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

Buckholtz, Don

Am. Math. Mon. 104, No. 1, 60-61 (1997).

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)

Keywords:

difference of Hilbert space projections; orthogonal projections; invertible