Wiener-Hopf Factorization Of Matrix Functions And Related Properties Of Singular Integral Operators

The matrix functions factorization theory is an important branch of algebra and analysis. TheWiener-Hopf factorization of matrix functions has a wide range of applications in mathematics,physics and elasticity. Combining with the Wiener-Hopf factorization theory and boundary valueproblem theory for analytic functions, this dissertation discusses the nature of the associatedsingular integral operators and then studies the factorization for a class group of exponentials ofnilpotent matrix function on the unbounded curve, and eventually draws some conclusions. Thecontent is summarized as follows:Firstly, when matrix functions satisfy the factorization condition, the general solutions of thecorresponding Riemann-Hilbert problem are obtained through the factorization factors. Byimproving the acting domain of the Cauchy integral operator, the operator S which is anidempotent operator is certified. The compactness of the commutable operator S aI aS underthe space ofH,0is obtained and proved. The properties of corresponding Toeplitz operator.The relation of singular integral operators and dimension of kernel space of Toeplitz operator aswell as partial indices are charactered respectively.Secondly, the problem of the canonical factorization for a group of exponentials of nilpotentmatrix function on unbounded curve is researched by the dissertation. By changing the structure ofnilpotent matrix function, the problem of matrix function factorization is transformed into theboundary value problem. And the necessary and sufficient condition for the existence of a canonicalfactorization for a group of exponentials of the nilpotent matrix functions is obtained.Thirdly, the explicit formulas of its meromorphic factorization factors of exponentials ofnilpotent matrix function are obtained under the appropriate regularization conditions. Using themethod of poles separating, separate the poles of the factors of meromorphic decomposition. Andthen introduces the ideas and methods of significant factors of its canonical factorization.