Abstract
We investigate some necessary and sufficient conditions for the hybrid reverse order law \({(ab)^\# = b^{\dag} a^{\dag}}\) in rings with involution. Assuming that a and b are Moore–Penrose invertible, we present an equivalent condition for the product ab to be an EP element.
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References
Ben-Israel A., Greville T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)
Cao C., Zhang X., Tang X.: Reverse order law of group inverses of products of two matrices. Appl. Math. Comput. 158, 489–495 (2004)
Cline R.E.: Note on the generalized inverse of the product of matrices. SIAM Rev. 6, 57–58 (1964)
Deng C.Y.: Reverse order law for the group inverses. J. Math. Anal. Appl. 382(2), 663–671 (2011)
Djordjević D.S.: Unified approach to the reverse order rule for generalized inverses. Acta Sci. Math. (Szeged) 67, 761–776 (2001)
Greville T.N.E.: Note on the generalized inverse of a matrix product. SIAM Rev. 8, 518–521 (1966)
Izumino S.: The product of operators with closed range and an extension of the reverse order law. Tohoku Math. J. 34, 43–52 (1982)
Koliha J.J., Djordjević D.S., Cvetković Ilić D.: Moore-Penrose inverse in rings with involution. Linear Algebra Appl. 426, 371–381 (2007)
Koliha J.J., Patrício P.: Elements of rings with equal spectral idempotents. J. Australian Math. Soc. 72, 137–152 (2002)
Mosić D., Djordjević D.S.: Further results on the reverse order law for the Moore-Penrose inverse in rings with involution. Appl. Math. Comput. 218, 1478–1483 (2011)
Mosić D., Djordjević D.S.: Reverse order law for the Moore–Penrose inverse in C*-algebras. Electron. J. Linear Algebra 22, 92–111 (2011)
Mosić, D., Djordjević, D.S.: Reverse order laws for the group inverses in rings with involution. Preprint
Takane Y., Tian Y., Yanai H.: On reverse-order laws for least-squares g-inverses and minimum norm g-inverses of a matrix product. Aequat. Math. 73, 56–70 (2007)
Tian Y.: Using rank formulas to characterize equalities for Moore-Penrose inverses of matrix products. Appl. Math. Comput. 147, 581–600 (2004)
Tian Y.: The equivalence between \({(AB)^{\dag} = B^{\dag} A^{\dag}}\) and other mixed-type reverse-order laws. Int. J. Math. Edu. Sci. Technol. 37(3), 331–339 (2006)
Xiong, Z., Qin, Y.: Note on the weighted generalized inverse of the product of matrices. J. Appl. Math. Comput. doi:10.1007/s12190-009-0370-2
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The authors are supported by the Ministry of Education and Science, Republic of Serbia, grant no. 174007.
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Mosić, D., Djordjević, D.S. Some results on the reverse order law in rings with involution. Aequat. Math. 83, 271–282 (2012). https://doi.org/10.1007/s00010-012-0125-2
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DOI: https://doi.org/10.1007/s00010-012-0125-2