Du, Hongke; Yao, Xiyan; Deng, Chunyuan Invertibility of linear combinations of two idempotents. (English) Zbl 1089.47002 Proc. Am. Math. Soc. 134, No. 5, 1451-1457 (2006). 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 \(\lambda_1+\lambda_2\neq 0.\) Cited in 1 ReviewCited in 20 Documents MSC: 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47L07 Convex sets and cones of operators Keywords:Hilbert space; invertibility; linear combination PDF BibTeX XML Cite \textit{H. Du} et al., Proc. Am. Math. Soc. 134, No. 5, 1451--1457 (2006; Zbl 1089.47002) Full Text: DOI OpenURL References: [1] Jerzy K. Baksalary and Oskar Maria Baksalary, Nonsingularity of linear combinations of idempotent matrices, Linear Algebra Appl. 388 (2004), 25 – 29. · Zbl 1081.15017 [2] Oskar Maria Baksalary, Idempotency of linear combinations of three idempotent matrices, two of which are disjoint, Linear Algebra Appl. 388 (2004), 67 – 78. · Zbl 1081.15019 [3] Man Duen Choi and Pei Yuan Wu, Convex combinations of projections, Linear Algebra Appl. 136 (1990), 25 – 42. · Zbl 0721.15007 [4] G. Corach, A. Maestripieri, and D. Stojanoff, Generalized Schur complements and oblique projections, Linear Algebra Appl. 341 (2002), 259 – 272. Special issue dedicated to Professor T. Ando. · Zbl 1015.47014 [5] Jürgen Groß and Götz Trenkler, Nonsingularity of the difference of two oblique projectors, SIAM J. Matrix Anal. Appl. 21 (1999), no. 2, 390 – 395. · Zbl 0946.15020 [6] Stanislav Kruglyak, Vyacheslav Rabanovich, and Yuriĭ Samoĭlenko, Decomposition of a scalar matrix into a sum of orthogonal projections, Linear Algebra Appl. 370 (2003), 217 – 225. · Zbl 1028.15010 [7] J. J. Koliha, V. Rakočević, and I. Straškraba, The difference and sum of projectors, Linear Algebra Appl. 388 (2004), 279 – 288. · Zbl 1060.15011 [8] Eugene Spiegel, Sums of projections, Linear Algebra Appl. 187 (1993), 239 – 249. · Zbl 0776.15015 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.