Let E be an idempotent on the separable Hilbert space \({\mathcal {H}}.\) We firstly introduce the definition of a quasi-self-adjoint pair and characterize all self-adjoint operators S such that (S, E) is a quasi-self-adjoint pair. Then, we present some equivalent conditions for the orthogonal projection P such that (P, E) is a quasi-projection pair. Moreover, the minimum and maximum of the sets of \((2P-I)(I-kE)\) and \((I-kE)(2P-I)\) are obtained, when (P, E) is a quasi-projection pair and \(k\ge 1.\) Particularly, a new characterization of the matched projection for E is given.
