On linear combinations of two commuting 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 in [J. Groß, G. Trenkler, Generalized and hypergeneralized projectors, Linear Algebra Appl. 264 (1997) 463–474]. Generalizing substantially preliminary observations given therein, Baksalary et al. in [J.K. Baksalary, O.M. Baksalary, J. Groß, On some linear combinations of hypergeneralized projectors, Linear Algebra Appl. 413 (2006) 264–273] characterized some situations in which a linear combination c1H1+c2H2, where c1,c2∈Cc1,c2∈C and H1, H2 are hypergeneralized projectors such that H1H2=η1H12+η2H22=H2H1 for some η1,η2∈Cη1,η2∈C, inherits the hypergenerality property. In the present paper, the problem considered in the latter paper is revisited and solved completely under the essentially weaker assumption that H1H2=H2H1.