On invertibility of combinations of k-potent operators

Some properties of combinations , where P1 and P2 are two different nonzero k-potents, c1, c2 and c3 are three nonzero complex numbers and positive integers and , are obtained. Furthermore, the invertibility, the group invertibility and the k-potency of the linear combinations of k-potents are investigated, under certain commutativity properties imposed on them.