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。这包括：(1)三角算子的补全，(2)插值，(3)不定内积空间，(4)概周期分解，(5)多变量正交多项式及其推广。对于补全问题，研究了三角群作用的不变量和半不变量，并将其应用于Jordan形式和算子谱分配问题。正常完井和确定的2D Toeplitz完井也将被研究。本文将发展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的插值方面取得进一步的进展。将探讨非平稳插值的应用。将发展不定内积空间上正规算子的分类，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。他们还将继续研究多变量几乎周期矩阵函数的正扩展和收缩扩展问题，并进一步强调计算方面的问题。相关的问题包括具有概周期矩阵符号的toeplitz算子的可逆性和Fredholmness准则，这些算子在Besikovitch空间上的有限分段方法，以及概周期矩阵函数在一个或多个变量中的显式分解。本文将利用pi最近提出的带法，研究多变量正交多项式、相关最小化多项式及其与黎曼-希尔伯特问题的关系。提议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程序的影响，也以应用程序为目标。例如，几个变量的正交多项式理论和相关的补全问题的预期结果将用于滤波器设计、图像压缩和分析、纹理建模和多变量随机过程。经典（Wiener-Hopf）分解已被用作解积分方程、偏微分方程和衍射理论的有力工具。pi将继续研究它的自然推广到几乎周期矩阵值函数（一个或几个变量），考虑到有限区间上的积分方程和逆散射和数学物理其他部分的相关问题。另一个例子涉及由偏振光的线性光学驱动的作用于Krein空间的算子的极性分解。预计将与科学家和工程师进行互动。此外，pi也会让本科生参与到他们的研究中来。

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.