Operator Algebras, Operator Theory and Applications

算子代数、算子理论与应用

• 批准号: 9706810
• 负责人: David Larson
• 金额: $ 16.11万
• 依托单位: Texas A&M Research Foundation
• 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: 1997-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706810&HistoricalAwards=false
• 关键词: Operator; Algebras; Theory; Applications

Abstract Larson This project has three areas of emphasis. The first focuses on problems related to the structural theory of non-selfadjoint operator algebras. Prior work by the investigator on quasitriangularity considerations within von Neumann algebras has impacted some work of others outside the area in applied control theory and in function theory. He will continue this direction of research. The second ongoing project concerns a new functional- analytic approach to some basic issues in wavelet theory. The PI has shown that certain classical orthonormal wavelets could be derived using operator-algebraic techniques, and he also proved the existence of single-function dyadic orthonormal wavelets in all dimensions greater than one. This was a surprise to some other researchers because it contradicted a wavelet folklore which indicated that such wavelets were impossible. He has also shown that the unitary group of a von Neumann algebra can in some cases be used to parameterize a norm-path-connected family of wavelets. This suggests the possibility of perturbation techniques. The third thrust extends prior work of this investigator and collaborators in which counterexamples were obtained to some old open problems concerning operator-algebraic reflexivity and related properties of single operators. In prior supported work this investigator solved, with collaborators, a conjecture in the area of non-selfadjoint operator algebras that had been posed about ten years earlier by another researcher. This led to an unsuspected development in the applied area of control theory which impacted work of others outside of mathematics. He will continue pursuing this line of research. In a second ongoing direction, this investigator and a former student have shown that some aspects of wavelet theory are amenable to operator algebraic computations of a nature that were previously unsuspected. Wavelet analysis and wavelet oriented technology has been the scene of a tremendous research drive in mathematics and engineering during the past few years. Primary applications have been to signal processing and data compression. They showed that certain classical wavelets could be derived using their techniques, and they also proved the existence of certain wavelets in higher dimensions which were previously thought to be impossible by many specialists. In a third direction the PI has recently extended some prior work in which counterexamples were obtained to some old open problems concerning properties of single operators. Several doctoral students have been involved in all of this work.

抽象拉森 该项目有三个重点领域。 第一部分主要讨论非自伴算子代数的结构理论。 以前的工作调查员对拟三角考虑冯诺依曼代数影响了一些工作的其他领域以外的应用控制理论和函数理论。 他将继续这一研究方向。 第二个正在进行的项目是关于小波理论中一些基本问题的新的泛函分析方法。 PI证明了某些经典的正交小波可以使用算子代数技术导出，他还证明了在大于1的所有维度上存在单函数并元正交小波。 这对其他一些研究人员来说是一个惊喜，因为它与小波民间传说相矛盾，该民间传说表明这种小波是不可能的。 他还表明，酉群的冯诺依曼代数可以在某些情况下被用来parameterising一个范数路径连接家庭的小波。 这表明微扰技术的可能性。 第三个推力扩展了以前的工作，这个调查员和合作者，其中获得了反例，一些老的开放问题，算子代数自反性和相关性能的单一运营商。 在先前的支持工作中，这位研究者与合作者一起解决了一个关于非自伴算子代数的猜想，这个猜想在十年前由另一位研究者提出。 这导致了一个意想不到的发展应用领域的控制理论，影响工作的其他以外的数学。 他将继续从事这方面的研究。 在第二个正在进行的方向，这个调查员和以前的学生已经表明，小波理论的某些方面是服从算子代数计算的性质，以前没有怀疑。 小波分析和面向小波的技术在过去的几年里一直是数学和工程领域的一个巨大的研究驱动力。 主要应用于信号处理和数据压缩。 他们表明，某些经典小波可以得出使用他们的技术，他们还证明了存在某些小波在更高的维度这是以前认为是不可能的许多专家。 在第三个方向的PI最近扩展了一些以前的工作中，反例获得了一些旧的开放问题有关的性质单一的运营商。 几位博士生参与了所有这些工作。