Key wordsThis paper contains first steps towards a Szegö theory of orthogonal rational matrix-valued functions on the unit circle T. Hereby we are guided by former work of Bultheel, González-Vera, Hendriksen, and Njåstad on scalar orthogonal rational functions on the one side and by investigations of Delsarte, Genin, and Kamp on orthogonal matrix polynomials on the other side. An essential characteristic of our matricial orthogonalization procedure is marked by an intensive interplay between left and right matrix-valued inner products generated by a nonnegative Hermitian Borel measure on the unit circle. The main feature of our approach is the distinguished role of Christoffel-Darboux formulas. We consider pairs of rational matrix-valued functions linked via ChristoffelDarboux type relations as an own subject. IntroductionThis paper continues the line of investigations started in [14]. Our main goal is to work out further steps on the way towards a Szegö theory of orthogonal rational matrix-valued functions on the unit circle T. First considerations on scalar orthogonal rational functions occur in the work of Djrbashian [8]- [11]. We are guided by the work of Bultheel, González-Vera, Hendriksen, and Njåstad who created in the 1990's a comprehensive theory of scalar orthogonal rational functions on T (see [1]-[5]). Another important source for our approach is the theory of orthogonal matrix polynomials on the unit circle which is due to Delsarte, Genin, and Kamp [6], [7] (see also [12, Section 3.6]). Concerning alternate treatments of the theory of orthogonal matrix polynomials we refer the reader to Youla and Kazanjian [27] and Fuhrmann [19]. A common feature of all these activities is that they are mainly inspired by the classical work of Szegö [26] and Geronimus [20]-[22] on the theory of orthogonal polynomials on the unit circle. A crucial part in the generalization from the scalar to the matrix case is the definition of the spaces of matrix-valued rational functions for which an orthogonal basis is to be constructed. This topic was handled in [14]. The main feature of the matrix case is that we have to consider the corresponding spaces of rational matrix-valued functions simultaneously as left and right modules over the algebra C q×q of complex q × q matrices. In Section 2, we will work out those parts of calculus in these modules which are needed for our subsequent considerations. This is a matricial generalization of the scalar calculus stated in Section 2.2 of the monograph [4]. In a slight generalization of the concept of Bultheel, González-Vera, Hendriksen, and Njåstad we consider modules of rational matrix-valued functions whose elements have a richer pole structure. More precisely, these functions are also allowed to have poles inside the unit disk D. We consider rational matrix-valued functions which are holomorphic in all points not belonging to a prescribed sequence of complex numbers which do not belong to the unit circle T. Starting from a sequence of complex numbers which are not located at T we...

Key wordsIn this paper, we combine both generalizations by studying orthogonal rational matrix-valued functions on the unit circle. In this way, we continue the line of investigations started in [18]-[20] where we realized first steps towards a Szegö theory for orthogonal rational matrix-valued functions on the unit circle.The main feature of our conception of this kind of Szegö theory is the distinguished role of the ChristoffelDarboux-formulas. A careful analysis of the scalar rational theory and the matrix polynomial case indicates that the Christoffel-Darboux formulas provide the key for extracting large amounts of essential information contained in orthonormal function systems. For this reason, we decided to place the Christoffel-Darboux formulas in the center of our conception. As a first cornerstone of our approach we introduced in [20, Section 6] the notions of left and right Christoffel-Darboux pairs of rational matrix-valued functions. The recursion formulas for left and right Christoffel-Darboux pairs which are obtained in Section 2 realize a crucial step in the way to our Szegö theory conception. Namely, these recursion formulas indicate that there is a one-to-one correspondence between Christoffel-Darboux pairs and sequences of matrices which are connected to particular signature matrices. In this way, we are led to study pairs of sequences of so-called j qq -recursively connected rational matrix-valued *

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Key words Nonnegative Hermitian-valued Borel measures on the unit circle, matrix-valued rational functionals MSC (2000) 47A57This paper provides first tools for generalizing the theory of orthogonal rational functions on the unit circle T created by Bultheel, González-Vera, Hendriksen and Njåstad to the matrix case. A crucial part in this generalization is the definition of the spaces of matrix-valued rational functions for which an orthogonal basis is to be constructed. An important feature of the matrix case is that these spaces will be considered simultaneously as left and right modules over the algebra C q×q . In this modules we will define simultaneously left and right matrix-valued inner products with the aid of a nonnegative Hermitian-valued q × q Borel measure on the unit circle. Given a sequence (αj) j∈N of complex numbers located in C \ T (especially in "good position" with respect to the unit circle) we will introduce a concept of rank for nonnegative Hermitian-valued q × q Borel measures on the unit circle which is based on the Gramian matrix of particular rational matrix-valued functions with prescribed pole structure. A main result of this paper is that this concept of rank is universal. More precisely, it turns out that the rank of a matrix measure does not depend on the given sequence (αj ) j∈N . IntroductionThis paper is aimed at developing the first steps of a theory of orthogonal rational matrix-valued functions on the unit circle. We are guided by the work of Bultheel, González-Vera, Hendriksen and Njåstad who created in the 1990's a comprehensive theory of scalar orthogonal rational functions on the unit circle. In a series of research papers (see, e.g., [1]-[5]) they worked out basic parts of a concept of generalizing essential parts of the classical theory of orthogonal polynomials on the unit circle which goes back mainly to Szegö [20] and Geronimus [14]- [16]. These investigations culminated in the monograph [4] which brings together the single pieces of the puzzle to the full picture. Concerning the history of the development of a Szegö theory of orthogonal rational functions on the unit circle it has to be mentioned that the first rational version of Szegö polynomials was given by in the 1960s for the sole purpose of pure mathematical generalization. The real process began only when Bultheel, González-Vera, Hendriksen and Njåstad started their collaboration at the end of the 1980s. Motivated by problems from different fields ranging from multipoint Padé approximation, moment problems, continued fractions, quadrature formulas, rational approximation, orthogonal Laurent polynomials, and Schur's algorithm to Nevanlinna-Pick interpolation they produced a large collection of results that formed a systematic generalization of Szegö polynomials to their rational counterparts.Our goal is to generalize essential features of the theory of orthogonal rational functions to the matrix case. The present paper contains first contributions in this direction. A crucial part in the generalization from...

The main theme of this paper is a thorough study of a parametrized family ( w ) w∈D of solutions of a nondegenerate matricial Carathéodory problem. If w = 0 the function w coincides with the so-called central solution which has several extremal properties amongst the set of all solutions. This observation inspired us to look for corresponding extremal properties of the functions w in the case of an arbitrary w ∈ D. It turns out that the functions w are extremal in several directions (semi-radii of Weyl matrix balls, extremal entropy etc.). Our methods are based on a combination of the theory of orthogonal matrix polynomials with respect to a nonnegative Hermitian matrix-valued measure on the unit circle developed by Delsarte, Genin, and Kamp with the authors' former investigations on the matricial Carathéodory problem.