Weighted Moore-Penrose Inverses and Weighted Core Inverses in Rings with Involution

Abstract

In this paper, the authors derive the existence criteria and the formulae of the weighted Moore-Penrose inverse, the e-core inverse and the f-dual core inverse in rings. Also, new characterizations between weighted Moore-Penrose inverses and one-sided inverses along an element are given.

Keywords

Weighted Moore-Penrose inverses / One-sided inverses along an element / Inverses along an element / e-Core inverses / f-Dual core inverses

References

[1] Benítez J, Boasso E. The inverse along an element in rings with an involution, Banach algebras and C*-algebras. Linear Multilinear Algebra, 2017, 65: 284-299

[2] Chen J L, Zhu H H, Patrício P, Zhang Y L. Characterizations and representations of core and dual core inverses. Canad. Math. Bull., 2017, 60: 269-282

[3] Li T T, Chen J L. Characterizations of core and dual core inverses in rings with involution. Linear Multilinear Algebra, 2018, 66: 717-730

[4] Mary X, Patrício P. Generalized inverses modulo ${\cal H}$ in semigroups and rings. Linear Multilinear Algebra, 2013, 61: 1130-1135

[5] Mosić D, Deng C Y, Ma H F. On a weighted core inverse in a ring with involution. Comm. Algebra, 2018, 46: 2332-2345

[6] Rakić D S, Dinčić N C, Djordjević D S. Group, Moore-Penrose, core and dual core inverse in rings with involution. Linear Algebra Appl., 2014, 463: 115-133

[7] Zhu H H, Chen J L, Patrício P. Further results on the inverse along an element in semigroups and rings. Linear Multilinear Algebra, 2016, 64: 393-403

[8] Zhu H H, Chen J L, Patrício P. Reverse order law for the inverse along an element. Linear Multilinear Algebra, 2017, 65: 166-177

[9] Zhu H H, Patrício P, Chen J L, Zhang YL. The inverse along a product and its applications. Linear Multilinear Algebra, 2016, 64: 834-841

[10] Zhu H H, Wang Q-W. Weighted pseudo core inverses in rings. Linear Multilinear Algebra, 2020, 68: 2434-2447