也球定义的约束称为线性矩阵不等式（LMI）。我们在这里所做的只是理解线性矩阵不等式在变量的非对易变化时的表现这一更大难题的一小部分。
In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems. In this paper we characterize NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call “NC ball maps”. We find that up to normalization, an NC ball map is the direct sum of the identity map with an NC analytic map of the ball into the ball. That is, “NC ball maps” are very simple, in contrast to the classical result of D'Angelo on such analytic maps in C. Another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary, but on these our results are limited. We shall be interested in several types of noncommutative balls, conventional ones, but also balls defined by constraints called Linear Matrix Inequalities (LMI). What we do here is a small piece of the bigger puzzle of understanding how LMIs behave with respect to noncommutative change of variables.