Mathematical Sciences: Abstract Spectral Decompositions and Transference Methods in Modern Analysis

Berkson教授进行了界面研究， 调和分析、遍历理论和算子理论。 所有 三个是现代分析的重要分支学科。 谐波 分析，粗略地说，体现了相互关系， 数学分析和群体行动。 历史上，Harmonic 分析起源于傅立叶级数和部分 微分方程 遍历理论的相关领域 起源于统计力学。 它的推力是在 理解和利用递归的概念。 遍历理论和调和分析，贯穿所有的 数学分析，利用方法和操作， 易受广泛的抽象原则的影响。 现代运营商 理论，它产生于数学的发展， 量子物理学的基础，旨在识别和解释 这些抽象的原则，以统一和扩大边界 科学。 本提案的主题是运营商 理论在这个概念意义上，强调应用， 遍历理论和调和分析。 伯克森教授建议继续他的发展， 抽象算子理论，包含弱形式的 正交性（如傅立叶逆和三角 非交换分析中的截断）。 该方法是基于 基于三角良有界算子和 光谱家庭的投影，这是用在一般 分析的设置，以取代，分别， 希尔伯特空间中心的算子和投影值测度 算子理论及其物理应用 于过往 合作研究，Berkson教授开发了这些 概念，沿着广泛应用于交换和 非交换分析 目前的调查将扩大 沿着沿着这些路线， 它的同伴向量值迁移方法，同时扩展 它的范围，以这样的主题，表示局部紧 交换群，傅立叶乘子转移，遍历理论， 广义解析性。

Professor Berkson conducts research on the interface of harmonic analysis, ergodic theory, and operator theory. All three are important subdisciplines of Modern Analysis. Harmonic analysis, roughly speaking, embodies the interrelationships of mathematical analysis and group actions. Historically, harmonic analysis has its origins in Fourier series and partial differential equations. The related field of ergodic theory originated in statistical mechanics. Its thrust is in the understanding and exploitation of the concept of recurrence. Both ergodic theory and harmonic analysis, as throughout all of mathematical analysis, utilize methods and operations that are susceptible to broad abstract principles. Modern operator theory, which arose in the development of the mathematical foundation of quantum physics, seeks to identify and explicate these abstract principles so as to unify and expand the frontiers of science. The subject matter of this proposal is operator theory in this conceptual sense, with emphasis on applications to ergodic theory and harmonic analysis. Professor Berkson proposes to continue his development of an abstract operator theory that encompasses weakened forms of orthogonality (such as Fourier inversion and triangular truncation in noncommutative analysis). The approach is based upon the notions of trigonometrically well-bounded operator and spectral family of projections which are used in the general settings of analysis to replace, respectively, the unitary operators and projection-valued measures central to Hilbert space operator theory and its physical applications. In prior collaborative research, Professor Berkson has developed these notions, along with extensive applications to commutative and noncommutative analysis. The current investigations will expand and sharpen the abstract operator theory along these lines and its companion vector-valued transference methods, while extending its scope to such topics as representations of locally compact abelian groups, Fourier multiplier transference, ergodic theory, and generalized analyticity.