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.

摘要Bercovici Hari Bercovici计划在两个基本方向上工作。第一个与Voiculescu的自由概率理论和谐波分析有关。这里的一个重要方向是乘法自由卷积的极限定律的sudy。与经典的谐波分析不同，乘法卷积的研究不能简单地通过对数减少到添加剂。众所周知，与乘法自由卷积相关的一些意外的“泊松”概率分布。一个重要的障碍是在自由乘法情况下，正确的经典限制定理，甚至中心限制定理的正确公式。 Bercovici研究的第二行是操作者理论及其与功能理论，控制理论和其他领域的相互作用。 在该领域的具体问题之一是操作员限制和压缩的模型理论，以及对运营商家庭产生的双重代数的研究。他还计划返回对偏斜的Toeplitz运营商的研究，这是广为人知的标量案例。 为了在本提案中为研究提供一些动力，我想回忆说，在当前对亚原子过程的描述中，人们必须处理概率过程。因此，例如，粒子的位置在任何给定时刻都不是精确的。相反，该粒子出现在给定的空间区域中的可能性。 例如，某些版本的非公共概率理论涉及量子化学的某些方面。 自由概率理论可能会发生在自然量子描述中的某个时刻。 此外，这是一个有趣的研究对象，因为它在某种意义上提供了“唯一”替代概率理论。 该项目的经营者理论部分在很大程度上源于大型结构控制理论所提出的问题。提出了该理论中的一些早期结果，这是NASP的主要控制机理。 虽然Bercovici在该理论的实际方面的专业知识是有限的，但控制理论家可以从理论上提出问题，从而使操作员的理论方法的应用成为可能。