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：罗德曼、斯皮特科夫斯基和沃尔德曼教授将继续研究算子和矩阵分析中的各种问题及其在科学和工程中的应用。 这些包括：(1) 三角算子的完成，(2) 插值，(3) 不定内积空间，(4) 几乎周期因式分解，(5) 多个变量的正交多项式和泛化。 对于完成问题，将研究三角群作用的不变量和半不变量，然后将其用于乔丹形式和算子谱分配问题。还将研究正常补全和正定 2D Toeplitz 补全。 将开发矩阵值函数的多点插值，并在具有对称性的矩阵和算子函数插值方面取得进一步进展。将探索非平稳插值的应用。将开发不定内积空间上的正规算子的分类，并进一步推广到可交换 J-自伴随算子集的谱理论。极性分解和J谱分解的研究将继续进行。 PI 还将继续研究几个变量中的几乎周期矩阵函数的正扩张问题和收缩扩张问题，并特别强调计算方面。 相关问题包括具有几乎周期矩阵符号的托普利茨算子的可逆性和 Fredholmness 准则、Besikovitch 空间上这些算子的有限截面方法以及一个或多个变量中的几乎周期矩阵函数的显式分解。多变量的正交多项式、相关的最小化多项式以及它们与黎曼-希尔伯特问题的联系将利用PI最近开发的新频带方法进行研究。所提出的研究涉及分析和算子理论的经典领域。主题的选择既受应用的影响，又是针对应用的。例如，多变量正交多项式理论和相关完成问题的预期结果将用于滤波器设计、图像压缩和分析、纹理建模和多元随机过程。 经典（维纳-霍普夫）分解已被用作积分方程、偏微分方程和衍射理论中的强大工具。 PI 将继续研究其对几乎周期矩阵值函数（一个或多个变量）的自然推广，该函数是在考虑有限区间积分方程以及逆散射和数学物理学其他部分的相关问题时产生的。另一个例子涉及由偏振光线性光学驱动的作用于 Kerin 空间的算子的极分解。预计将与科学家和工程师进行互动。此外，PI 还将让本科生参与他们的研究。