Mathematical Sciences: Topics in Operator Theory

小行星9401848 该项目涉及算子理论和矩阵理论的工作。五个具体项目概述：1。具有一般Hunt-Muckenhoupt-Wheeden权空间上的Toeplitz、Weiner-Hopf和伪微分算子的Fredholm理论。矩阵函数的概周期分解及其在分析中的延拓和插值问题中的应用。3.衍射理论、弹性力学和热力学应用中出现的积分方程和边界问题的显式解。4.进一步分析了由两个投影生成的Banach代数的结构。5.厄米特矩阵对的矛盾标准形及其应用。 该项目围绕奇异积分方程系统的理论。从历史上看，这些方程是从翼型方程的研究和材料力学中的其他应用中发展出来的。求解这些奇异积分方程的基本工具是Wiener-Hopf分解法。这种分解的矩阵形式由电气工程师独立开发，用于研究线性系统理论。该项目的一部分涉及显式因式分解，这将在工程问题中得到应用。完成后，该项目的理论部分将是矩阵函数分解理论的重大进展。 ***

9401848 Spitkovsky The project involves work on operator theory and matrix theory. Five specific projects are outlined: 1. Fredholm theory of Toeplitz, Weiner-Hopf and pseudodifferential operators on spaces with general Hunt-Muckenhoupt-Wheeden weights. Almost periodic factorization of matrix functions and its applications to extension and interpolation problems of Analysis. 3. Explicit solutions of integral equations and boundary problems emerging from applications to diffraction theory, elasticity and thermodynamics. 4. Further analysis of the structure of Banach algebras generated by two projections. 5. Contragredient canonical forms of pairs of Hermitian matrices and their applications. The project centers around the theory of systems of singular integral equations. Historically, these equations grew out of the study of the equation of the airfoil and other applications in the mechanics of materials. The basic tool for solving these singular integral equations is the method of Wiener-Hopf factorization. The matrix form of this factorization was developed independently by electrical engineers to study linear systems theory. A part of the project deals with explicit factorization and this would have application in engineering problems. When completed, the theoretical portion of the project will be a significant advance in the theory of matrix function factorization. ***