System Theory, the Schur Algorithm and Multidimensional Analysis

A Truncated Matricial Moment Problem on a Finite Interval. The Case of an Odd Number of Prescribed Moments (p. 99-100)

Abdon E. Choque Rivero, Yuriy M. Dyukarev, Bernd Fritzsche and Bernd Kirstein

Abstract. The main goal of this paper is to study the truncated matricial moment problem on a .nite closed interval in the case of an odd number of prescribed moments by using of the FMI method of V.P. Potapov. The solvability of this problem is characterized by the fact that two block Hankel matrices built from the data of the problem are nonnegative Hermitian (Theorem 1.3). An essential step to solve the problem under consideration is to derive an e.ective coupling identity between both block Hankel matrices (Proposition 2.5). In the case that these Hankel matrices are both positive Hermitian we parametrize the set of solutions via a linear fractional transformation the generating matrix-valued function of which is a matrix polynomial whereas the set of parameters consists of distinguished pairs of meromorphic matrix-valued functions.

Mathematics Subject Classification (2000). Primary 44A60, 47A57, 30E05. Keywords. Matricial moment problem, system of fundamental matrix inequalities of Potapov-type, Stieltjes transform.

0. Introduction

This paper continues the authors’ investigations on the truncated matricial moment problem on a .nite closed interval of the real axis. The scalar version of this problem was treated by M.G. Krein (see [K], [KN, Chapter III]) by di.erent methods. A closer look at this work shows that the cases of an even or odd number of prescribed moments were handled separately. These cases turned out to be intimately related with di.erent classes of functions holomorphic outside the .xed interval [a, b].

The same situation can be met in the matricial version of the moment problem under consideration. After studying the case of an even number of prescribed moments in [CDFK], we handle in this paper the case that the number of prescribed moments is odd. It is not quite unexpected that there are several features which are common to our treatment of both cases. This concern particularly our basic strategy. As in [CDFK] we will use V.P. Potapov’s so-called Fundamental Matrix Inequality (FMI) approach in combination with L.A. Sakhnovich’s method of operator identities. (According to applications of V.P. Potapov’s approach to matrix versions of classical moment and interpolation problems we refer, e.g., to the papers Dubovoj [Du], Dyukarev/Katsnelson [DK], Dyukarev [Dy1], Golinskii [G1], [G2], Katsnelson [Ka1]–[Ka3], Kovalishina [Ko1], [Ko2]. Concerning the operator identity method, e.g., the works [IS], [S1], [S2], [BS], and [AB] are mentioned. Roughly speaking, at a first view the comparison between the even and odd cases shows that the principal steps are similar whereas the detailed realizations of these steps are rather different. More precisely, the odd case turned out to be much more diffiult. Some reasons for this will be listed in the following more concrete description of the contents of this paper. Similarly as in [CDFK], we will again meet the situation that our matrix moment problem has solutions if and only if two block Hankel matrices built from the given data are nonnegative Hermitian (see Theorem 1.3). Each of these block Hankel matrices satis.es a certain Ljapunov type identity (see Proposition 2.1). An essential point in the paper is to find an effective algebraic coupling between both block Hankel matrices. The desired coupling identity will be realized in Proposition 2.5. The comparison with the analogous coupling identity in the even case (see [CDFK, Proposition 2.2]) shows that the coupling identity in Proposition 2.5 is by far more complicated. This is caused by the fact that in contrast with [CDFK] now two block Hankel matrices of di.erent sizes and much more involved structure have to be coupled. A first main result (see Theorem 1.2) indicates that (after Stieltjes transform) the original matrix moment problem is equivalent to a system of two fundamental matrix inequalities (FMI) of Potapov type. Our proof of Theorem 1.2 is completely di.erent from the proof of the corresponding result in the even case (see [CDFK, Theorem 1.2]). Whereas the proof of Theorem 1.2 in [CDFK] is mainly based on an appropriate application of generalized inversion formula of Stieltjes-Perron type, in this paper we prefer a much more algebraically orientated approach. Hereby, in particular Lemma 4.5 should be mentioned. It occupies a key role in the proof of Theorem 1.2.