New characterizations of weak CMP inverses

Abstract In 2025, Mosić defined the weak CMP inverse utilizing a minimal rank weak Drazin inverse instead of the Drazin inverse. The weak CMP inverse is a new broader class of generalized inverses, with the CMP and MPCEP inverses as particular cases, and it has applications in solving systems of linear equations. In this paper, we provide several expressions, along with a number of new characterizations and properties of the weak CMP inverse by means of Hartwig–Spindelböck decomposition. Moreover, we investigate the relationships between the weak CMP inverse and some well-known generalized inverses, such as the Moore–Penrose inverse and the weak MPD inverse. Finally, we show that the weak CMP inverse, weak MPD inverse and weak DMP inverse are all strong Bott–Duffin ( e , f ) {(e,f)} -inverses.