A Truncated Matricial Moment Problem on a Finite Interval

A Schur-Nevanlinna type algorithm for the truncated matricial Hausdorff moment problem

“…It will turn out that the case of the matricial Hausdorff moment problem is much more difficult in comparison with the matricial problems named after Hamburger and Stieltjes. This phenomenon could be already observed in the discussion of the so-called non-degenerate case, where the Hamburger problem (see [14,41]) and the Stieltjes problem (see [15,16,18,19]) were studied considerably earlier than the Hausdorff problem (see [10,11]). The main reason for the greater complexity of the Hausdorff moment problem is caused by the fact that the localization of the measure in a prescribed compact interval of the real axis requires to satisfy simultaneously more conditions.…”

“…Continuing the work done in [5,10,11] A. E. Choque-Rivero [6][7][8][9] investigated further aspects of the non-degenerate truncated matricial Hausdorff moment problem. As in [5,10,11] he distinguished between the case of an odd or even number of prescribed matricial moments. The approach used in [5,10,11] is based on V. P. Potapov's method of Fundamental Matrix Inequalities.…”

On matrix‐valued Herglotz‐Nevanlinna functions with an emphasis on particular subclasses

“…For each z ∈ Π + , Theorem 6.3 yields \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$F(z)=\int _\mathbb {R}\frac{1}{t-z}\sigma _{F}(\mbox{d}t)$\end{document}. Then F belongs to \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {R}_{q}(\Pi _+)$\end{document} and fulfills \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\sup _{y\in [1,+\infty )}y\Vert F(\mathrm{i}y)\Vert _{\mathrm{S}}<+\infty \ ($\end{document}see, e. g., 7, Theorem 8.7] for each y ∈ [1, +∞)).…”

“…Conversely, we now consider an arbirary \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$F\in \mathcal {R}_{q}(\Pi _+)$\end{document} which satisfies \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\sup _{y\in [1,+\infty )}y\Vert F(\mathrm{i}y)\Vert _{\mathrm{S}}<+\infty$\end{document}. Then7, Theorem 8.7] yields the existence of a \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\sigma \in \mathcal {M}_\ge ^{q}(\mathbb {R})$\end{document} such that for each z ∈ Π + . Thus, Theorem 6.3 shows that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$F\in \mathcal {R}_{q}^{[0]}(\Pi _+)$\end{document} and γ F = 0 q × q , i. e., that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$F\in \mathcal {R}_{0,q}(\Pi _+)$\end{document}.…”

“…The recent monograph3 is devoted to conservative realizations of various classes of Stieltjes, inverse Stieltjes, and general Herglotz‐Nevanlinna functions as impedance functions of linear systems. For comprehensive surveys of basic facts on Herglotz‐Nevanlinna functions and their matrix‐valued generalizations, we refer the reader to1, 7, 10, 20. Generalizations of integral representations to the case of operator‐valued Herglotz‐Nevanlinna functions have been studied by Šmul′jan22.…”