Strongly EP elements and the solutions of equations in rings

Year 2022, Volume: 32 Issue: 32, 129 - 140, 16.07.2022

Yue Suı Yimin Huang Junchao Weı

https://doi.org/10.24330/ieja.1143740

Cited By: 1

Abstract

In this paper, we present many new characterizations of strongly EP elements in rings with involution. We especially investigate the strongly EP elements by constructing certain equations and considering the solutions of equations, revealing the existence of solutions of certain equations and the general solutions of some binary equations that play a role in characterizing strongly EP elements. Proofs of relevant conclusions are also given.

Keywords

EP element, strongly EP element, partial isometry, solutions of equations, general solutions

References

• A. Ben-Israel and T. N. E. Greville , Generalized Inverses: Theory and Applications, CMS Books in Mathematics, 2nd edition, Springer, Heidelberg, 2003.
• E. Boasso and V. Rakocevic, Characterizations of EP and normal Banach algebra elements, Linear Algebra Appl., 435(2) (2011), 342-353.
• W. Chen, On EP elements, normal elements and partial isometries in rings with involution, Electron. J. Linear Algebra, 23 (2012), 553-561.
• A. Dajic, Common solutions of linear equations in a ring, with applications, Electron. J. Linear Algebra, 30 (2015), 66-79.
• A. Dajic and J. J. Koliha, Equations $ax=c$ and $xb=d$ in rings and rings with involution with applications to Hilbert space operators, Linear Algebra Appl., 429(7) (2008), 1779-1809.
• R. E. Hartwig, Block generalized inverses, Arch. Ration. Mech. Anal., 61(3) (1976), 197-251.
• S. Karanasios, EP elements in rings and semigroup with involution and C*-algebras, Serdica Math. J., 41 (2015), 83-116.
• J. J. Koliha and D. Djordjevic, D. Cvetkovic, Moore-Penrose inverse in rings with involution, Linear Algebra Appl., 426(2-3) (2007), 371-381.
• M. Mbekhta, Partial isometries and generalized inverses, Acta Sci. Math. (Szeged), 70 (2004), 767-781.
• D. Mosic and D. S. Djordjevic, Further results on partial isometries and EP elements in rings with involution, Math. and Comput. Modelling, 54(1-2) (2011), 460-465.
• D. Mosic, D. S. Djordjevic and J. J. Koliha, EP elements in rings, Linear Algebra Appl., 431(5-7) (2009), 527-535.
• D. Mosic and D. S. Djordjevic, Partial isometries and EP elements in rings with involution, Electron. J. Linear Algebra, 18 (2009), 761-772.
• R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc., 51 (1955), 406-413.
• S. Z. Xu, J. L. Chen and J. Benitez, EP elements in rings with involution, Bull. Malays. Math. Sci. Soc., 42(6) (2019), 3409-3426.
• Z. C. Xu, R. J. Tang and J. C. Wei, Strongly EP elements in a ring with involution, Filomat, 34(6) (2020), 2101-2107. Q. W. Wang and Z. H. He, Solvability conditions and general solution for the fixed Sylvester equations, Automatica, 49 (2013), 2713-2719.
• X. Zhang, A system of generalized sylvester quaternion matrix equations and its applications, Appl. Math. Comput., 273 (2016), 74-81.
• R. J. Zhao, H. Yao and J. C. Wei, Characterizations of partial isometries and two special kinds of EP elements, Czechoslovak Math. J., 70(2) (2020), 539-551.