On the star and minus orders on Hilbert space operators

Abstract

We study the star and minus partial orders on the set B(H,K) 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 A<∗B with f(AA∗)A<∗f(BB∗)B for a wide class of continuous functions f. Also, we prove that A<∗B if and only if A<−B and A2<−B2 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.