ABSTRACT
A weak group element is introduced in a proper *-ring. Several equivalent conditions of weak group elements are investigated. We prove that an element is pseudo core invertible if it is both partial isometry and weak group invertible. Reverse order law and additive property of the weak group inverse are presented. Finally, under certain assumption on a, equivalent conditions of are presented by using the normality of the group invertible part of an element in its group-EP decomposition.
Acknowledgments
We would like to thank the Referees for their constructive suggestions which help us to improve considerably the presentation of the paper.
This research was supported by the National Natural Science Foundation of China (No. 11771076, 11871145), the Postgraduate Research and Practice Innovation Program of Jiangsu Province (No. KYCX18−0053), the China Scholarship Council (File No. 201906090122), the Qing Lan Project of Jiangsu Province. The fourth author was partially supported by Ministerio de Economía y Competitividad of Spain (grant Red de Excelencia MTM2017-90682-REDT) and by Universidad Nacional de Río Cuarto, Argentina, Res. Rectoral N 083/2020.
Disclosure statement
No potential conflict of interest was reported by the author(s).