Subspaces with a common complement in a separable Hilbert space
type: Item
instance of: scholarly article
title: Subspaces with a common complement in a separable Hilbert space
zbMATH Open document ID: 1182.46012
author: Dimosthenis Drivaliaris
author: Nikos Yannakakis
published in: Integral Equations and Operator Theory
publication date: 2009-10-19T00:00:00Z
review text: Two closed subspaces \(M\) and \(N\) of a Hilbert space \(H\) are said to have a common complement in \(H\) if there exists a closed subspace \(K\) of \(H\) such that \(M\oplus K = N \oplus K = H\). If \(H\) is finite-dimensional, then \(M\) and \(N\) have a common complement if and only if \(\dim(M)=\dim(N)\). The situation in the infinite-dimensional case is different. \textit{M.\,Lauzon} and \textit{S.\,Treil} [J.~Funct.\ Anal.\ 212, No.\,~2, 500--512 (2004; Zbl 1065.46018)] proved that if \(H\) is a separable Hilbert space, then the subspaces \(M\) and \(N\) of \(H\) have a common complement in \(H\) if and only if \(\dim(M\cap N^\perp)=\dim(M^\perp\cap N)\) or the restriction of the operator \(I-G^*G\) to \(\ker(G)\) is non-compact. The authors of the paper under review give an alternative proof of this result. They also present a simple example of a double triangle subspace lattice, which is not similar to an operator double triangle [see also \textit{W.\,E.\thinspace Longstaff} and \textit{O.\,Panaia}, Proc.\ Am.\ Math.\ Soc.\ 126, No.\,10, 3019--3026 (1998; Zbl 0902.46009)] as well as a characterization of pairs of subspaces in generic position which are not completely asymptotic to each other.
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 46C05
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 5617574
zbMATH Keywords: common complement
zbMATH Keywords: algebraic complement
zbMATH Keywords: pair of subspaces
zbMATH Keywords: relative position
zbMATH Keywords: generic position
zbMATH Keywords: equivalently positioned subspaces
zbMATH Keywords: completely asymptotic subspaces
zbMATH Keywords: Hilbert space geometry
zbMATH Keywords: double triangle subspace lattice
MaRDI profile type: Publication
full work available at URL: s00020-008-1622-5
OpenAlex ID: W1979633409
DOI: 10.1007/S00020-008-1622-5
description: scientific article; zbMATH DE number 5617574