On some linear combinations of hypergeneralized projectors

The concept of a hypergeneralized projector as a matrix H satisfying H2 = H†, where H† denotes the Moore–Penrose inverse of H, was introduced by Groß and Trenkler [Generalized and hypergeneralized projectors, Linear Algebra Appl. 264 (1997) 463–474]. In the present paper, the problem of when a linear combination c1H1 + c2H2 of two hypergeneralized projectors H1, H2 is also a hypergeneralized projector is considered. Although, a complete solution to this problem remains unknown, this article provides characterizations of situations in which (c1H1 + c2H2)2 = (c1H1 + c2H2)† derived under certain commutativity property imposed on matrices H1 and H2. The results obtained substantially generalize those given in the above mentioned paper by Groß and Trenkler.