Research ArticleOpen AccessGoogle Scholar indexed
Mixed-Type Reverse Order Laws for Generalized Inverses over Hilbert Space
College of Mathematics and Satistics, Shangqiu Normal University, Shangqiu, China
College of Mathematics and Satistics, Shangqiu Normal University, Shangqiu, China
- 1 College of Mathematics and Satistics, Shangqiu Normal University, Shangqiu, China
- 2 College of Mathematics and Satistics, Shangqiu Normal University, Shangqiu, China
Copy link · social · email
Abstract
In this paper, by using a block-operator matrix technique, we study mixed-type reverse order laws for {1,3}-, {1,2,3}- and {1,3,4}-generalized inverses over Hilbert spaces. It is shown that and when the ranges of are closed. Moreover, a new equivalent condition of is given.
Keywords{123}-Reverse{134}-ReverseReverse Order LawBlock-Operator Matrix
- Cvetkovic-Ilic, D.S. and Djikic, M. (2016) Various Solutions to Reverse Order Law Problems. Linear and Multilinear Algebra, 64, 1207-1219. https://doi.org/10.1080/03081087.2015.1082956
- Greville, T.N.E. (1966) Note on the Generalized Inverse of a Matrix Product. Society for Industrial and Applied Mathematics Review, 8, 518-521. https://doi.org/10.1137/1008107
- Galperin, A.M. and Waksman, Z. (1980) On Pseudo Inverse of Operator Products. Linear Algebra and Applications, 33, 123-131.
- Izumino, S. (1982) The Product of Operators with Closed Range and an Extension of the Reverse Order Law. Tohoku Mathematical Journal, 34, 43-52. https://doi.org/10.2748/tmj/1178229307
- Liu, X.J., Huang, S. and Cvetkovic-Ilic, D.S. (2012) Mixed-Type Reverse-Order Laws for {1,3,4}-Generalized Inverses over Hilbert Spaces. Applied Mathematics and Computation, 218, 8570-8577.
- Liu, X.J., Wu, S.X. and Cvetkovic-Ilic, D.S. (2013) New Results on Reverse Order Law for {1,2,3}- and {1,2,4}-Inverses of Bounded Operators. Mathematics of Computation, 82, 1597-1607. https://doi.org/10.1090/S0025-5718-2013-02660-9
- Wang, M., Wei, M. and Jia, Z. (2009) Mixed-Type Reverse-Order Law of . Linear Algebra and Applications, 430, 1691-1699.
- Wang, J., Zhang, H.Y. and Ji, G.X. (2010) A Generalized Reverse Order Law for the Products of Two Operators. Journal of Shaanxi Normal University (Natural Science), 38, 13-17.
- Yang, H. and Lu, X.F. (2011) Mixed-Type Reverse-Order Laws of and . Applied Mathematics and Computation, 217, 10361-10367. https://doi.org/10.1016/j.amc.2011.05.046
- Zhang, H.Y. and Si, H.Y. (2016) Reverse Order Laws for {1,2,3}-Inverse of Two-Operator Product. Journal of Sichuan Normal University (Natural Science), 39, 671-677.