Harm Bart

Factorizations of Transfer Functions

This paper is concerned with minimal factorizations of rational matrix functions. The treatment is based on a new geometrical principle. In fact, it is shown that there is a one-to-one correspondence between minimal factorizations on the one hand and certain projections on the other. Considerable attention is given to the problem of stability of a minimal factorization. Also the numerical aspects are discussed. Along the way, a stability theorem for solutions of the matrix Riccati equation is obtained.

Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators

Abstract For a certain class of matrix functions that are analytic on the real line but not at infinity an explicit method of factorization and of constructing inverse Fourier transforms is developed. This method is applied to invert Wiener-Hopf integral equations on the half line and the full line. The results obtained extend analogous results of the authors for rational matrix functions and for functions that are analytic on the real line and at infinity. The analysis is based on an infinite dimensional realization theorem which involves operators that are a direct sum of two infinitesimal generators of C 0 -semigroups of negative exponential type, one of which has support on the negative half line and the other on the positive half line. The latter operators are called exponentially dichotomous and the study of their properties forms an essential part of the paper.

Zero sums of idempotents in Banach algebras

The problem treated in this paper is the following.Let p1,...,pkbe idempotents in a Banach algebra B, and assume p1+...+pk=0.Does it follow that pj=0,j=1,..., k? For important classes of Banach algebras the answer turns out to be positive; in general, however, it is negative. A counterexample is given involving five nonzero bounded projections on infinite-dimensional separable Hilbert space. The number five is critical here: in Banach algebras nontrivial zero sums of four idempotents are impossible. In a purely algebraic context (no norm), the situation is different. There the critical number is four.

A state space approach to canonical factorization with applications

Convolution equations, canonical factorization and the state space method.- The role of canonical factorization in solving convolution equations.- The state space method and factorization.- Convolution equations with rational matrix symbols.- Explicit solutions using realizations.- Factorization of non-proper rational matrix functions.- Equations with non-rational symbols.- Factorization of matrix functions analytic in a strip.- Convolution equations and the transport equation.- Wiener-Hopf factorization and factorization indices.- Factorization of selfadjoint rational matrix functions.- Preliminaries concerning minimal factorization.- Factorization of positive definite rational matrix functions.- Pseudo-spectral factorizations of selfadjoint rational matrix functions.- Review of the theory of matrices in indefinite inner product spaces.- Riccati equations and factorization.- Canonical factorization and Riccati equations.- The symmetric algebraic Riccati equation.- J-spectral factorization.- Factorizations and symmetries.- Factorization of positive real rational matrix functions.- Contractive rational matrix functions.- J-unitary rational matrix functions.- Applications of J-spectral factorizations.- Application to the rational Nehari problem.- Review of some control theory for linear systems.- H-infinity control applications.

Logarithmic residues in Banach algebras

Letf be an analytic Banach algebra valued function and suppose that the contour integral of the logarithmic derivativef′f−1 around a Cauchy domainD vanishes. Does it follow thatf takes invertible values on all ofD? For important classes of Banach algebras, the answer is positive. In general, however, it is negative. The counterexample showing this involves a (nontrivial) zero sum of logarithmic residues (that are in fact idempotents). The analysis of such zero sums leads to results about the convex cone generated by the logarithmic residues.

Stable factorizations of monic matrix polynomials and stable invariant subspaces

The stable factorizations of a monic matrix polynomial are characterized in terms of spectral properties. Proofs are based on the divisibility theory developed by I. Gohberg, P. Lancaster and L. Rodman. A large part of the paper is devoted to a detailed analysis of stable invariant subspaces of a matrix. The results are also used to describe all stable solutions of the operator Riccati equation.

Logarithmic residues, generalized idempotents, and sums of idempotents in Banach algebras

In a commutative Banach algebraB the set of logarithmic residues (i.e., the elements that can be written as a contour integral of the logarithmic derivative of an analyticB-valued function), the set of generalized idempotents (i.e., the elements that are annihilated by a polynomial with non-negative integer simple zeros), and the set of sums of idempotents are all the same. Also, these (coinciding) sets consist of isolated points only and are closed under the operations of addition and multiplication. Counterexamples show that the commutativity condition onB is essential. The results extend to logarithmic residues of meromorphicB-valued functions.

Fredholm theory of Wiener-Hopf equations in terms of realization of their symbols

The Fredholm properties (index, kernel, image, etc.) of Wiener-Hopf integral operators are described in terms of realization of the symbol for a class of matrix symbols that are analytic on the real line but not at infinity. The realizations are given in terms of exponentially dichotomous operators. The results obtained give a complete analogue of the earlier results for rational symbols.

Logarithmic residues of analytic Banach algebra valued functions possessing a simply meromorphic inverse

Abstract A logarithmic residue is a contour integral of a logarithmic derivative (left or right) of an analytic Banach algebra valued function. For functions possessing a meromorphic inverse with simple poles only, the logarithmic residues are identified as the sums of idempotents. With the help of this observation, the issue of left versus right logarithmic residues is investigated, both for connected and nonconnected underlying Cauchy domains. Examples are given to elucidate the subject matter.

Upper triangularization of matrices by lower triangular similarities

Abstract This paper is concerned with the following questions. Given a square matrix A , when does there exist an invertible lower triangular matrix L such that L -1 AL is upper triangular? And if so, what can be said about the order in which the eigenvalues of A may appear on the diagonal of L -1 AL ? The motivation for considering these questions comes from systems theory. In fact they arise in the study of complete factorizations of rational matrix functions. There is also an intimate connection with the problem of complementary triangularization of pairs of matrices discussed elsewhere by the first author.