Problems in Operator and Matrix Analysis

摘要：Rodman、Spitkovsky和Woerdeman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。它们包括：(1)三角算子的完备化，(2)内插，(3)不定内积空间，(4)概周期因式分解，(5)数列变量的正交多项式及其推广。对于完备化问题，将研究三角群作用的不变量和半不变量，并将其用于Jordan形式和算子谱分配问题。正规完备化和正定2D Toeplitz完备化也将深入研究。将发展矩阵值函数的多点内插，并在具有对称性的矩阵和算子函数的内插方面取得进一步的进展。将探索非平稳插补的应用。发展了不定内积空间上正规算子的分类，并进一步推广到交换J-自伴算子集的谱理论。极分解和J-谱因式分解的研究将继续进行。PI还将继续研究几个变量的概周期矩阵函数的正扩张和压缩扩张问题，并将重点放在计算方面。相关问题包括具有几乎周期矩阵符号的Toeplitz算子的可逆性和Fredholm性准则，这些算子在Besikovitch空间上的有限截面方法，以及一元和几元概周期矩阵函数的显式分解。利用PI最近发展的新的带法，研究多元正交多项式，相关的极小多项式，以及它们与Riemann-Hilbert问题的关系。选题既受应用的影响，又针对应用。例如，多元正方多项式理论中的预期结果和相关的完成问题将被用于滤波器设计、图像的压缩和分析、纹理建模和多变量随机过程。经典(Wiener-Hopf)分解在积分方程组、偏微分方程组和绕射理论中已成为一种强有力的工具。PI将继续研究它对概周期矩阵值函数(一元和多变量)的自然推广，这将考虑有限区间上的积分方程和逆散射中的相关问题以及数学物理的其他部分。另一个例子涉及作用于Krein空间的算子的极分解，这些算子是由偏振光的线性光学驱动的。预计将与科学家和工程师进行互动。此外，PIs还将让本科生参与他们的研究。

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.