Characterizations and linear combinations of k-generalized projectors

We extend generalized projectors (introduced by Groß and Trenkler [Generalized and hypergeneralized projectors, Linear Algebra Appl. 264 (1997) 463-474]) to k-generalized projectors and we characterize them obtaining results in the aforesaid paper as a consequence. Moreover, we list all situations when a linear combination of commuting k-generalized projectors is a k-generalized projector. The method for obtaining this result permits to give a revisited version of the main result by Baksalary and Baksalary [On linear combinations of generalized projectors, Linear Algebra Appl. 388 (2004) 17-24]. In addition, the case of orthogonal projectors is also analyzed.