On linear combinations of two commuting hypergeneralized projectors

The concept of a hypergeneralized projector as a matrix H satisfying H = H†, where H† denotes the Moore–Penrose inverse of H, was introduced by Groß and Trenkler in [Generalized and hypergeneralized projectors, Linear Algebra Appl. 264 (1997) 463-474]. Generalizing substantially some preliminary observations given therein, Baksalary et al. in [On some linear combinations of hypergeneralized projectors, Linear Algebra Appl. 413 (2006) 264-273] characterized all situations in which a linear combination c1H1+c2H2, where c1, c2 ∈ C andH1, H2 are hypergeneralized projectors such that H1H2 = η1H 2 1 + η2H 2 2 = H2H1 for some η1, η2 ∈ C, inherits the hypergenerality property. In the present paper, the problem considered in the latter paper is revisited and solved under the essentially weaker assumption that H1H2 = H2H1. AMS classification: 15A57; 15A09; 15A27