Generalizations of a property of orthogonal projectors

Generalizing the result in Lemma of Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Commutativity of projectors, Linear Algebra Appl. 341 (2002) 129–142], Baksalary et al. [J.K. Baksalary, O.M. Baksalary, T. Szulc, Linear Algebra Appl. 354 (2002) 35–39] have shown that if P1 and P2 are orthogonal projectors, then, in all nontrivial situations, a product of any length having P1 and P2 as its factors occurring alternately is equal to another such product if and only if P1 and P2 commute, in which case all products involving P1 and P2 reduce to the orthogonal projector P1P2 (= P2P1). In the present paper, further generalizations of this property are established. They consist in replacing a product of the type specified above, appearing on the left-hand side (say) of the equality under considerations, by an affine combination of two or three such products. Comments on the problem when the number of components in a combination exceeds three are also given.