## 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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