On invertibility of combinations of k-potent operators

Abstract

Some properties of combinations c1P1+c2P2-c3P1sP2k-1-s, where P1 and P2 are two different nonzero k-potents, c1, c2 and c3 are three nonzero complex numbers and positive integers k⩾2 and s⩽k-1, 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.