Mathematical Sciences: Operator Theory and Noncommutative Harmonic Analysis

9401380贝尔科维奇拟议的工作有三个方向。第一个是基本算符理论。对偶代数理论将应用于交换算子对的研究。这些技术在单一操作员的情况下最为成功。第二个方向是与结构奇异值概念相关的内插和提升问题的研究。这些问题是算子论中有趣的问题，但它们也适用于控制理论。第三个方向是在沃库列斯库自由简谐分析框架内早期工作的发展。这一分析中的卷积理论仍有许多方面没有得到很好的理解。例如，中心极限定理中的半圆分布的吸引域是未知的。关于大偏差的问题还没有在这方面进行研究。这个项目是几个数学分支(算子论、概率论、函数论)的交汇点。该项目在很大程度上涉及对这些学科的更好理解和新成果。此外，该项目的某些部分与控制理论和系统理论有很大的相互作用，其结果可能会对与大系统控制有关的计算产生影响。***

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. ***