Properties of core-EP order in rings with involution

Abstract

We study properties of a relation in *-rings, called the core-EP (pre)order which was introduced by H. Wang on the set of all n × n complex matrices [Linear Algebra Appl., 2016, 508: 289–300] and has been recently generalized by Y. Gao, J. Chen, and Y. Ke to *-rings [Filomat, 2018, 32: 3073–3085]. We present new characterizations of the core-EP order in *-rings with identity and introduce the notions of the dual core-EP decomposition and the dual core-EP order in-rings.

Keywords

Drazin inverse / core-EP decomposition / pre-order / ring

References

[1] Baksalary O M, Trenkler G. Core inverse of matrices. Linear Multilinear Algebra, 2010, 58(6): 681–697

[2] Campbell S L, Meyer C D. Generalized Inverse of Linear Transformations. Classics Appl Math, Vol 56. Philadelphia: SIAM, 2009

[3] Drazin M P. Pseudo-inverse in associative rings and semigroups. Amer Math Monthly, 1958, 65: 506–514

[4] Gao Y, Chen J. Pseudo core inverses in rings with involution. Comm Algebra, 2018, 46(1): 38–50

[5] Gao Y, Chen J, Ke Y. *-DMP elements in-semigroups and *-rings. Filomat, 2018, 32: 3073–3085

[6] Harte R, Mbekhta M. On generalized inverses in C*-algebras. Studia Math, 1992, 103(1): 71–77

[7] Hartwig R E, Levine J. Applications of the Drazin inverse to the Hill cryptographic system, Part III. Cryptologia, 1981, 5(2): 67–77

[8] Manjunatha Prasad K, Mohana K S. Core-EP inverse. Linear Multilinear Algebra, 2014, 62(6): 792–802

[9] Marovt J. Orders in rings based on the core-nilpotent decomposition. Linear Multi-linear Algebra, 2018, 66(4): 803–820

[10] Miller V A, Neumann M. Successive overrelaxation methods for solving the rank deficient linear least squares problem. Linear Algebra Appl, 1987, 88-89: 533–557

[11] Mitra S K, Bhimasankaram P, Malik S B. Matrix Partial Orders, Shorted Operators and Applications. Series in Algebra, Vol 10. London: World Scientific, 2010

[12] Rakić D S, Dinčić N Č, Djordjević D S. Group, Moore-Penrose, core and dual core inverse in rings with involution. Linear Algebra Appl, 2014, 463: 115–133

[13] Rakić D S, Djordjević D S. Star, sharp, core and dual core partial order in rings with involution. Appl Math Comput, 2015, 259: 800–818

[14] Rao C R, Mitra S K. Generalized Inverse of Matrices and Its Application. New York: Wiley, 1971

[15] Wang H. Core-EP decomposition and its applications. Linear Algebra Appl, 2016, 508: 289–300

[16] Xu S, Chen J, Zhang X. New characterizations for core inverses in rings with involution. Front Math China, 2017, 12(1): 231–246

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