Mathematical Sciences: Operator Algebras and Operator Theory

[401544] Larson研究计划的一个领域涉及冯·诺伊曼代数中准三角形考虑的先前工作。这位研究者和他的同事在准三角性方面的工作导致了一个次优化公式的发展，这个公式影响了最近在控制理论方面的其他人的工作，并应用于电气和航空工程。我们的计划是继续朝这个方向努力。该计划的第二个领域涉及与三角形和半三角形算子有关的问题，以及反身性，这些问题产生于本研究者和同事之前的工作，这些工作产生了反例，以解决这些主题的开放性问题。该计划的第三个领域涉及正交小波理论某些方面的算子理论方法，通过将正交小波解释为希尔伯特空间上一组特殊的酉算子的流浪向量而得到。采用的新思想是，实线上平方可积复函数希尔伯特空间上常用的膨胀平移小波系统（D，T）的所有正交小波的集合可以用单个小波v和系统在x处的局部交换子上的所有酉算子的集合自然地参数化。对这个酉集的分析得到关于所有正交小波集合W（D，T）的信息。研究计划包括三个独立但相关的项目。第一个是建立在这位研究者之前在算子代数领域的工作基础上的。三年前为解决纯数学中的一个开放问题而导出的一个公式最近影响了一些从事控制理论工作的数学家和工程师的面向应用的研究项目。正在进行这方面的进一步工作。第二个项目继续先前关于单算子和对偶算子代数的性质的工作。这里最近的成功包括反例，这些反例影响了其他几位数学家的工作。第三个项目涉及小波。在过去的几年中，小波分析在数学和工程领域已经成为一个相当大的研究领域。本项目采用的新概念是利用单算子理论和算子代数的现有技术，通过分析某一自然相关算子集的结构性质来确定新的小波和小波的泛函性质。本总体方案可供4名博士生进行论文研究。＊＊＊

9401544 Larson One area of the research plan concerns prior work on quasitriangularity considerations within von Neumann algebras. Work by this investigator and co-workers on quasitriangularity led to the development of a sub-optimization formula which has impacted recent work by others in control theory with applications to electrical and aeronautical engineering. The plan is to continue work in this direction. A second area of the plan concerns problems related to triangular and semitriangular operators, and to reflexivity, that arose out of prior work by this investigator and co-workers which yielded counterexamples to open questions on these topics. The third area of the plan concerns an operator-theoretic approach to certain aspects of orthogonal wavelet theory obtained by interpreting orthogonal wavelets as wandering vectors for a special set of unitary operators on Hilbert space. The new idea used is that the set of all orthogonal wavelets for the usual dilation-translation wavelet system (D,T) on the Hilbert space of square- integrable complex functions on the real line can be parameterized in a natural way by a single wavelet v together with the set of all unitary operators in the local commutant of the system at x. Analysis of this unitary set yields information concerning the set W(D,T) of all orthogonal wavelets. The research plan involves three separate but related projects. The first builds onto previous work by this investigator in the area of operator algebras. A formula which was derived three years ago to solve an open problem in pure mathematics has recently impacted applications-oriented research projects of several mathematicians and engineers working incontrol theory. Further work in this connection is being pursued. The second project continues previous work on properties of single operators and dual operator algebras. Recent successes here include counter examples which have impacted work of several other mathemati cians. The third project concerns wavelets. Wavelet analysis has been the scene of a sizable research drive in mathematics and engineering during the past few years. The new concept utilized in this project is the idea of determining new wavelets and functional properties of wavelets by analyzing structural properties of a certain naturally related set of operators, using established techniques of single operator theory and operator algebras. This overall plan accomodates the dissertation research of four doctoral students. ***