Operator Algebras, Operator Theory and Applications

摘要拉尔森这个项目有三个重点领域。第一部分是关于非自伴算子代数结构理论的相关问题。这位研究者先前在von Neumann代数中的拟三角考虑方面的工作已经影响了该领域以外的其他人在应用控制理论和函数论方面的一些工作。他将继续这一研究方向。第二个正在进行的项目涉及一种新的泛函分析方法，以解决小波理论中的一些基本问题。PI证明了某些经典的正交化小波可以用算符-代数方法得到，他还证明了在所有维度大于1的单函数二进正交化小波的存在性。这让其他一些研究人员感到惊讶，因为它与一个小波民间传说相矛盾，后者表明这样的小波是不可能的。他还证明了von Neumann代数的酉群在某些情况下可以用来参数化范数路径连通的小波族。这暗示了微扰技术的可能性。第三个推力扩展了这位研究者和合作者以前的工作，其中得到了关于算子-代数自反性和单算子的相关性质的一些古老的公开问题的反例。在之前支持的工作中，这位研究人员与合作者一起解决了大约十年前由另一位研究人员提出的关于非自伴算子代数领域的一个猜想。这导致了控制理论应用领域的一项意想不到的发展，影响了数学以外的其他人的工作。他将继续从事这方面的研究。在第二个正在进行的方向上，这位研究人员和一名前学生已经证明，小波理论的某些方面可以服从于以前未曾怀疑的性质的算子代数计算。在过去的几年里，小波分析和面向小波的技术在数学和工程领域引起了巨大的研究热潮。主要应用领域是信号处理和数据压缩。他们证明了某些经典小波可以用他们的技术推导出来，他们还证明了某些高维小波的存在，这在以前被许多专家认为是不可能的。在第三个方向上，PI最近扩展了一些先前的工作，其中反例被获得到一些关于单算子性质的旧的公开问题。几名博士生参与了所有这项工作。

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.