Hugo J. Woerdeman

The band method for positive and strictly contractive extension problems: An alternative version and new applications

The band method for positive and strictly contractive extension problems is deduced from a new set of axioms. New applications concern extension problems for operator-valued functions in the Wiener class and for certain infinite operator matrices.

Minimal rank completions for block matrices

Abstract This paper deals with the following problem of completion of block matrices. Let Aij, i ⩾ j, be given matrices. Find additional matrices Aij such that the completion A = (Aij)ni, j = 1 has lowest possible rank. The structure of the set of all minimal rank completions is studied. Special attention is paid to the Toeplitz case together with its connection to the partial realization problem.

Ranks of Completions of Partial Matrices

For an n-by-m array with some entries specified and the remainder free to be chosen from a given field, we study the possible ranks occurring among all completions. For any such partial matrix the maximum rank may be nicely characterized and all possible ranks between the minimum and maximum are attained. The minimum is more delicate and is not in general determined just by the ranks of fully specified submatrices. This focusses attention upon the patterns of specified entries for which the minimum is so determined. It is shown that it is necessary that the graph of the pattern be (bipartite) chordal, and some evidence is given for the conjecture that this is also sufficient.

The higher rank numerical range is convex

In this short note we provide the final step in showing that the higher rank numerical range is convex. The previous steps appear in the paper “Geometry of Higher-Rank Numerical Ranges” by Choi, M.-D., Giesinger, M., Holbrook, J.A. and Kribs, D.W.

Spectral Factorizations and Sums of Squares Representations via Semidefinite Programming

In this paper we find a characterization for when a multivariable trigonometric polynomial can be written as a sum of squares. In addition, the truncated moment problem is addressed. A numerical algorithm for finding a sum of squares representation is presented as well. In the one-variable case, the algorithm finds a spectral factorization. The latter may also be used to find inner-outer factorizations.

A maximum entropy principle in the general framework of the band method

Abstract A maximum entropy principle is developed in the general framework of the band method. Known maximum entropy principles appear as corollaries and new ones are derived.

Carathéodory-Toeplitz and Nehari problems for matrix valued almost periodic functions

In this paper the positive and strictly contractive extension problems for almost periodic matrix functions are treated. We present nece ssary and sufficient conditions for the existence of extensions in terms of Toeplit;z and Hankel operators on Besicovitch spaces and Lebesgue spaces. Furthermore, when a solution exists a special extension (the band extension) is constructed which enjoys a maximum entropy property. A linear fractional parameterization of the set of all extensions is also provided. The techniques used xn the proofs include factorizations of matrix valued almost periodic functions and a general algebraic scheme called the band method.

Minimal rank completions of partial banded matrices

It is proven that the minimal rank of a partial banded matrix equals the maximum of the minimal ranks of all triangular subpatterns. This proves partly the minimal rank conjecture in a paper by N. Cohen, C. R. Johnson, L. Rodman and H. J. Woerdeman (Operator Theory: Advances and Applications 40 (1989), 165-185). The results are applied to the problem of simultaneously completing a matrix and its inverse.

Two-variable polynomials: intersecting zeros and stability

In order to construct two-variable polynomials with a certain zero behavior, the notion of intersecting zeros is studied. We show that generically two-variable polynomials have a finite set of intersecting zeros, and give an algorithm on how to construct a polynomial with the desired intersecting zeros. Relations with the Cayley-Bacharach theorem are addressed. In addition, we will also address the case when stable polynomials are sought.

Strictly contractive and positive completions for block matrices

Abstract This paper deals with the following two problems of completion of block matrices: (1) Let A ij , i ⩾ j − q , be given matrices. Find additional matrices A ij such that the block matrix A =( A ij ) n i =1 , m j =1 has norm less than one. (2) Let B ij , | j − i | ⩽ q , be given matrices. Find additional B ij such that the completion B = ( B ij ) n i , j =1 is positive definite. The analysis is based on a study of certain linear fractional maps. This approach leads to an explicit description of all solutions by linear fractional maps of which the coefficients are given directly in terms of the given data. The maximum entropy principle appears as a corollary of a general formula for the determinant of a completion. Special attention is paid to the Toeplitz case.