The star order for idempotent operators

Abstract

The purpose of this note is to consider the lattice properties of $\mathcal{Q}(\mathcal{H})$ (the set of all idempotent operators on a Hilbert space $\mathcal{H}$) with respect to the star partial order. In the domain of $\mathcal{Q}(\mathcal{H}),$ we prove that the star infimum always exists for an arbitrary nonempty subset of $\mathcal{Q}(H)$. Also, we present the necessary and sufficient conditions for the existence of the star supremum for an arbitrary nonempty subset of $\mathcal{Q}(H)$. In particular, an explicit representation of the star supremum is established for two arbitrary idempotent operators in $\mathcal{Q}(\mathcal{H})$.

Keywords:

• Idempotent
• star-order
• minus order
• supremum

Mathematics Subject Classifications:

• 47A05
• 47B65
• 46C20

Acknowledgments

The authors would like to express their heartfelt thanks to the anonymous referees. In particular, the reviewer provided Lemmas 3.2, 3.3 and valuable suggestions on the structure of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant numbers 11671242, 11601339], the Fundamental Research Funds for the Central Universities [grant number GK201801011] and the Fundamental Research Funds for the Development of Young and Middle-aged Talent of Shanghai Institute of Technology [grang number ZQ2020-20].