论文标题
关于“非交通分布”的讲义
Lecture Notes on "Non-Commutative Distributions"
论文作者
论文摘要
这在非交通运算符或随机矩阵的非交通分布理论的介绍中。从基本问题开始，找到一种良好的方法来解决“非交易分布”的含义，尤其将涵盖：免费分析，这是几个非公认变量的复杂分析的一种版本；自由概率理论的操作员值（组合和分析方面）；将非线性标量问题减少到线性操作员值问题的线性化技巧；运算符值卷积和线性化的组合，以计算自由变量中多项式的分布；非交通理性函数的基本理论。一方面，这是关于“自由概率”的讲义的延续，Arxiv：1908.08125。另一方面，自由概率的理论再次发展，但在更一般的操作员值得情况下。因此，原则上并采取了一些额外的努力，应该可以阅读本注释，而无需对自由概率的先验知识。材料的大部分也不是很大的自由变量，而是最大程度地涉及最大非交换变量的分析和代数方面。这里的材料于2019年夏季学期在萨尔兰大学（Saarland University）介绍了20分钟的20分钟讲座。记录了讲座，可以在https://www.math.uni-sb.de/ag/speicher/web_video/index.html上在线找到。
This in an introduction to the theory of non-commutative distributions of non-commuting operators or random matrices. Starting from the basic problem to find a good approach to the meaning of "non-commutative distribution" we will, in particular, cover: free analysis, which is a version of complex analysis for several non-commuting variables; the operator-valued version of free probability theory (combinatorial but also analytic aspects); the linearization trick to reduce non-linear scalar problems to linear operator-valued problems; the combination of operator-valued convolution and linearization to calculate the distribution of polynomials in free variables; the basic theory of non-commutative rational functions. On one hand, this is a continuation of the Lecture Notes on "Free Probability", arXiv:1908.08125. On the other hand, the theory of free probability is developed again, but in a more general, operator-valued context. So, in principle and with some additional efforts, it should be possible to read the present notes without having a prior knowledge on free probability. Big parts of the material do also not deal so much with free variables, but more general with analytic and algebraic aspects of maximal non-commuting variables. The material here was presented in the summer term 2019 at Saarland University in 20 lectures of 90 minutes each. The lectures were recorded and can be found online at https://www.math.uni-sb.de/ag/speicher/web_video/index.html.
