Subspaces with a common complement in a separable Hilbert space
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.
scientific article; zbMATH DE number 5617574