×
Xu, Sanzhang; Chen, Jianlong; Benítez, Julio

EP elements in rings with involution. (English) Zbl 1432.16035

Bull. Malays. Math. Sci. Soc. (2) 42, No. 6, 3409-3426 (2019).
Summary: Let \(R\) be a with involution. We first show that the EP elements in \(R\) can be characterized by three equations. Namely, let \(a\in R\), then \(a\) is EP if and only if there exists \(x\in R\) such that \((xa)^\ast=xa, xa^2=a\) and \(ax^2=x.\) Any EP element in \(R\) is core invertible and Moore-Penrose invertible. We give more equivalent conditions for a core (Moore-Penrose) to be an EP element. Finally, any EP element is characterized in terms of the \(n\)-EP property, which is a generalization of the bi-EP property.

Cited in 15 Documents

MSC:

16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16U90 Generalized inverses (associative rings and algebras)

Keywords:

core inverse; ep; bi-EP; \(n\)-EP
Cite Review PDF
Full Text: DOI arXiv Link

References:

[1] Baksalary, O.M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58(6), 681-697 (2010) · Zbl 1202.15009 · doi:10.1080/03081080902778222
[2] Benítez, J.: Moore-Penrose inverses and commuting elements of \(C^*\)-algebras. J. Math. Anal. Appl. 345(2), 766-770 (2008) · Zbl 1152.46046 · doi:10.1016/j.jmaa.2008.04.062
[3] Bhaskara Rao, K.P.S.: The Theory of Generalized Inverses Over Commutative Rings. Taylor and Francis, London (2002) · Zbl 0992.15003
[4] Boasso, E.: On the Moore-Penrose inverse, EP Banach space operators, and EP Banach algebra elements. J. Math. Anal. Appl. 339(2), 1003-1014 (2008) · Zbl 1129.47001 · doi:10.1016/j.jmaa.2007.07.059
[5] Chen, W.X.: On EP elements, normal elements and partial isometries in rings with involution. Electron. J. Linear Algebra 23, 553-561 (2012) · Zbl 1266.16044
[6] Drivaliaris, D., Karanasios, S., Pappas, D.: Factorizations of EP operators. Linear Algebra Appl. 429, 1555-1567 (2008) · Zbl 1155.47001 · doi:10.1016/j.laa.2008.04.026
[7] Hartwig, R.E.: Block generalized inverses. Arch. Retion. Mech. Anal. 61(3), 197-251 (1976) · Zbl 0335.15004 · doi:10.1007/BF00281485
[8] Hartwig, R.E., Spindelböck, K.: Matrices for which \(A^*\) and \(A^{\dagger }\) commute. Linear Multilinear Algebra 14(3), 241-256 (1983) · Zbl 0525.15006 · doi:10.1080/03081088308817561
[9] Koliha, J.J., Patrício, P.: Elements of rings with equal spectral idempotents. J. Aust. Math. Soc. 72(1), 137-152 (2002) · Zbl 0999.16025 · doi:10.1017/S1446788700003657
[10] Mosić, D., Djordjević, D.S., Koliha, J.J.: EP elements in rings. Linear Algebra Appl. 431, 527-535 (2009) · Zbl 1186.16047 · doi:10.1016/j.laa.2009.02.032
[11] Mosić, D., Djordjević, D.S.: New characterizations of EP, generalized normal and generalized Hermitian elements in rings. Appl. Math. Comput. 218, 6702-6710 (2012) · Zbl 1251.15008
[12] Patrício, P., Puystjens, R.: Drazin-Moore-Penrose invertiblity in rings. Linear Algebra Appl. 389, 159-173 (2004) · Zbl 1080.15004 · doi:10.1016/j.laa.2004.04.006
[13] Rakić, D.S., Dinčić, Nebojša Č., Djordjević, D.S.: Group, Moore-Penrose, core and dual core inverse in rings with involution. Linear Algebra Appl. 463, 115-133 (2014) · Zbl 1297.15006 · doi:10.1016/j.laa.2014.09.003
[14] von Neumann, J.: On regular rings. Proc. Natl. Acad. Sci. USA 22(12), 707-713 (1936) · Zbl 0015.38802 · doi:10.1073/pnas.22.12.707
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.