Problems in Operator and Matrix Analysis

College Of William And Mary, Williamsburg VA

Investigators

Abstract

ABSTRACT: Professors Rodman, Spitkovsky and Woerdeman will continue their study of a variety of problems in operator and matrix analysis and their applications in science and engineering. These include: (1) Completion of Triangular Operators, (2) Interpolation, (3) Indefinite Inner Product Spaces, (4) Almost Periodic Factorization, (5) Orthogonal Polynomials of Several Variables and Generalizations. For completion problems, invariants and semiinvariants of the triangular group action will be studied and then used in the Jordan form and operator spectrum assignment problems. Normal completions and positive definite 2D Toeplitz completions will also be looked into. Multipoint interpolation for matrix valued functions will be developed, and further advances in interpolation of matrix and operator functions with symmetries will be obtained. Applications of non-stationary interpolation will be explored. Classification of normal operators on indefinite inner product spaces will be developed, and generalized further to a spectral theory for sets of commuting J-selfadjoint operators. Study of polar decomposition and J-spectral factorization will be continued. The PIs will also continue their research on positive and contractive extension problems for almost periodic matrix functions in several variables, with additional emphasis on the computational aspects. The related issues include invertibility and Fredholmness criteria for Toeplitz operators with almost periodic matrix symbols, finite section methods for these operators on Besikovitch spaces, and explicit factorization of almost periodic matrix functions in one and several variables. 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 operator theory. The choice of topics is both influenced by and aimed to applications. For example, the expected results in the theory of orthogonal polynomials for several variables and related completion problems will be used in filter design, compression and analysis of images, texture modeling, and multivariate stochastic processes. Classical (Wiener-Hopf) factorization has been used as a powerful tool in integral 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) which arises in consideration of integral equations on finite intervals and related problems in inverse scattering and other parts of mathematical physics. 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 will also involve undergraduate students in their research.