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.
In an earlier paper 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 B_L, called a "pencil ball", associated with a homogeneous linear pencil L(x):= A_1 x_1 + ... + A_m x_m, where A_j are complex matrices. For an m-tuple X of square matrices of the same size, define L(X):=\sum A_j \otimes X_j and let B_L denote the set of all such tuples X satisfying ||L(X)||<1.
We study the generalization of NC ball maps to these pencil balls B_L, and call them "pencil ball maps". We show that every B_L has 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 on such analytic maps in C^m. To prove our main theorem, this paper uses the results of our previous paper mentioned above plus entirely different techniques, namely, those of completely contractive maps.
给定g元幺一线性束L，设D_L为其正性域，即使L（X）半正定的所有大小的对称矩阵的所有g元组X的集合。由于L是幺一线性束，所以DL是具有内部的凸集，反之，具有内部的凸有界非交换半代数集都是DL形式的。本文的主要结果建立了凸半代数集上的完备非交换Nichtnegativstellenetrix。也就是说，一个非交换多项式p在D_L上是半正定的当且仅当它有一个带最优次数界的加权平方和表示：p = s^* s + \sum_j f_j^* L f_j，其中s，f_j是次数不大于大于1/2 deg（p）的非交换多项式的向量。这个非对易的结果与对易的情况形成了鲜明的对比，在对易的情况下，对s，f_j的度数没有控制，并且假设只有p是非负的，而不是p严格为正，产生一个干净的Positivstellenance是如此罕见，以至于这种情况是值得注意的。
本文得到了如下算子值多元Bohr型不等式：一类非交换全纯函数，推广了开单位圆盘上的解析函数;非交换圆盘代数和非交换解析Toeplitz代数;一类非交换自伴调和函数，推广了开单位圆盘上的实值调和函数;具有n个生成元的自由群的约化（或全）群C* 代数; C^n的开单位球上的一类解析函数。证明了经典的Bohr不等式是正三角多项式系数的Fejer不等式和幂零算子幂零算子的Haagerup-de la Harpe不等式的一个推论.此外，我们提供了一个不等式，它是开放的单位圆盘上的解析多项式，比玻尔的不等式更尖锐。
Operator-Valued Positivstellensätze on Matrix Convex Sets and Free Products of Finite Abelian Groups
University of Ljubljana（卢布尔雅那大学）；Famnit, University of Primorska（波摩拉大学 FAMNIT）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；University of Florida（佛罗里达大学）
