Operators related to subordination for free multiplicative convolutions
Operators related to subordination for free multiplicative convolutions
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与自由乘法卷积的从属相关的运算符
DOI：
10.1512/iumj.2008.57.3228
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发表时间：
2007
影响因子：
1.1
通讯作者：
R. Lenczewski
中科院分区：
文献类型：
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作者：
R. Lenczewski
Voiculescu 和 Biane 已经证明，解析从属性质对于自由加法和乘法卷积成立。在本文中，我们提出了一种自由乘法卷积的从属操作方法。这项研究基于最近在自由加性卷积的背景下引入的“自由与从属”或“自由独立”和“正交独立”的概念。特别是，我们介绍和研究了相关的乘法卷积并构造了相关的算子，称为“从属算子”和“从属分支”。利用正交独立性，我们推导了从属分支的分解以及 s-free 和 free 乘法卷积的相关分解。这些操作方法产生了几种新类型的图产品，称为“循环产品”，与不同的独立概念（单调、布尔、正交、s-free）相关。我们还证明了图的循环乘积和图的自由乘积上的有根“交替双返回游走”的枚举给出了相应的乘法卷积的矩。
It has been shown by Voiculescu and Biane that the analytic subordina- tion property holds for free additive and multiplicative convolutions. In this paper, we present an operatorial approach to subordination for free multiplicative convolutions. This study is based on the concepts of 'freeness with subordination', or 's-free inde- pendence', and 'orthogonal independence', introduced recently in the context of free additive convolutions. In particular, we introduce and study the associated multiplica- tive convolutions and construct related operators, called 'subordination operators' and 'subordination branches'. Using orthogonal independence, we derive decompositions of subordination branches and related decompositions of s-free and free multiplicative convolutions. The operatorial methods lead to several new types of graph products, called 'loop products', associated with different notions of independence (monotone, boolean, orthogonal, s-free). We also prove that the enumeration of rooted 'alternat- ing double return walks' on the loop products of graphs and on the free product of graphs gives the moments of the corresponding multiplicative convolutions.