Perturbation analysis, including perturbation bounds, is developed for nonlinear operator equations of the form $X = Q \pm A^* F(X) A$, under perturbations of the given operators Q (which is assumed to be positive definite) and A and of the given operator function F(X) which takes self-adjoint operator values. Stability of fixed points under suitable map perturbations serves as the main technical tool. More detailed analysis is provided in the particular cases where F(X) is a power map.

Motivated by state space realizations of transfer functions from system theory, a number of operations on Schur complements are introduced and studied. These operations are equivalence, extension, multiplication, inversion, and factorization. Together they form an algebraic framework which is of independent interest, and also useful in solving problems in analysis.

This paper deals with two interrelated issues. One is an invariant subspace approach to finding solutions for the algebraic Riccati equation for a class of infinite dimensional systems. The second is approximation of the solution of the algebraic Riccati equation by finite dimensional approximants. The theory of exponentially dichotomous operators and bisemigroups is instrumental in our approach.

An analogue of Banach's fixed point theorem in partially ordered sets is proved in this paper, and several applications to linear and nonlinear matrix equations are discussed.

In this paper we study the matrix equation X = Q + A* (X- C)(-1) A, where Q is an n x n positive definite matrix, C is an mn x mn positive semidefinite matrix, A is an arbitrary mn x n matrix and X is the m x m block diagonal matrix with on each diagonal entry the n x it matrix X. We are interested in the existence and uniqueness of solutions which are contained in a certain subset of the set of the positive definite matrices, under the condition that C < Q. These solutions play a role in an optimal interpolation theory problem [Interpolation Theory and Its Applications, Mathematics and Its Applications, vol. 428, Kluwer Academic, Dordrecht, 1997 (Chapter 7)]. (C) 2003 Elsevier Inc. All rights reserved.

It is shown that, for any given polynomially normal matrix with respect to an indefinite inner product, a nonnegative (with respect to the indefinite inner product) invariant subspace always admits an extension to an invariant maximal nonnegative subspace. Such an extension property is known to hold true for general normal matrices if the nonnegative invariant subspace is actually neutral. An example is constructed showing that the extension property does not generally hold true for normal matrices, even when the nonnegative invariant subspace is assumed to be positive. On the other hand, it is proved that the extension property holds true for hyponormal (with respect to the indefinite inner product) matrices under certain additional hypotheses.

In this paper we show that any rational matrix function having hermitian values on the imaginary axis, and with constant signature and constant pole signature admits a minimal symmetric factorization with possibly nonsquare factors. Our proof is based on a construction which shows that any such function can be extended (preserving its McMillan degree) to a function that admits J-symmetric factorization with square factors. Also, we consider other properties of the factors in J-symmetric factorizations. Particular attention is given to the study of the common invariant zero structure of these factors.

We consider the problem of parametrizing the set of all nonsquare minimal spectral factors of a rational matrix function taking positive semidefinite values on the imaginary axis.

This paper treats a set of equations of the form $X+A^{\star}{\cal F}(X)A =Q$, where ${\cal F}$ maps positive definite matrices either into positive definite matrices or into negative definite matrices, and satisfies some monotonicity property. Here A is arbitrary and Q is a positive definite matrix. It is shown that under some conditions an iteration method converges to a positive definite solution. An estimate for the rate of convergence is given under additional conditions, and some numerical results are given. Special cases are considered, which cover also particular cases of the discrete algebraic Riccati equation.

In this paper, the nonlinear matrix equation X+A∗F(X)A=Q is discussed. Sufficient conditions for the existence and uniqueness of a positive semidefinite solution are derived. Also conditions are given under which the solution depends continuously on the matrices A and Q.

We consider the problem of parametrizing the set of square minimal spectral factors of a rational matrix function taking positive semidefinite values on the imaginary axis in terms of invariant subspaces and of the minimal unitary left divisors of a certain unitary function. We shall use an approach which involves null-pole triples.

For a given real invertible skew-symmetric matrix H, we characterize the real $2n\times 2n$ matrices X that allow an H-Hamiltonian polar decomposition of the type X= UA, where U is a real H-symplectic matrix ( UTHU = H) and A is a real H-Hamiltonian matrix ( HA=- A T H ).

In this paper we consider transport equations with accretive collision operators. We characterize when the equation has a unique solution and show that in this case the solution is stable under small perturbations of the collision operator and the initial value. In one case in which there is more than one solution we show how to make a special selection of a solution, which is then stable again under small perturbations of both the collision operator and the initial value. The results obtained here parallel those obtained earlier for the case where the collision operator is positive semidefinite.

In this paper factorization results for transfer functions of Pritchard–Salamon systems are obtained. In particular, transfer functions sufficiently close to the identity operator are shown to have a canonical Wiener–Hopf factorization. Moreover, the Bounded Real Lemma is generalized to Pritchard–Salamon systems and applied to relate left and right canonical Wiener–Hopf factorizations of their transfer functions.

Nonnegative Hermitian solutions of various types of continuous and discrete algebraic Riccati equations are studied. The Hamiltonian is considered with respect to two different indefinite scalar products. For the set of nonnegative solutions the order structure and the topology of the set and the stability of solutions is treated. For general Hermitian solutions a method to compute the inertia is given. Although most attention is payed to the classical types arising from LQ optimal control theory, the case where the quadratic term has an indefinite coefficient is studied as well.

Witt's theorem on the extension of H-isometries to H-unitary matrices with respect to the scalar product generated by a self-adjoint nonsingular matrix H is studied in detail. All possible extensions are given, and their structure as a real analytic manifold is described. Analogous problems with respect to skew-symmetric scalar products are studied as well.The main motivation to study these problems, as well as the main applications of the results obtained, concerns polar decompositions in indefinite scalar product spaces. As another application, for given B all solutions of the matrix equation XA = B with H-unitary X and upper triangular A are described. Equations of this type are of vital importance in hyperbolic QR decompositions.

Several classes of polar decompositions of real and complex matrices with respect to a given indefinite scalar product are studied. Matrices that admit such polar decompositions are described in various ways. In particular, a full description of all polar decompositions of a given matrix up to the natural similarity between polar decompositions is given.

Recently there has been renewed interest in the problem of spectral factorization and in particular, the problem of parametrizing all square minimal spectral factors of a given spectrum. For instance, let us mention two recent papers in this journal, one of which dealing with more computational aspects of this problem (Clements, 1993), the other giving a parametrization under additional constraints (Ferrante et al., 1993). This paper is motivated in a large part by (Ferrante et al., 1993), in fact we shall show that a parametrization similar to the one given there can be achieved without one of the additional constraints imposed in (Ferrante et al., 1993). A secondary aim of this paper is to give an overview of several parametrizations available in the literature.

Given a stable invariant subspace M of a real matrix A, we study the rate of convergence to M of invariant subspaces of nearby matrices. Several related concepts of stability (studied previously for complex matrices) are studied for real matrices.

An $n \times n$ matrix polynomial $L( \lambda )$ (with real or complex coefficients) is called self-adjoint if $L( \lambda ) = ( L( \bar \lambda ) )^ * $ and symmetric if $L( \lambda ) = ( L( \pm \lambda ) )^T $. Factorizations of selfadjoint and symmetric matrix polynomials of the form $L( \lambda ) = ( M( \bar \lambda ) )^ * DM( \lambda )$ or $L( \lambda ) = ( M( \pm \lambda ) )^T DM( \lambda )$ are studied, where D is a constant matrix and $M( \lambda )$ is a matrix polynomial. In particular, the minimal possible size of D is described in terms of the elementary divisors of $L( \lambda )$ and (sometimes) signature of the Hermitian values of $L( \lambda )$.