Common properties among various generalized inverses and constrained binary relations

Abstract

The common characterizations and various individual properties of different generalized inverses are established. Several equivalent conditions for the core-EP, weak group inverse, m-weak group inverse and weak core inverse are presented. The brand new explicit expressions for the operator binary relations defined by various generalized inverses are obtained. We derive properties and study the relationship between two different elements $A,B\in \mathcal{B}\mathcal{(}\mathcal{H}\mathcal{)}$ and their $(T,S)$-outer generalized inverses $A_{T,S}^{(2)}$ and $B_{T,S}^{(2)}$ through which the reverse order law results about $(AB{)}^{◯\mathrm{w}}=B^{◯\mathrm{w}}{A}^{◯\mathrm{w}}$ and $(AB{)}^{◯\mathrm{d}}=B^{◯\mathrm{d}}{A}^{◯\mathrm{d}}$ are obtained.

Keywords:

• Generalized inverse
• $(T,S)$-outer generalized inverse
• reverse order law
• binary relation

2000 Mathematics Subject Classifications:

• 15A09
• 47A05

Disclosure statement

No potential conflict of interest was reported by the author(s).