On several types of resolvent matrices of nondegenerate matricial Carathéodory problems
For a Carathéodory problem, one is given a set of \(q\times q\) matrices \((\Gamma_i)_{i=0}^n\) and one has to find a function \(\Omega\) such that \(\Omega^{(k)}(0)=k!\Gamma_k\), \(k=0,\ldots,n\), and such that it is a Carathéodory function, i.e., it has to be analytic and have a positive real part in the open unit disk. A Schur problem is similar, except that the function has to belong to the Schur class, i.e., it has to be analytic and contractive in the open unit disk. The solution set of a Carathéodory problem is characterized as a linear fractional transform of the set of Schur functions. The four matrix coefficients of the linear fractional transform can be arranged in a \(2q\times 2q\) matrix. This is called the resolvent matrix for the truncated Carathéodory problem. By a Cayley transform one can derive a corresponding Schur problem with associated left and right resolvent matrices. Also a matrix valued Schur function can be described as the ``ratio'' of two matrix valued functions with particular properties. There is another linear fractional transform which maps this pair of functions to a solution of the Carathéodory problem, hence another resolvent matrix arises. This paper discusses the relations between all these different ways of describing the solution sets of these problems and the corresponding resolvent matrices.
scientific article; zbMATH DE number 1370029
scientific article