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Invertibility of linear combinations of two idempotents
Authors:Hongke Du  Xiyan Yao  Chunyuan Deng
Institution:College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, People's Republic of China ; College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, People's Republic of China ; College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, People's Republic of China
Abstract: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.$

Keywords:Idempotent  invertibility  linear combination of two idempotents
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