On the star and minus orders on Hilbert space operators

Abstract

We study the star and minus partial orders on the set $\mathcal{B}(H,K)$ 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 $A{<}^{\ast }B$ with $f(A{A}^{\ast })A{<}^{\ast }f(B{B}^{\ast })B$ for a wide class of continuous functions f. Also, we prove that $A{<}^{\ast }B$ if and only if $A{<}^{-}B$ and $A^2{<}^{-}{B}^2$ 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.

KEYWORDS:

• Star partial order
• minus partial order
• Moore–Penrose inverse
• bounded linear operator

AMS SUBJECT CLASSIFICATIONS:

• 06A06
• 47A05
• 15A45
• 15A09

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).

Additional information

Funding

The research is financially supported by the Ministry of Education, Science and Technological Development, Republic of Serbia, Grant No. 451-03-68/2022-14/200109 and by the bilateral project between Serbia and Slovenia (Generalized inverses, operator equations and applications, Grant No. 337-00-21/2020-09/32).