Mathematical Sciences: RUI: Research on Operators in Hilbert Space

9400566 Fialkow这个项目涉及到Hilbert空间上算子的几个项目。这些问题涉及以下主题：联合次异常，特别是加权移位的k-次异常和弱k-次异常；矩理论中的相关问题，包括复截矩问题；C*-代数理论中的多数化、因子分解和可逆因子定理。本研究的一个方面涉及亚正规，k-次正规和弱k-次正规算子的结构和计算模型，重点是单边移位。本文还研究了多维幂矩问题，包括截断复矩问题和正矩矩阵的扩展问题。本研究的另一个方面涉及作用于素数C*-代数上的初等算子的谱图的描述。本研究还涉及方程a = bx在C*-代数中的溶解度，包括x可逆的溶解度的情况。算子理论是线性代数和矩阵理论的无限维版本。线性代数的一个重要应用是求解有限个线性方程组。求解算子方程的问题可以看作是试图求解一个包含大量或无限数量方程的系统。这个项目的一部分涉及求解算子方程。项目的其他部分包括使用算子理论来解决经典矩问题。这里的基本问题是从不同的质量矩中确定一个度量或质量分布。值得注意的是，本研究将在本科院校进行，并将对数学教育产生重大影响。* * *

9400566 Fialkow The project involves work on several projects concerning operators on Hilbert spaces. These problems involve the following topics: joint hyponormality, especially k-hyponormality and weak k-hyponormality for weighted shifts; related problems in the theory of moments, including the complex truncated moment problem; majorization, factorization and invertible factor theorems in the theory of C*-algebras. One aspect of this research concerns structural and computational models for subnormal, k-hyponormal, and weakly k-hyponormal operators, with emphasis on unilateral shifts. This study also concerns multidimensional power moment problems, including the truncated complex moment problem and the extension problem for positive moment matrices. Another facet of this research concerns the description of the spectral picture of an elementary operator acting on a prime C*-algebra. This research also concerns solubility of the equation a = bx in a C*-algebra, including the case of solubility with x invertible. Operator theory is an infinite dimensional version of linear algebra and matrix theory. One of the important applications of linear algebra is in solving a system of a finite number of linear equations. The problem of solving operator equations can be considered as an attempt to solve a system involving a very large or infinite number of equations. Part of this project involves solving operator equations. Other portions of the project involves using operator theory to solve classical moment problems. The basic problem here is to determine a measure or mass distribution from various moments of mass. It is important to note that this research will be conducted at an undergraduate institution and will have significant impact in mathematics education. ***