The minus order for projections

Abstract

Let $\mathcal{B(H)}^{Id}$ be the set of all projections on a Hilbert space $\mathcal{H}.$ The necessary and sufficient conditions are presented for the existence of the supremum, as well as the infimum, of two arbitrary projections in $\mathcal{B(H)}^{Id}$ with respect to the minus order $\preceq.$ For a projection $Q$ in $\mathcal{B(H)}^{Id},$ the properties of the sets $\{P: P  \hbox{ is an orthogonal projection on } \mathcal{H} \hbox{ and }Q \preceq P\}$ and $\{P: P  \hbox{ is an orthogonal projection on } \mathcal{H} \hbox{ and } P\preceq Q\}$ are further explored.