Rational matrix functions with coisometric values on the imaginary line
type: Item
instance of: scholarly article
title: Rational matrix functions with coisometric values on the imaginary line
zbMATH Open document ID: 0896.47007
author: Daniel Alpay
author: Marek Rakowski
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-09-23T00:00:00Z
review text: Given two signature matrices \(J_1\) and \(J_2\), the authors provide a necessary and sufficient condition for a rational matrix \(W\), and holomorphic at infinity, to satisfy the equation \(J_1= W(z)J_2 W(z)^*\) on the imaginary axis. Here a signature matrix is a matrix \(J\) that satisfies \(J= J^*= J^{-1}\). The present condition is based on a Lyapunov equation involving matrices \((A,B,C,D)\) in an observable realization \((A,B,C,D)\) of \(W\), and it extends a more known condition when \(J_1= J_2= J\), in which case \(W(z)\) is \(J\)-unitary on the imaginary line. The fact that \(J_1\neq J_2\) is possible means, in particular, that \(J_1\) and \(J_2\) may have different sizes, and hence \(W\) is allowed to be rectangular. When the condition is satisfied, then for each observable realization \((A,B,C,D)\) of \(W\) there exists a unique, possibly singular, Hermitian matrix \(G\) which satisfies the Lyapunov equation and such that \(CG= -DJ_2B^*\). Finally, the authors also indicate that the imaginary axis, where the equation is satisfied, can be replaced by the unit circle or a more general curve as considered in \textit{D. Alpay} and \textit{H. Dym} [J. Funct. Anal. 111, No. 1, 1-28 (1993; Zbl 0813.46018)].
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 32A30
Mathematics Subject Classification ID: 47N70
Mathematics Subject Classification ID: 30D50
zbMATH DE Number: 811234
zbMATH Keywords: Krein space
zbMATH Keywords: coisometry
zbMATH Keywords: controllable realizations
zbMATH Keywords: rational matrix function
zbMATH Keywords: signature matrices
zbMATH Keywords: Lyapunov equation
zbMATH Keywords: observable realization
reviewed by: Jacob Burbea
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1298
OpenAlex ID: W1992113102
DOI: 10.1006/JMAA.1995.1298
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description: scientific article; zbMATH DE number 811234