The Moore-Penrose inverse of differences and products of projectors in a ring with involution

The Moore-Penrose inverse of differences and products of projectors in a ring with involution

In this paper, we study the Moore Penrose inverses of differences and products of projectors in a ring with involution. Some necessary and sufficient conditions for the existence of the Moore Penrose inverse are given. Moreover, the expressions of the Moore Penrose inverses of differences and products of projectors are presented.

___

  • [1] Cheng SZ, Tian YG. Moore-Penrose inverses of products and differences of orthogonal projectors. Acta Sci Math 2003; 69: 533-542.
  • [2] Deng CY. The Drazin inverses of products and differences of orthogonal projections. J Math Anal Appl 2007; 335: 64-71.
  • [3] Deng CY, Wei YM. Further results on the Moore-Penrose invertibility of projectors and its applications. Linear Multilinear Algebra 2012; 60: 109-129.
  • [4] Drazin MP. Commuting properties of generalized inverses. Linear Multilinear Algebra 2013; 61: 1675-1681.
  • [5] Harte RE, Mbekhta M. On generalized inverses in C ∗ -algebras. Studia Math 1992; 103: 71-77.
  • [6] Koliha JJ. The Drazin and Moore-Penrose inverse in C ∗ -algebras. Math Proc R Ir Acad 1999; 99A: 17-27.
  • [7] Koliha JJ, Djordjevi´c D, Cvetkovi´c D. Moore-Penrose inverse in rings with involution. Linear Algebra Appl 2007; 426: 371-381.
  • [8] Koliha JJ, Patr´ıcio P. Elements of rings with equal spectral idempotents. J Austral Math Soc 2002; 72: 137-152.
  • [9] Koliha JJ, Rakoˇcevi´c V, Straˇskraba I. The difference and sum of projectors. Linear Algebra Appl 2004; 388: 279-288.
  • [10] Li Y. The Moore-Penrose inverses of products and differences of projections in a C ∗ -algebra. Linear Algebra Appl 2008; 428: 1169-1177.
  • [11] Penrose R. A generalized inverse for matrices. Proc Cambridge Philos Soc 1955; 51: 406-413.
  • [12] Zhang XX, Zhang SS, Chen JL, Wang L. Moore-Penrose invertibility of differences and products of projections in rings with involution. Linear Algebra Appl 2013; 439: 4101-4109.