Mathematical Sciences: Operator Theory and Noncommutative Harmonic Analysis

小行星9401380 拟议的工作有三个方向。首先是基本算子理论。对偶代数的理论将被应用于研究交换算子对。这些技术在单个操作者的情况下最成功。第二个方向是与结构奇异值概念相关的插值和提升问题的研究。这些问题作为算子理论中的问题是有趣的，但它们也有控制理论的应用。第三个方向是一个发展的早期工作的框架Voiculescu的自由谐波分析。在这个分析中的卷积理论仍然有许多方面没有很好地理解。例如，在中心极限定理中，吸引到圆分布的域是未知的。在这方面尚未研究大偏差问题。 这个项目在于数学的几个分支（算子理论，概率论，函数论）的接口。该项目的大部分内容涉及对这些学科的更好理解和新成果。此外，该项目的部分内容与控制理论和系统理论有着重要的相互作用，其结果可能会对与大型系统控制相关的计算产生影响。 ***

9401380 Bercovici The proposed work is in three directions. The first is in basic operator theory. The theory of dual algebras will be applied to study pairs of commuting operators. These techniques were most successful in the case of single operators. The second direction is related to the study of interpolation and lifting problems associated with the concept of structured singular value. These issues are interesting as problems in operator theory but they also have applications for control theory. The third direction is a development of earlier work in the framework of Voiculescu's free harmonic analysis. The convolution theory in this analysis still has many aspects which are not well understood. For instance, the domain of attraction to the semicircle distribution in the central limit theorem is not known. Problems on large deviations have not been studied in this context. This project lies at the interface of several branches of mathematics (operator theory, probability, function theory). Much of the project involves a better understanding and new results in these disciplines. Further, parts of the project have a significant interaction with control theory and systems theory, and the results might have an impact on calculations related to the control of large systems. ***