Noncommutative Plurisubharmonic Polynomials Part I: Global Assumptions
Noncommutative Plurisubharmonic Polynomials Part I: Global Assumptions
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非交换多次谐波多项式第一部分：全局假设
DOI：
10.1016/j.jfa.2011.08.006
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
V. Vinnikov
中科院分区：
文献类型：
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作者：
J. M. Greene;J. Helton;V. Vinnikov
我们考虑非对易(NC)自由变量{x1，x2，…中的对称多项式p、xG}。我们将p的NC复Hesse定义为二阶方向导数(用y代替xT)，如果一个NC对称多项式有一个NC复Hesse，且对任意大小n的n×n矩阵的所有元组求值都是半正定的，即对每n个⩾1[公式：见文本]，我们称它为NC多项式。也就是说，NC对称多项式p是NC毛绒的当且仅当它具有和有限且Fj、Kj、F都是NC解析的形式。本文还给出了NC多项式的非对易积分理论，并证明了Frobenius定理的一个非对易形式。随后的一篇论文(J.M.Greene，Preprint[6])证明了：如果p的NC复形q在“NC开集”上取半正定值，则q在所有元组X，H上取半正定值。因此，p具有式中的形式。(0.1)。J.M.Greene(预印本)[6]中的证明借鉴了本文中的大多数定理，以及一种涉及非交换二次函数表示的非常不同的技术。
We consider symmetric polynomials, p, in the noncommutative (nc) free variables {x1,x2,…,xg}. We define the nc complex hessian of p as the second directional derivative (replacing xTby y) We call an nc symmetric polynomial nc plurisubharmonic (nc plush) if it has an nc complex hessian that is positive semidefinite when evaluated on all tuples of n×n matrices for every size n; i.e., for all [Formula: see text] for every n⩾1. In this paper, we classify all symmetric nc plush polynomials as convex polynomials with an nc analytic change of variables; i.e., an nc symmetric polynomial p is nc plush if and only if it has the form where the sums are finite and fj, kj, F are all nc analytic. In this paper, we also present a theory of noncommutative integration for nc polynomials and we prove a noncommutative version of the Frobenius theorem. A subsequent paper (J.M. Greene, preprint [6]), proves that if the nc complex hessian, q, of p takes positive semidefinite values on an “nc open set” then q takes positive semidefinite values on all tuples X, H. Thus, p has the form in Eq. (0.1). The proof, in J.M. Greene (preprint) [6], draws on most of the theorems in this paper together with a very different technique involving representations of noncommutative quadratic functions.