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Invertibility of linear combinations of two idempotents. (English) Zbl 1089.47002

Summary: Let \(P\) and \(Q\) be two idempotents on a Hilbert space. In this note, we prove that the invertibility of the linear combination \(\lambda_1P+\lambda_2Q\) is independent of the choice of \(\lambda_i\), \(i=1,2,\) if \(\lambda_1\lambda_2\neq 0\) and \(\lambda_1+\lambda_2\neq 0.\)

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47L07 Convex sets and cones of operators
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