F. van Schagen

Eigenvalues of completions of submatrices

This paper concerns the following problem:Given a submatrix of an n× nmatrix Ztogether with its position in Z, determine the restrictions imposed on the eigenvalues of Zand their elementary divisors by this prescribed part. The terms on which the final results depend are identified as the block similarity invariants. Special cases, which were considered before, are reviewed, and the case of off-diagonal blocks is solved.

On the local theory of regular analytic matrix functions

Abstract A concise exposition is given of the local spectral theory of regular analytic matrix functions without any reference to elements from the global theory.

Szegö-Kac-Achiezer formulas in terms of realizations of the symbol

Abstract For rational and analytic matrix functions new formulas are obtained for the limits in the Szego-Kac-Achiezer limit theorems. In the rational case the new expressions are given in terms of finite matrices which come from special representations of the matrix functions. These representations are known as realizations in mathematical systems theory.

Rational contractive and unitary interpolants in realized form

A solution of the rational Nehari problem is given in terms of a realization. Other aspects of this problem, like one step extension, maximum entropy interpolants and unitary interpolants, are also analyzed for the rational case. The results are based on earlier work of H. Dym and I. Gohberg.

Similarity of operator blocks and canonical forms. I. General results, feedback equivalence and kronecker indices

In the present paper the problem of classifying blocks of matrices up to similarity is considered. The notion of block similarity used here is a natural generalization of similarity for matrices. The invariants are described and canonical forms are given. This theory of block-similarity provides a general framework, which includes the state feedback theory for systems, the theory of Kronecker equivalence and a similarity theory for non-everywhere defined operators. New applications, in particular to factorization problems, are also obtained.

Rational matrix and operator functions with prescribed singularities

In this paper it is shown how a rational matrix function may be reconstructed when complete information about its zeros and poles is given. The analogous problem for infinite dimensional operator functions is also solved.

Similarity of Operator Blocks and Canonical Forms. II. Infinite Dimensional Case and Wiener-Hopf Factorization

The concept of block-similarity introduced in part I and its extension to the infinite dimensional case developed here provide a unified approach to state feedback theory for systems, the theory of Kronecker indices and Wiener-Hopf factorization problems. In this part we concentrate on the connections with the factorization theory.

Finite Section Method for Difference Equations

A finite section method is developed for linear difference equations over an infinite time interval. A necessary and sufficient condition is given in order that the solutions of such equations may be obtained as limits of solutions of corresponding equations over a finite time interval. Both the time-variant and the time-invariant case are considered. For the time-invariant case the condition reduces to the requirement that two subspaces defined in terms of the equations should be complementary. The results obtained extend those derived earlier for linear ordinary differential equations.

The discrete twofold Ellis-Gohberg inverse problem

In this paper a twofold inverse problem for orthogonal matrix functions in the Wiener class is considered. The scalar-valued version of this problem was solved by Ellis and Gohberg in 1992. Under reasonable conditions, the problem is reduced to an invertibility condition on an operator that is defined using the Hankel and Toeplitz operators associated to the Wiener class functions that comprise the data set of the inverse problem. It is also shown that in this case the solution is unique. Special attention is given to the case that the Hankel operator of the solution is a strict contraction and the case where the functions are matrix polynomials.

On Inversion of Certain Structured Linear Transformations Related to Block Toeplitz Matrices

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.