Problems in Operator and Matrix Analysis

摘要：罗德曼教授、斯皮特科夫斯基教授和沃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。 其中包括：（1）三角算子的完备化，（2）插值，（3）不定内积空间，（4）概周期分解，（5）多元正交多项式及其推广。 在完备化问题中，研究了三角群作用的不变量和半不变量，并将其应用于Jordan形式和算子谱配置问题。正规完备化和正定的2DToeplitz完备化也将被研究。 将发展矩阵值函数的多点插值，并进一步发展矩阵和算子函数的对称插值。将探讨非stationaryinterpolation的应用。本文将发展不定内积空间上正规算子的分类，并推广可交换J-自伴算子集的谱理论。极分解和J-谱分解的研究将继续进行。该PI也将继续他们的研究积极和contractiveextension问题几乎周期矩阵函数在several variables，与额外的重点放在计算方面。 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Fredholness准则，Besikovitch空间上这些算子的有限段方法，以及一元和多元概周期矩阵函数的显式因子分解。本文将利用近年来国际数学家们发展起来的新的带方法研究多元正交多项式、相关的极小化多项式及其与Riemann-Hilbert问题的联系，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，又以应用为目的。例如，多元正交多项式理论和相关的完备化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。 经典的Wiener-Hopf分解在积分方程、偏微分方程和衍射理论中有着广泛的应用。PI将继续研究其自然推广到几乎周期矩阵值函数（一个和几个变量），其中考虑到有限区间上的积分方程以及逆散射和物理学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激发的作用于Krein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。

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.