Weak group inverses and partial isometries in proper *-rings

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 $a^{◯\mathrm{W}}{a}^{\ast }=a^{\ast }{a}^{◯\mathrm{W}}$ are presented by using the normality of the group invertible part of an element in its group-EP decomposition.

Keywords:

• Weak group inverse
• weak group element
• pseudo core inverse
• partial isometry

AMS subject classifications:

• 15A09
• 16W10

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).

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant number 11771076], [grant number 11871145] and the China Scholarship Council [grant number 201906090122] and the Qing Lan Project of Jiangsu Province and Ministerio de Economia y Competitividad of Spain [Red de Excelencia MTM2017-90682-REDT] and Universidad Nacional de Rio Cuarto, Argentina [Res. Rectoral N 083/2020] and the Postgraduate Research and Practice Innovation Program of Jiangsu Province [grant number KYCX18-0053].