Meromorphic Matrix Functions Research Articles

Efficient solution of parameter‐dependent quasiseparable systems and computation of meromorphic matrix functions

SummaryIn this paper, we focus on the solution of shifted quasiseparable systems and of more general parameter‐dependent matrix equations with quasiseparable representations. We propose an efficient algorithm exploiting the invariance of the quasiseparable structure under diagonal shifting and inversion. This algorithm is applied to compute various functions of matrices. Numerical experiments show that this approach is fast and numerically robust.

An inverse spectral problem for the matrix Sturm-Liouville operator on the half-line

The matrix Sturm-Liouville operator with an integrable potential on the half-line is considered. The inverse spectral problem is studied, which consists in recovering of this operator by the Weyl matrix. The author provides necessary and sufficient conditions for a meromorphic matrix function being a Weyl matrix of the non-self-adjoint matrix Sturm-Liouville operator. We also investigate the self-adjoint case and obtain the characterization of the spectral data as a corollary of our general result.

Factorization of meromorphic matrix functions

The paper considers a class of matrix-functions defined on some contour in the complex plane that have meromorphic continuations in the interior or exterior domain of that contour. These matrix-functions generally do not admit Wiener-Hopf standard factorization. The paper studies the problem of index factorization, which is a version of Wiener-Hopf factorization. Some criterions for index factorization and exact formulas for particular indices are found along with a constructive method which applies the factorization to solution of an explicit, finite system of linear algebraic equations.

Spectral theory and spectral gaps for periodic Schrödinger operators on product graphs

Floquet theory and its applications to spectral theory are developed for periodic Schrödinger operators on product graphs 𝔾×ℤ, where 𝔾 is a finite graph. The resolvent and the spectrum have detailed descriptions which involve the eigenvalues and singularities of the meromorphic Floquet matrix function. Existence and size estimates for sequences of spectral gaps are established.

A rank criterion for the order of a pole of a matrix function

By treating the null chains of a meromorphic matrix function, A( z), as the solutions of systems of linear equations, this paper gives a computational criterion for the order of a pole of A( z) −1.

Interpolation and Divisibility of Meromorphic Matrix Functions

We construct a meromorphic matrix function with given spectral data in the form of null and pole functions, coupling matrix, and left annihilator at every point in the domain of definition of the function. Based on these results, descriptions are given of minimal divisibility of meromorphic matrix functions in terms of restrictions (appropriately understood) of their spectral data.

Interpolation with Meromorphic Matrix Functions

Interpolation with Meromorphic Matrix Functions

Interpolation with meromorphic matrix functions

A complete solution is given to a first-order pole-zero meromorphic matrix function interpolation problem on a closed Riemann surface. The solution to the interpolation problem is constructed from the solution to a natural linear homogeneous system.

On Wiener-Hopf factorization of meromorphic matrix functions

Integral Equations and Operator Theory Vol. 14 (1991) 0378-620X/91/060767-0851.50+0.20/0 (c) 1991Birkh~user Verlag, Basel ON WIENER-HOP? FACTORIZATION OF MEROMORPHIC MATRIX FUNCTIONS V. M. Adukov In this paper notions 9 f indices and essential polynomials of a finite sequenc2 ol matrices are. introauced. In terms of this notions explicit solution of the Wlener-Hopf factorlzatlon problem for meromorphlc matrix functions is presented.~ In

Multiplicities of Nonsimple Zeros of Meromorphic Matrix Functions of the Class Nn×nx

Multiplicities of Nonsimple Zeros of Meromorphic Matrix Functions of the Class <i>N</i>^{<i>n</i>×<i>n</i>}_x

Minimal factorization of meromorphic matrix functions in terms of local data

Description de tous les diviseurs minimaux en un point donne d'une fonction matricielle meromorphe en termes de paires spectrales locales et de matrice de couplage locale

Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions: Parametrization of the set of all solutions

We consider a general matrix version of a Pick-Loewner interpolation problem on the closed unit disk. Solutions are allowed to have a finite numberl of free poles in the open disk. We show that the smallestl for which a solution to the problem exists is the number of negative eigenvalues of an appropriately defined “Pick matrix,” and for this value ofl we obtain a linear fractional map parametrization of the class of all solutions. The idea is to adapt the Grassmannian approach involving Krein space geometry and invariant subspace representations of the authors; this was successful previously for the case where all interpolating points are inside the disk. Also an appendix includes an errata to earlier work together with simplified proofs.

Interpolation and local data for meromorphic matrix and operator functions

The main result of this paper is a generalization of the Mittag-Leffler theorem to matrix and operator valued meromorphic functions. Namely, a meromorphic matrix or operator valued function is constructed when the singular parts of the function and if its inverse are given in all singular points (which are assumed to be isolated). The paper contains also interpolation theorems based on other forms of local data (Jordan chains from left and right of the function and its inverse). An analysis of the local data, which is used in the proofs of these theorems is also included.

Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions

Classical interpolation problems are concerned with the problem of finding an analytic function on the unit disk bounded by one which takes on prescribed values at certain prescribed points inside the disk (Pick-Nevanlinna) or on the boundary of the disk (Loewner). We consider the problem for matrix-valued functions having as many asl poles (l a nonnegative integer) inside the disk but still uniformly bounded by one on the boundary of the disk. Our technique is an adaptation of that of Sz.-Nagy and Koranyi to spaces with an indefinite inner product. The problem arises in the broadband matching problem for electrical circuits and certain multichannel scattering problems in physics.

A sign characteristic for selfadjoint meromorphic matrix functions

We introduce the notion of sign characteristic for selfadjoint meromorphic matrix functions which generalizes this notion for matrix polynomials introduced by the authors in [4]. The purpose of this paper is to show equivalence of various definitions of the sign characteristic for meromorphic matrix functions