Perturbations of direct complements in Hilbert spaces (Q1001108)

Let \(H\) be an infinite-dimensional complex Hilbert space and \(B(H)\) the set of all bounded linear operators on \(H\). If \(M\) is a closed subspace of \(H\), then \(P_M \in B(H)\) denotes the orthogonal projector on \(M\). Let \(W\) and \(L\) be closed subspaces of a Hilbert space \(H\). Then the set of subspaces of \(H\) is a matrix space if the distance between \(W\) and \(L\) is measured by the gap \(\Theta(W,L)=\|P_W -P_L\|\). In this paper, the largest number \(r_i\) such that all subspaces \(M\) of \(H\) for which \(\Theta(L, M)<r_i\) implies \(\text{dim}(M\cap W) < i\) are determined when \(W\) and \(L\) are subspaces of \(H\) such that \(H=W\otimes L\). This is a generalization of some recent results of \textit{H.\,K.\thinspace Wimmer} [Linear Algebra Appl.\ 287, No.\,1--3, 373--379 (1999; Zbl 0937.15002)]. These new results are obtained with the help of some stability results from Fredholm theory.