Invertibility of linear combinations of two idempotents

Abstract. Let P and Q be two idempotents on a Hilbert space. In this note, we prove that the invertibility of the linear combination λ1P + λ2Q is independent of the choice of λi, i = 1, 2, if λ1λ2 = 0 and λ1 + λ2 = 0. Let H be a Hilbert space, and let all bounded linear operators on H be denoted by B(H). An operator P ∈ B(H) is said to be idempotent if P 2 = P. The set P of all idempotents in B(H) is invariant under similarity; that is, if P ∈ P and S ∈ B(H) is an invertible operator, then S−1PS is still an idempotent since (S−1PS)2 = S−1PSS−1PS = S−1P 2S = S−1PS. An idempotent P is called an orthogonal projection if P 2 = P = P ∗, where P ∗ is the adjoint of P. Moreover, for an idempotent P ∈ P, there exists an invertible operator U ∈ B(H) such that U−1PU is an orthogonal projection. In fact, if P ∈ P, then P can be written in the form of P