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