Problems in Operator and Matrix Analysis

算子和矩阵分析中的问题

• 批准号: 9988579
• 负责人: Leiba Rodman
• 金额: $ 20.15万
• 依托单位: College of William and Mary
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9988579&HistoricalAwards=false
• 关键词: Problems; Operator; Matrix; Analysis

ABSTRACT:Professors Rodman, Spitkovsky and Woerdeman will continue their study of avariety of problems in operator and matrix analysis and their applicationsin science and engineering. These include: (1) Completion of TriangularOperators, (2) Interpolation, (3) Indefinite Inner Product Spaces, (4)Almost Periodic Factorization, (5) Orthogonal Polynomials of SeveralVariables and Generalizations. For completion problems, invariants andsemiinvariants of the triangular group action will be studied and thenused in the Jordan form and operator spectrum assignment problems. Normalcompletions and positive definite 2D Toeplitz completions will also belooked into. Multipoint interpolation for matrix valued functions will bedeveloped, and further advances in interpolation of matrix and operatorfunctions with symmetries will be obtained. Applications of non-stationaryinterpolation will be explored. Classification of normal operators onindefinite inner product spaces will be developed, and generalized furtherto a spectral theory for sets of commuting J-selfadjoint operators. Studyof polar decomposition and J-spectral factorization will be continued. ThePIs will also continue their research on positive and contractiveextension problems for almost periodic matrix functions in severalvariables, with additional emphasis on the computational aspects. Therelated issues include invertibility and Fredholmness criteria forToeplitz operators with almost periodic matrix symbols, finite sectionmethods for these operators on Besikovitch spaces, and explicitfactorization of almost periodic matrix functions in one and severalvariables. Orthogonal polynomials of several variables, related minimizing polynomials, and their connections with Riemann-Hilbert problems will be investigated with the use of the new band method developed by the PIs recently.The proposed research concerns classical areas of analysis and operatortheory. The choice of topics is both influenced by and aimed toapplications. For example, the expected results in the theory oforthogonal polynomials for several variables and related completionproblems will be used in filter design, compression and analysis ofimages, texture modeling, and multivariate stochastic processes. Classical (Wiener-Hopf) factorization has been used as a powerful tool inintegral equations, partial differential equations and diffraction theory. The PIs will continue their study of its natural generalization to almost periodic matrix valued functions (of one and several variables) whicharises in consideration of integral equations on finite intervals andrelated problems in inverse scattering and other parts of mathematicalphysics. Another example concerns polar decompositions of operators acting on Krein spaces motivated by linear optics of polarized light. Interaction with scientists and engineers is anticipated. In addition, the PIs willalso involve undergraduate students in their research.

摘要：Rodman，Spitkovsky和Woerdeman教授将继续研究操作员和Matrix分析中的问题及其在科学与工程中的应用。 其中包括：（1）完成三角运动剂，（2）插值，（3）无限期的内部产物空间，（4）几乎是周期性分解，（5）几个变量和概括的正交多项式。 对于完成问题，将研究三角形组动作的不变式和emiinvariants，然后以约旦形式和操作员频谱分配问题进行研究。正常的结算和正确定的2D toeplitz完成也将陷入困境。 矩阵值函数的多点插值将被构成障碍，并将获得具有对称性的基质和操作员插值的进一步进步。将探索非平稳插入的应用。将开发正常运算符的分类内部产品空间，并将其概括为通勤J-selfadchoint运算符集的光谱理论。极性分解和J谱分解的研究将继续进行。 ThePI还将继续研究几乎几乎几个变量的矩阵函数的积极和合同延迟问题，并进一步强调了计算方面。 与之相关的问题包括具有几乎周期性矩阵符号的运营商的可逆性和Fredholmness标准，这些操作员在Besikovitch空间上为这些操作员提供有限的部分，以及在一个和几个数字中的几乎周期性矩阵函数的显式效果。通过使用PIS最近开发的新频段方法，将研究几个变量的正交多项式，相关的最小化多项式以及它们与Riemann-Hilbert问题的联系。拟议的研究涉及分析和操作者的经典研究领域。主题的选择既受到针对性的影响，又受到瞄准。例如，对于多个变量和相关完成问题的正交理论中的预期结果将用于滤波器设计，压缩和分析图像，纹理建模和多元随机过程。 经典（Wiener-HOPF）分解已被用作强大的工具不集成方程，部分微分方程和衍射理论。 PI将继续研究其自然概括（一个和几个变量）几乎是周期性矩阵有价值的函数，这些函数在有限的间隔中考虑了积分方程，并在数学物理学的反向散射和其他部分中存在了相关问题。另一个示例涉及作用于凯林空间的操作员的极性分解，该空间是由偏振光线性光学元件动机的。预计将与科学家和工程师的互动。此外，PIS Willalso涉及本科生的研究。