Projections in Krein spaces
type: Item
instance of: scholarly article
title: Projections in Krein spaces
zbMATH Open document ID: 1188.47002
author: T. Ando
published in: Linear Algebra and its Applications
publication date: 2009-11-23T00:00:00Z
review text: The problem when the sum of two regular subspaces of a Krein space \((\mathcal K,[\cdot,\cdot])\) is again regular is treated. A regular subspace in a Krein space is a closed subspace which is, equipped with the indefinite inner product \([\cdot,\cdot]\), again a Krein space, or, what is the same, the range of a \(J\)-selfadjoint projection. Here, a \(J\)-selfadjoint projection \(P\) is a bounded linear idempotent operator (\(P^2=P\)) which is selfadjoint with respect to the indefinite inner product \([\cdot,\cdot]\). It is shown that the sum of a uniformly positive closed subspace and a uniformly negative closed subspace is always regular. Moreover, it is shown that every \(J\)-selfadjoint projection \(P\) can be written as the sum of a \(J\)-positive projection \(P_1\) and a \(J\)-negative projection \(P_2\) such that \(P_1\) commutes with \(P_2\) and with \(P_2^*\). Furthermore, the problem when the closure of the range of the product of a \(J\)-bicontraction \(A\) and a \(J\)-biexpansion \(B\) becomes regular is discussed. If \(A\) commutes with \(B\) and \(B^+\), where \(B^+\) denotes the adjoint of \(B\) with respect to the indefinite inner product \([\cdot,\cdot]\), then the closure of the range of \(AB\) is regular.
reviewed by: Carsten Trunk
Mathematics Subject Classification ID: 47A07
Mathematics Subject Classification ID: 46C20
Mathematics Subject Classification ID: 46C50
zbMATH DE Number: 5637030
zbMATH Keywords: \(J\)-projection
zbMATH Keywords: \(J\)-positive projection
zbMATH Keywords: \(J\)-contractive operator
zbMATH Keywords: regular subspace
zbMATH Keywords: Krein space
MaRDI profile type: Publication
full work available at URL: j.laa.2009.03.008
OpenAlex ID: W2031656503
cites work: Q3866685
cites work: Q4692935
cites work: Q4702980
cites work: Interpolation zwischen den Klassen 𝔖p von Operatoren in Hilberträumen
cites work: On projections in a space with an indefinite metric
cites work: Q3436106
DOI: 10.1016/J.LAA.2009.03.008
description: scientific article; zbMATH DE number 5637030