Andreas Lasarow

The Matricial Carathéodory Problem in Both Nondegenerate and Degenerate Cases

The main goal of this paper is to present a new approach to both the nondegenerate and degenerate case of the matricial Caratheodory problem. This approach is based on the analysis of central matrix-valued Caratheodory functions which was started in [FK1] and then continued in [FK3]. In the nondegenerate situation we will see that the parametrization of the solution set obtained here coincides with the well-known formula of D.Z. Arov and M.G. Krein for that case (see [AK]).

On the Weyl Matrix Balls Corresponding to the Matricial Carathéodory Problem in Both Nondegenerate and Degenerate Cases

The main goal of the paper is to determine the Weyl matrix balls associated with an arbitrary matricial Caratheodory problem. For the special case of a nondegenerate matricial Caratheodory problem the corresponding Weyl matrix balls were computed by I.V. Kovalishina [Ko] and alternatively by the first and the second authors in [FK1, Parts IV and V]. Mathematics Subject Classification (2000). Primary: 44A60, 47A57, 30E05 Secondary: 47A56.

On Canonical Solutions of a Moment Problem for Rational Matrix-valued Functions

We discuss extremal solutions of a certain finite moment problem for rational matrix functions which satisfy an additional rank condition.W e will see, among other things, that these solutions are molecular nonnegative Hermitian matrix-valued Borel measures on the unit circle and that these measures have a particular structure.W e study the above-mentioned solutions in all generality, but later focus on the nondegenerate case.In this case, the family of these special solutions can be parametrized by the set of unitary matrices.T his realization allows us to further examine the structure of these solutions.H ere, the analysis of the structural properties relies, to a great extent, on the theory of orthogonal rational matrix functions on the unit circle.

On a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case II

The main theme of this paper is the discussion of a family of extremal solutions of a finite moment problem for rational matrix functions in the nondegenerate case. We will point out that each member of this family is extremal in several directions. Thereby, the investigations below continue the studies in Fritzsche et al. (in press) [1]. In doing so, an application of the theory of orthogonal rational matrix functions with respect to a nonnegative Hermitian matrix Borel measure on the unit circle is used to get some insights into the structure of the extremal solutions in question. In particular, we explain characterizations of these solutions in the whole solution set in terms of orthogonal rational matrix functions. We will also show that the associated Riesz-Herglotz transform of such a particular solution admits specific representations, where orthogonal rational matrix functions are involved.

SCHUR–NEVANLINNA SEQUENCES OF RATIONAL FUNCTIONS

We study certain sequences of rational functions with poles outside the unit circle. Such kinds of sequences are recursively constructed based on sequences of complex numbers with norm less than one. In fact, such sequences are closely related to the Schur–Nevanlinna algorithm for Schur functions on the one hand, and to orthogonal rational functions on the unit circle on the other. We shall see that rational functions belonging to a Schur–Nevanlinna sequence can be used to parametrize the set of all solutions of an interpolation problem of Nevanlinna–Pick type for Schur functions.

More on a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case

The main subject of the paper is an in-depth analysis of Weyl matrix balls which are associated with a finite moment problem for rational matrix functions in the nondegenerate case. Thereby, the investigations tie in with preceding studies on a class of extremal solutions of the moment problem in question. We will point out that each member of this class is also extremal concerning the parameters of Weyl matrix balls. The considerations lead to characterizations of these particular solutions within the whole solution set of the problem. In doing so, an application of the theory of orthogonal rational matrix functions with respect to a nonnegative Hermitian matrix Borel measure on the unit circle is used to get that insight.

Szegő Pairs of Orthogonal Rational Matrix-valued Functions on the Unit Circle

We study distinguished pairs of orthonormal systems of rational matrix-valued functions on the unit circle, namely the so-called Szegő pairs. These pairs are determined by an initial condition and a sequence of strictly contractive q × q matrices, which is called the sequence of Szegő parameters. The Szegő parameters contain essential information on the underlying q × q nonnegative Hermitian Borel measure on the unit circle.

Schur-Nevanlinna-Potapov sequences of rational matrix functions

We study particular sequences of rational matrix functions with poles outside the unit circle. These Schur-Nevanlinna-Potapov sequences are recursively constructed based on some complex numbers with norm less than one and some strictly contractive matrices. The main theme of this paper is a thorough analysis of the matrix functions belonging to the sequences in question. Essentially, such sequences are closely related to the theory of orthogonal rational matrix functions on the unit circle. As a further crosslink, we explain that the functions belonging to Schur-Nevanlinna-Potapov sequences can be used to describe the solution set of an interpolation problem of Nevanlinna-Pick type for matricial Schur functions.

Solution of a multiple Nevanlinna–Pick problem for Schur functions via orthogonal rational functions

An interpolation problem of Nevanlinna–Pick type for complex-valued Schur functions in the open unit disk is considered. We prescribe the values of the function and its derivatives up to a certain order at finitely many points. Primarily, we study the case that there exist many Schur functions fulfilling the required conditions. For this situation, an application of the theory of orthogonal rational functions is used to characterize the set of all solutions of the problem in question. Moreover, we treat briefly the case of exactly one solution and present an explicit description of the unique solution in that case.

An Operator-theoretic Approach to a Multiple Point Nevanlinna-Pick Problem for Generalized Carathéodory Functions

We study a Nevanlinna-Pick type interpolation problem for matrix-valued generalized Caratheodory functions, where the values of the function and the values of its derivatives up to a certain order are prescribed at finitely many points of the open unit disk. Under the assumption that the generalized Schwarz-Pick-Potapov block matrix, which is associated to the given data, is non-singular we establish a correspondence between the set of solutions of the problem and the set of minimal unitary extensions of a certain isometry in a Pontryagin space, which is one-to-one modulo unitary equivalence.