Abstract
In this paper, necessary and sufficient conditions of the one-sided reverse order law \((ab)^{\tiny {\textcircled {\tiny \#}}}=b^{\tiny {\textcircled {\tiny \#}}}a^{\tiny {\textcircled {\tiny \#}}}\), the two-sided reverse order law \((ab)^{\tiny {\textcircled {\tiny \#}}}=b^{\tiny {\textcircled {\tiny \#}}}a^{\tiny {\textcircled {\tiny \#}}}\) and \((ba)^{\tiny {\textcircled {\tiny \#}}}=a^{\tiny {\textcircled {\tiny \#}}}b^{\tiny {\textcircled {\tiny \#}}}\) for the core inverse are given in rings with involution. In addition, the mixed-type reverse order laws, such as \((ab)^{\#}=b^{\tiny {\textcircled {\tiny \#}}}(abb^{\tiny {\textcircled {\tiny \#}}})^{\tiny {\textcircled {\tiny \#}}}\), \(a^{\tiny {\textcircled {\tiny \#}}}=b(ab)^{\#}\) and \((ab)^{\#}=b^{\tiny {\textcircled {\tiny \#}}}a^{\tiny {\textcircled {\tiny \#}}}\), are also considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baksalary, O.M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58, 681–697 (2010)
Baksalary, O.M., Trenkler, G.: Problem 48–1: reverse order law for the core inverse. Image 48, 40 (2012)
Benítez, J., Liu, X.J., Zhong, J.: Some results on matrix partial orderings and reverse order law. Electron. J. Linear Algebra 20, 254–273 (2010)
Chen, J.L., Zhu, H.H., Patrício, P., Zhang, Y.L.: Characterizations and representations of core and dual core inverses. Can. Math. Bull. 60, 269–282 (2017)
Cline, R.E.: An application of representation for the generalized inverse of a matrix, MRC Technical Report 592, (1965)
Cohen, N., Herman, E.A., Jayaraman, S.: Solution to problem 48–1: reverse order law for the core inverse. Image 49, 46–47 (2012)
Cvetković-Ilić, D.S., Djikić, M.: Various solutions to reverse order law problems. Linear Multilinear Algebra 64, 1207–1219 (2016)
Cvetković-Ilić, D.S., Harte, R.: Reverse order laws in \(C^{\ast }\)-algebras. Linear Algebra Appl. 434, 1388–1394 (2011)
Cvetković-Ilić, D.S., Pavlović, V.: A comment on some recent results concerning the reverse order law for \(\{1, 3, 4\}\)-inverses. Appl. Math. Comput. 217, 105–109 (2010)
Deng, C.Y.: Reverse order law for the group inverses. J. Math. Anal. Appl. 382, 663–671 (2011)
Dinčić, N.Č., Djordjević, D.S.: Hartwig’s triple reverse order law revisited. Linear Multilinear Algebra 62, 918–924 (2014)
Drazin, M.P.: Pseudo-inverses in associative rings and semigroups. Am. Math. Mon. 65, 506–514 (1958)
Greville, T.N.E.: Note on the generalized inverse of a matrix product. SIAM Rev. 8, 518–521 (1966)
Hartwig, R.E.: Block generalized inverses, Arch. Ration. Mech. Anal. 61, 197–251 (1976)
Hartwig, R.E., Luh, J.: A note on the group structure of unit regular ring elements. Pac. J. Math. 71, 449–461 (1977)
Izumino, S.: The product of operators with closed range and an extension of the reverse order law. Tohoku Math. J. 34, 43–52 (1982)
Liu, X.J., Fu, S.Q., Yu, Y.M.: An invariance property of operator products related to the mixed-type reverse order laws. Linear Multilinear Algebra 64, 885–896 (2016)
Liu, X.J., Zhang, M., Benítez, J.: Further results on the reverse order law for the group inverse in rings. Appl. Math. Comput. 229, 316–326 (2014)
Mary, X.: Reverse order law for the group inverse in semigroups and rings. Commun. Algebra 43, 2492–2508 (2015)
Mosić, D., Dinčić, N.Č.: Reverse order law \((ab)^{\dagger }=b^{\dagger }(a^{\dagger }abb^{\dagger })^{\dagger }a^{\dagger }\) in rings with involution. Filomat 28, 1791–1815 (2014)
Mosić, D., Djordjević, D.S.: Reverse order law for the Moore-Penrose inverse in \(C^{\ast }\)-algebras. Electron. J. Linear Algebra 22, 92–111 (2011)
Mosić, D., Djordjević, D.S.: Some results on the reverse order law in rings with involution. Aequ. Math. 83, 271–282 (2012)
Mosić, D., Djordjević, D.S.: Reverse order law for the group inverse in rings. Appl. Math. Comput. 219, 2526–2534 (2012)
Mosić, D., Djordjević, D.S.: The reverse order law \((ab)^{\#}=b^{\dagger }(a^{\dagger }abb^{\dagger })^{\dagger }a^{\dagger }\) in rings with involution, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A Matemáticas 109, 257–265 (2015)
Pavlović, V., Cvetković-Ilić, D.S.: Applications of completions of operator matrices to reverse order law for \(\{1\}\)-inverses of operators on Hilbert spaces. Linear Algebra Appl. 484, 219–236 (2015)
Penrose, R.: A generalized inverse for matrices. Proc. Camb. Philos. Soc. 51, 406–413 (1955)
Puystjens, P., Hartwig, R.E.: The group inverse of a companion matrix. Linear Multilinear Algebra 43, 137–150 (1997)
Rakić, D.S., Dinčić, N.Č., Djordjević, D.S.: Group, Moore–Penrose, core and dual core inverse in rings with involution. Linear Algebra Appl. 463, 115–133 (2014)
Tian, Y.: On mixed-type reverse-order laws for the Moore–Penrose inverse of a matrix product. Int. J. Math. Math. Sci. 58, 3103–3116 (2004)
Wang, H.X., Liu, X.J.: Characterizations of the core inverse and the core ordering. Linear Multilinear Algebra 63, 1829–1836 (2015)
Xu, S.Z., Chen, J.L., Zhang, X.X.: New characterizations for core inverses in rings with involution. Front. Math. China 12, 231–246 (2017)
Acknowledgements
This research was supported by China Postdoctoral Science Foundation (No. 2018M632385), the National Natural Science Foundation of China (No.11771076), the Natural Science Foundation of Jiangsu Province (No.BK20141327), the Portuguese Funds through FCT-“Fundação para a Ciência e a Tecnologia”, within the project UID/MAT/00013/2013.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zou, H., Chen, J. & Patrício, P. Reverse Order Law for the Core Inverse in Rings. Mediterr. J. Math. 15, 145 (2018). https://doi.org/10.1007/s00009-018-1189-6
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-018-1189-6