Operator Theory and Free Harmonic Analysis

算子理论与自由谐波分析

• 批准号: 9706557
• 负责人: Hari Bercovici
• 金额: $ 8.5万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-06-01 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706557&HistoricalAwards=false
• 关键词: Operator; Theory; Free; Harmonic; Analysis

Abstract Bercovici Hari Bercovici plans to work in two basic directions. The first is related with Voiculescu's free probability theory and harmonic analysis. An important direction here is the sudy of limit laws for the multiplicative free convolution. Unlike classical harmonic analysis, the study of multiplicative convolution cannot simply be reduced to the additive case by taking logarithms. It is already known that there are some unexpected `Poisson like' probability distributions related with multiplicative free convolution. An important obstacle is the correct formulation of the classical limit theorems, even the central limit theorem, in the free multiplicative case. The second line of Bercovici's research is in operator theory and its interactions with function theory, control theory, and other areas. Among the specific problems in this area are the model theories of restrictions and compressions of operators, and the study of dual algebras generated by commuting families of operators. He also plans to return to the study of skew Toeplitz operators beyond the fairly well understood scalar case. In order to give some motivation for the research in this proposal I would like to recall that in the current description of subatomic processes one has to deal with probabilistic processes. Thus, for instance, the location of a particle is not known with precision at any given moment. One has instead the probability that that particle appears in a given region of space. Some versions of noncommutative probability theory deal with some of the aspects of quantum chemistry for instance. Free probability theory is likely to occur at some point in the quantum description of nature. Besides, it is an interesting object of study because it provides, in some sense, the `only' alternative probability theory. The operator theoretical part of this project stems to a great extent from problems raised by the control theory of large structures. Some earlier results in this theory were proposed a s the main control mechanism of the NASP. While Bercovici's expertise in the practical side of this theory is limited, control theorists can formulate their problems in theoretical terms, thus making the application of operator theoretical methods possible.

抽象的贝尔科维奇 Hari Bercovici计划在两个基本方向上工作。第一个是Voiculescu的自由概率论和谐波分析。本文的一个重要方向是研究乘性自由卷积的极限律。与经典的调和分析不同，乘性卷积的研究不能简单地通过计算简化为加性卷积。众所周知，有一些意想不到的“泊松样”概率分布与乘法自由卷积。一个重要的障碍是经典极限定理的正确表述，甚至是中心极限定理，在自由乘法的情况下。 Bercovici的第二条研究路线是算子理论及其与函数论、控制论和其他领域的相互作用。 在这一领域的具体问题是模型理论的限制和压缩的运营商，并研究对偶代数所产生的交换家庭的运营商。他还计划回到研究斜Toeplitz算子超出了相当好的理解 标量情形 为了给一些动机的研究在这个建议中，我想回顾一下，在目前的描述亚原子过程中，一个必须处理的概率过程。因此，例如，在任何给定时刻，粒子的位置都不是精确的。相反，我们有这个粒子出现在给定空间区域的概率。 非对易概率论的某些版本涉及量子化学的某些方面。 自由概率论很可能出现在对自然的量子描述中的某个时刻。 此外，它是一个有趣的研究对象，因为它提供了，在某种意义上，“唯一的”替代概率理论。 这个项目的算子理论部分在很大程度上源于大型结构的控制理论所提出的问题。该理论的一些早期结果被认为是NASP的主要控制机制。 虽然Bercovici的专业知识在这一理论的实际方面是有限的，控制理论家可以制定他们的问题在理论方面，从而使应用运营商的理论方法成为可能。