Analytic mappings between noncommutative pencil balls
Analytic mappings between noncommutative pencil balls
复制标题
非交换铅笔球之间的解析映射
DOI：
10.1016/j.jmaa.2010.11.040
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发表时间：
2009
影响因子：
1.3
通讯作者：
S. McCullough
中科院分区：
文献类型：
--
作者：
J. Helton;I. Klep;S. McCullough
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 (Helton et al. (2009) [10], de Oliviera et al. (2009) [8]). In the earlier paper (Helton et al. (2009) [9]) we characterized 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”. In this paper we turn to a more general dimension-free ball BL, called a “pencil ball”, associated with a homogeneous linear pencil For [Formula: see text] , define L(X):=∑Aj⊗Xjand let We study the generalization of NC ball maps to these pencil balls BL, and call them “pencil ball maps”. We show that every BLhas a minimal dimensional (in a certain sense) defining pencil L˜. Up to normalization, a pencil ball map is the direct sum of L˜ with an NC analytic map of the pencil ball into the ball. That is, pencil ball maps are simple, in contrast to the classical result of D'Angelo (1993) [7, Chapter 5] showing there is a great variety of such analytic maps from Cgto Cmwhen g≪m. To prove our main theorem, this paper uses the results of our previous paper (Helton et al. (2009) [9]) plus entirely different techniques, namely, those of completely contractive maps. What we do here is a small piece of the bigger puzzle of understanding how Linear Matrix Inequalities (LMIs) behave with respect to noncommutative change of variables.
