The bitangential matrix Nevanlinna-Pick interpolation problem revisited (Q1626067)

The bitangential Nevanlinna-Pick interpolation problem for matrix-valued functions in the right half-plane (RHP) is revisited. Given in the RHP a set of points with left interpolation vectors and another set of points and right interpolation vectors, and in the common points the given bitangential data need to be consistent. First, a compact general operator theoretic formulation of the problem is given. Since the 1980s, different approaches to solve this interpolation problem have been considered in the literature. \begin{itemize} \item[(1)] The state space approach of Ball-Gohberg-Rodman [\textit{J. A. Ball} et al., Interpolation of rational matrix functions. Basel etc.: Birkhäuser (1990; Zbl 0708.15011)]; \item[(2)] The fundamental matrix inequality approach of Potapov [\textit{I. V. Kovalishina} and \textit{V. P. Potapov}, Seven papers translated from the Russian. Transl., Ser. 2, Am. Math. Soc. 138 (1988); \textit{I. V. Kovalishina}, Math. USSR, Izv. 22, 419--463 (1984; Zbl 0549.30026); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 47, No. 3, 455--497 (1983)]; \item[(3)] The reproducing kernel space interpretation of Dym [\textit{H. Dym}, \(J\) contractive matrix functions, reproducing kernel Hilbert spaces and interpolation. Providence, RI: American Mathematical Society (AMS) (1989; Zbl 0691.46013)]; and \item[(4)] The Grassmann-Kreĭn-space geometry approach of Ball-Helton [\textit{J. A. Ball} and \textit{J. W. Helton}, J. Oper. Theory 9, 107--142 (1983; Zbl 0505.47029)]. \end{itemize} These different approaches are first recalled, but the main purpose of this paper is to explain the connection between these approaches and to prove the equivalence of the solutions and consistency conditions. For the entire collection see [Zbl 1388.46001].