1MP and MP1 inverses and one-sided star orders in a ring with involution

TL;DR

This paper explores the properties of 1MP and MP1 inverses within rings with involution, linking them to star partial orders and extending their application to bounded Hilbert space operators.

Contribution

It generalizes the concepts of 1MP and MP1 inverses to rings with involution and connects them with star partial orders, extending known matrix results to a broader algebraic context.

Findings

1MP and MP1 inverses coincide with certain generalized inverses.

Binary relations based on these inverses are equivalent to star partial orders.

Results are applicable to bounded Hilbert space operators.

Abstract

The classes of 1MP-inverses and MP1-inverses are recently introduced classes of generalized inverses of complex matrix. Actually, they coincide with the classes of $\{1,2,3\}$ and $\{1,2,4\}$ inverses, respectively. We consider these inverses in the context of a ring with involution and prove that their most important characterizations and properties remain true. We show that the binary relations based on these inverses are in fact the well known left-star and right-star partial orders. We extend these relations to the ring case, connect them with the unified theory of partial order relations based on generalized inverses and provide several properties. Finally, we indicate how these results can be applied to bounded Hilbert space operators.