On the Courant--Fischer theory for Krein spaces
type: Item
instance of: scholarly article
title: On the Courant--Fischer theory for Krein spaces
zbMATH Open document ID: 1168.47004
author: Natalia Bebiano
author: Hiroshi Nakazato
author: João Da Providência
published in: Linear Algebra and its Applications
publication date: 2009-03-17T00:00:00Z
review text: If \(\mathbb C^n\) is equipped with a positive definite inner product \([\cdot,\cdot]\) and \(A\) is a Hermitian \(n\times n\)-matrix (i.e., \([Ax,x] = [x,Ax]\)), then, by the Courant--Fischer minimax principle, the values \[ \min_{\dim S=k}\max_{x\in S}R(x)\quad\text{with the Rayleigh ratio}\quad R(x):=\frac{[Ax,x]}{[x,x]}\tag{1} \] give the eigenvalues of \(A\). The present paper generalizes this result to a situation where \([\cdot,\cdot]\) is not positive definite anymore, but a Krein space inner product on \(\mathbb C^n\). In this case, according to a previous paper by the authors, additional conditions must be imposed assuring that the values of \(R(x)\) are semibounded at least for all \(x\) with \([x,x] > 0\) or for all \(x\) with \([x,x] < 0\). In particular, it is required that the spectrum is real and all eigenvalues of positive and of negative type (i.e., with a positive or negative eigenspace) are not ``interlacing'' (i.e., one group of eigenvalues is on the left of the other). Additionally, only one eigenvalue with a Jordan chain is allowed (increasing the difficulties). Here perhaps, the terminology of definitizable operators in Krein spaces and their critical points could have been helpful. The authors obtain characterizations of the eigenvalues by expressions like (1) with restrictions to \(x \in S\) where \([x,x] > 0\) (or \([x,x] < 0\), respectively). Finally, the results are applied to Hamiltonian dynamics. Note that there is a certain overlap with earlier results [cf.\ \textit{P.\,Binding, B.\,Najman} and \textit{Q.\,Ye}, Integral Equations Oper.\ Theory 35, No.\,4, 398--422 (1999; Zbl 0971.47008) and some of the references there].
reviewed by: Andreas Fleige
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 46C20
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 5529492
zbMATH Keywords: Kreĭn space
zbMATH Keywords: \(J\)-Hermitian matrix
zbMATH Keywords: Rayleigh ratio
zbMATH Keywords: Courant-Fischer theory
MaRDI profile type: Publication
full work available at URL: j.laa.2008.10.013
OpenAlex ID: W2167779970
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DOI: 10.1016/J.LAA.2008.10.013
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description: scientific article; zbMATH DE number 5529492