Quantum Physics Division

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Generalizations of a property of orthogonal projectors

J. K. Baksalary^{1}
,
O. M. Baksalary^{2}
,
P. Kik^{2}

^{1} Faculty of Mathematics, Informatics and Econometrics, Zielona Góra University, ul. Podgorna 50, 65-246 Zielona Góra, Poland

^{2} Institute of Physics, Adam Mickiewicz University, ul. Umultowska 85, 61-614 Poznań, Poland

**Linear Algebra and Its Applications, 420, 1-8 (2007)**

Abstract:

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 P_{1} and P_{2} are orthogonal projectors, then, in all nontrivial situations, a product of any length having P_{1} and P_{2} as its factors occurring alternately is equal to another such product if and only if P_{1} and P_{2} commute, in which case all products involving P_{1} and P_{2} reduce to the orthogonal projector P_{1}P_{2} (= P_{2}P_{1}). 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.

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