[1] Ma Yilin, Chen Jianlong, Han Ruizhu,. Characterizations of EP, normal and Hermitian elementsin rings using generalized inverses [J]. Journal of Southeast University (English Edition), 2017, 33 (2): 249-252. [doi:10.3969/j.issn.1003-7985.2017.02.020]

Copy

Characterizations of EP, normal and Hermitian elementsin rings using generalized inverses()

Share：

Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

33

Issue:

2017 2

Page:

249-252

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2017-06-30

Info

Title:

Characterizations of EP, normal and Hermitian elementsin rings using generalized inverses

Author(s):

Ma Yilin¹;  Chen Jianlong²;  Han Ruizhu¹

¹School of Economics and Management, Southeast University, Nanjing 211189, China
²School of Mathematics, Southeast University, Nanjing 211189, China

Keywords:

equal projection(EP)elements;  normal elements;  Hermitian elements;  core inverse;  Moore-Penrose inverse;  group inverse

PACS:

O151.2

DOI:

10.3969/j.issn.1003-7985.2017.02.020

Abstract:

The properties and some equivalent characterizations of equal projection(EP), normal and Hermitian elements in a ring are studied by the generalized inverse theory. Some equivalent conditions that an element is EP under the existence of core inverses are proposed. Let a∈R^, then a is EP if and only if aa^a#=a^#aa^. At the same time, the equivalent characterizations of a regular element to be EP are discussed. Let a∈R, then there exist b∈R such that a=aba and a is EP if and only if a∈R^, a⁼a^baba. Similarly, some equivalent conditions that an element is normal under the existence of core inverses are proposed. Let a∈R^, then a is normal if and only if a^*a⁼a^aa^*. Also, some equivalent conditions of normal and Hermitian elements in rings with involution involving powers of their group and Moore-Penrose inverses are presented. Let a∈R^∩R^#, n∈N, then a is normal if and only if a^*a⁽a^#)ⁿn=a^#a^*(a⁾ⁿn. The results generalize the conclusions of Mosi et al.

References:

[1] Baksalary O M, Trenkler G. Core inverse of matrices [J]. Linear and Multilinear Algebra, 2010, 58(6): 681-697. DOI:10.1080/03081080902778222.
[2] Raki D S, Dini N, Djordjevi D S. Group, Moore-Penrose, core and dual core inverse in rings with involution [J]. Linear Algebra and Its Applications, 2014, 463: 115-133. DOI:10.1016/j.laa.2014.09.003.
[3] Xu S Z, Chen J L, Zhang X X. New characterizations for core inverses in rings with involution [J]. Frontiers of Mathematics in China, 2016, 12(1): 231-246.DOI:10.1007/s11464-016-0591-2.
[4] Baksalary O M, Trenkler G. Characterizations of EP, normal and Hermitian matrices [J]. Linear and Multilinear Algebra, 2008, 56(3): 299-304. DOI:10.1080/03081080600872616.
[5] Cheng S, Tian Y. Two sets of new characterizations for normal and EP matrices [J]. Linear Algebra and Its Applications, 2003, 375: 181-195. DOI:10.1016/s0024-3795(03)00650-5.
[6] Djordjevi D S. Characterization of normal, hyponormal and EP operators [J]. Journal of Mathematical Analysis and Applications, 2007, 329(2): 1181-1190. DOI:10.1016/j.jmaa.2006.07.008.
[7] Djordjevi D S, Koliha J J. Characterizing Hermitian, normal and EP operators [J]. Filomat, 2007, 21(1): 39-54. DOI:10.2298/fil0701039d.
[8] Koliha J J. Elements of C^*-algebras commuting with their Moore-Penrose inverse [J]. Studia Math, 2000, 139: 81-90.
[9] Mosi D, Djordjevi D S, Koliha J J. EP elements in rings [J]. Linear Algebra and its Applications, 2009, 431(5): 527-535. DOI:10.1016/j.laa.2009.02.032.
[10] Mosi D, Djordjevi D S. Partial isometries and EP elements in rings with involution [J]. Electronic Journal of Linear Algebra, 2009, 18(1): 761-772. DOI:10.13001/1081-3810.1343.
[11] Elsner L, Ikramov K D. Normal matrices: An update [J]. Linear Algebra and Its Applications, 1998, 285(1): 291-303. DOI:10.1016/s0024-3795(98)10161-1.
[12] Grone R, Johnson C R, Sa E M, et al. Normal matrices [J]. Linear Algebra and Its Applications, 1987, 87: 213-225. DOI:10.1016/0024-3795(87)90168-6.
[13] Mosi D, Djordjevi D S. Moore-Penrose-invertible normal and Hermitian elements in rings [J]. Linear Algebra and Its Applications, 2009, 431(5/6/7): 732-745. DOI:10.1016/j.laa.2009.03.023.
[14] Mosi D, Djordjevi D S. New characterizations of EP, generalized normal and generalized Hermitian elements in rings [J]. Applied Mathematics and Computation, 2012, 218(12): 6702-6710. DOI:10.1016/j.amc.2011.12.030.
[15] Drazin M P. Pseudo-inverse in associative rings and semigroups [J]. The American Mathematical Monthly, 1958, 65(7): 506-514. DOI:10.2307/2308576.
[16] Harte R E, Mbekhta M. On generalized inverses in C^*-algebras [J]. Studia Mathematica, 1992, 103: 71-77.
[17] Xu S Z, Chen J L, Benítez J. EP elements in rings with involution [J]. arXiv preprint arXiv:1602.08184, 2016. https://arxiv.org/abs/1602.08184.

Memo

Memo:

Biographies: Ma Yilin(1992─), male, graduate; Chen Jianlong(corresponding author), male, doctor, professor, jlchen@seu.edu.cn.
Foundation items: The National Natural Science Foundation of China(No.11371089), the Specialized Research Fund for the Doctoral Program of Higher Education(No.20120092110020), the Natural Science Foundation of Jiangsu Province(No.BK20141327).
Citation: Ma Yilin, Chen Jianlong, Han Ruizhu. Characterizations of EP, normal and Hermitian elements in rings using generalized inverses[J].Journal of Southeast University(English Edition), 2017, 33(2):249-252.DOI:10.3969/j.issn.1003-7985.2017.02.020.

Common functions