On properties of generalized quadratic operators

In this paper, we investigate the set ω(P) of generalized quadratic operators A satisfying the equation A2=αA+βP for all complex numbers α and β and for an idempotent operator P such that AP=PA=A. Furthermore, the close relationship between the operator A∈ω(P) and the idempotent operator P are established and expressions for the inverse, the Moore–Penrose inverse and the Drazin inverse of A∈ω(P) are given. Some related results are also obtained.