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Inverting the difference of Hilbert space projections. (English) Zbl 0901.46019

Summary: Let \(R\) and \(K\) be subspaces of a Hilbert space \(H\), and let \(P_R\) and \(P_K\) denote the orthogonal projections of \(H\) onto these subspaces. When is the operator \(P_R-P_K\) invertible? We show here that the obvious necessary condition, \(H=R\oplus K\), is sufficient as well. We also find the inverse.

MSC:

46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
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