Left-star and right-star partial orderings

Abstract

Two partial orderings in the set of complex matrices are introduced by combining each of the conditions A^∗A=A^∗B and AA^∗=BA^∗, which define the star partial ordering, with one of the conditions M(A)⊆(B) and M(A^∗)⊆M(B^∗), which define the space preordering. Several properties of these orderings are examined, with main emphasis on comparing the new orderings with the star ordering, the minus ordering, and other related partial orderings. Moreover, some further characterizations of partial orderings in terms of inclusions of appropriate classes of generalized inverses are derived, with the main emphasis on characterizations involving reflexive generalized inverses.