The minus order for idempotents

Abstract

Let $P$ and $Q$ be idempotents on a Hilbert space $\mathcal{H}.$ The minus order $P⪯Q$ is defined by the equation $PQ=QP=P.$ In this note, we first present some necessary and sufficient conditions for which the supremum and infimum of idempotents $P$ and $Q$ exist with respect to the minus order. Also, some properties of the minimum $Q^{or}$ are characterized, where $Q^{or}$=min $\{P^{{}^{\prime }}:P^{{}^{\prime }}$ is an orthogonal projection on $\mathcal{H}$ with $Q⪯P^{{}^{\prime }}\}.$