Abstract
We study the star and minus partial orders on the set of all bounded operators acting from a Hilbert space H to a Hilbert space K. We extend and strengthen some results from matrix case to the case of general operators, which may not possess generalized inverses. By means of some norm inequalities, we give necessary and sufficient condition under which two operators have orthogonal ranges, and thus, we give a characterization of the star partial order. When A = PB for some projection P, we prove the equivalence of
with
for a wide class of continuous functions f. Also, we prove that
if and only if
and
when A is a weak EP operator and B is a self-adjoint operator. Finally, we consider the Moore–Penrose invertibility and the ordinary invertibility of a linear combination of operators when they are related with one of these two orders.
Acknowledgments
The authors thank the referees for their careful reading and many useful comments that improved the quality of the paper.
Disclosure statement
No potential conflict of interest was reported by the author(s).