On a Simultaneous Approach to the Even and Odd Truncated Matricial Hamburger Moment Problems

Abstract

The main goal of this paper is to achieve a simultaneous treatment of the even and odd truncated matricial Hamburger moment problems in the most general case. In the odd case, these results are completely new for the matrix case, whereas the scalar version was recently treated by V.A. Derkach, S. Hassi and H.S.V. de Snoo [12]. The even case was studied earlier by G.-N. Chen and Y.-J. Hu [9]. Our approach is based on Schur analysis methods. More precisely, we use two interrelated versions of Schurtype algorithms, namely an algebraic one and a function-theoretic one. The algebraic version was worked out in a former paper of the authors. It is an algorithm which is applied to finite or infinite sequences of complex matrices. The construction and investigation of the function-theoretic version of our Schur-type algorithm is a central theme of this paper. This algorithm will be applied to relevant subclasses of holomorphic matrix-valued functions of the Herglotz–Nevanlinna class. Using recent results on the holomorphicity of the Moore–Penrose inverse of matrix-valued Herglotz–Nevanlinna functions, we obtain a complete description of the solution set of the moment problem under consideration in the most general situation.

Dedicated to Lev Aronovich Sakhnovich on the occasion of his 80th birthday

Mathematics Subject Classification (2010). Primary 44A60, 47A57; Secondary 30E05.