We study two slightly different versions of the truncated matricial Hamburger moment problem. A central topic is the construction and investigation of distinguished solutions of both moment problems under consideration. These solutions turn out to be nonnegative Hermitian q × q Borel measures on the real axis which are concentrated on a finite number of points. These points and the corresponding masses will be explicitly described in terms of the given data. Furthermore, we investigate a particular class of sequences (sj) ∞ j=0 of complex q × q matrices for which the corresponding infinite matricial Hamburger moment problem has a unique solution. Our approach is mainly algebraic. It is based on the use of particular matrix polynomials constructed from a nonnegative Hermitian block Hankel matrix. These matrix polynomials are immediate generalizations of the monic orthogonal matrix polynomials associated with a positive Hermitian block Hankel matrix. We generalize a classical theorem due to Kronecker on infinite Hankel matrices of finite rank to block Hankel matrices and discuss its consequences for the nonnegative Hermitian case. Mathematics Subject Classification (2000). Primary 44A60, 47A57; Secondary 30E05.

IntroductioiiLet q, n be positive integers. Denote 31Z,Axp the set of complex ,nxq matrices aiid5XF the set ofnon-negativelierrnitianqxq matrices. If C,, C , , ..., C,,~-,~3lt,,, t.heii we put co C ; C? ... c;-[ (1) T,,=T,,(C1,, C , , ..., C , -, ) : = ("' c., i; ::: ; : ) * c,+, clc-I C;,-S ... C,] We shall consider the following Problem (P) : Let T,,=T,,(Co, C1, ..., C,-,) c X):. Describe the set of all C,CQII,x, suchWe shall show that this set can be represented as a matrix ball. Further, we shall give a description of all singular block extensions of a given non-negative Hermitian block TOEPLITZ matrix.The problem ( P ) is intimately connected to the trigonometric truncated matrix moment problem. This problem was solved by ANDO [l] using the NAI-NARK Dilation Theorem. Assuming a regularity condition DELSARTEIGENINI KAMP [Z] constructed an explicite solution. We shall give an alternate proof of ANDO'S Theorem.Finally, we shall consider the connection to the C A R A T H E~D~R Y problem which was completely treated by KOVALISHIXA [6]. It should be remarked that ADAMJAN~ A R O V~E I N[lo] considered a special one step extension procedure for infmite block HANEEL matrices and obtained an analogous matrix ball description of the set of solutions. Further, CONSTANTINESCU [ 111 gave a parametrization of strict positive definite block TOEPLITZ matrices using choice sequences. In EL forthcoming paper we shall show the connection with our parametrization. In that paper we shall also discuss some properties of that sequence which one obtains by choosing the centers of the permissible matrix balls. For instance, we shall verify that i t attains the minimum of entropy.that T,,+l=T,+l(Co, Ci, .

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...

No abstract

No abstract