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From the mid-1980s R.L. Ellis, I. Gohberg and D.C. Lay wrote several papers on systems of orthogonal matrix polynomials and matrix functions, culminating in the monograph [3] by Ellis and Gohberg, where additional background and further references can be found. Inverse problems related to these orthogonal systems were first considered in [2] for scalar-valued Wiener functions on the circle, both for unilateral systems (onefold problem) and bilateral systems (twofold problem). In later work extensions of the onefold inverse problem on the circle were considered for square matrix-valued polynomials in [5] and for square matrix-valued Wiener functions in [4]. Nonsquare versions were only recently dealt with in [12] and [13] for the onefold problems on the circle and real line, respectively, while nonsquare twofold problems on the circle and real line were solved in [9] and [10], respectively. In this paper we further develop the solution to the twofold inverse problem on the...

This paper presents a few additions to commutant lifting theory. An operator interpolation problem is introduced and shown to be equivalent to the relaxed commutant lifting problem. Using this connection a description of all solutions of the former problem is given. Also a new application, involving bounded operators induced by $H^2$ operator-valued functions, is presented., Comment: 14 pages

This paper presents an explicit inversion formula for certain structured linear transformations that are closely related to finite block Toeplitz matrices. The conditions of invertibility are illustrated by an example. State space techniques from mathematical system theory play an important role.

For the strictly positive case (the suboptimal case), given stable rational matrix functions $G$ and $K$, the set of all $H^\infty$ solutions $X$ to the Leech problem associated with $G$ and $K$, that is, $G(z)X(z)=K(z)$ and $\sup_{|z|\leq 1}\|X(z)\|\leq 1$, is presented as the range of a linear fractional representation of which the coefficients are presented in state space form. The matrices involved in the realizations are computed from state space realizations of the data functions $G$ and $K$. On the one hand the results are based on the commutant lifting theorem and on the other hand on stabilizing solutions of algebraic Riccati equations related to spectral factorizations., 28 pages

In a recent series of papers concrete procedures were derived to compute a stable rational matrix solution to the Leech equation with rational matrix data. In one of these papers the procedure involved a specific Douglas-Shapiro-Shields (DSS) factorization. In the present paper it is shown that one can take a class of DSS factorizations as the starting point of the procedure, leading to solutions which may have a lower McMillan degree as the one obtained in the original procedure http://fwn06.housing.rug.nl/mtns2014-papers/fullPapers/0071.pdf

In a 1990 paper Helton and Young showed that under certain conditions the optimal solution of the Nehari problem corresponding to a finite rank Hankel operator with scalar entries can be efficiently approximated by certain functions defined in terms of finite dimensional restrictions of the Hankel operator. In this paper it is shown that these approximants appear as optimal solutions to restricted Nehari problems. The latter problems can be solved using relaxed commutant lifting theory. This observation is used to extent the Helton and Young approximation result to a matrix-valued setting. As in the Helton and Young paper the rate of convergence depends on the choice of the initial space in the approximation scheme., 22 pages

A time-variant analogue of an interpolation problem equivalent to the relaxed commutant lifting problem is introduced and studied. In a somewhat less general form the problem already appears in the analysis of the set of all solutions to the three chain completion problem. The interpolants are upper triangular operator matrices of which the columns induce contractive operators. The set of all solutions of the problem is described explicitly. The results presented are time-variant analogues of the main theorems in [A.E. Frazho, S. ter Horst, and M.A. Kaashoek, All solutions to the relaxed commutant lifting problem, Acta Sci. Math. (Szeged) 72 (2006), 299--318].

In the mid-fifties, in a seminal paper, M. G. Krein introduced continuous analogs of Szego orthogonal polynomials on the unit circle and established their main properties. In this paper, we generalize these results and subsequent results that he obtained jointly with Langer to the case of matrix-valued functions. Our main theorems are much more involved than their scalar counterparts. They contain new conditions based on Jordan chains and root functions. The proofs require new techniques based on recent results in the theory of continuous analogs of resultant and Bezout matrices and solutions of certain equations in entire matrix functions. © 2007 Springer Science+Business Media, Inc.