Research On Toeplitz-Bezout Matrices And Related Matrices

The concept of Bezoutians is introduced, which have important applications in poly-nomial zero location, stability problems, control theory and the interpolation problems.The application of the Bezoutians have been extended especially to the field of operator theory since P.A.Fuhrmann introduced the concept of polynomial model. The inversion of Be-zoutians and its various generalizations play an essential role. Therefore, it has practical meanings for the study of T-Bezoutians and its inversion. This paper consists of four chapters:In chapter one, the backgrounds and significance of the problems and the main work in this paper are given. Some results are also presented, which are needed in the subsequent sections.In chapter two, we characterize a class of centro-symmetric Bezoutian that are given by a pair of symmetric polynomials. Then the sufficient and necessary conditions for the inversion of Bezoutians being themselves are obtained by means of operator methods. The specific constructions of this kind of matrices are also derived.The connection between polynomial Bezoutians and symmetric matrices are discussed in chapter three. We study the polynomials classes satisfying certain conditions from special cases and deduce some new results.The kernel structure problems of symmetric(centro-symmetric and centro-persymmetric) block Toeplitz plus Hankel matrices are discussed in chapter four.