Authors’ abstract: “The explicit factorization of matrix functions of the form \[ A_\gamma(b)= \begin{pmatrix} e & b\\ b^* & b^* b+\gamma e\end{pmatrix}, \] where \(b\) is an \(n\times n\) matrix function, \(e\) represents the identity matrix, and \(\gamma\) is a complex constant, is studied. To this purpose, some relations between a factorization of \(A_\gamma\) and the resolvents of the selfadjoint operators \[ N_+(b)= P_+ bP_- b^* P_+\quad\text{and}\quad N_-(b)= P_- b^* P_+ bP_- \] are analyzed. The main idea is to show that if \(b\) is a matrix function that can be represented through the decomposition \(b= b_- + b_+\) where at least one of the summands is a rational matrix, then it is possible to construct an algorithm that allows us to determine an effective canonical factorization of the matrix function \(A_\gamma\).”
For the entire collection see [Zbl 1012.00041].