The minus order for idempotents

TL;DR

This paper investigates the minus order relation among idempotents on a Hilbert space, providing conditions for the existence of their supremum and infimum, and characterizing the minimal orthogonal projection related to this order.

Contribution

It introduces necessary and sufficient conditions for the existence of supremum and infimum of idempotents under the minus order, and characterizes the minimal orthogonal projection associated with this order.

Findings

Conditions for supremum and infimum existence are established.

Properties of the minimal orthogonal projection are characterized.

The minus order relation among idempotents is systematically analyzed.

Abstract

Let $P$ and $Q$ be idempotents on a Hilbert space $\mathcal{H}.$ The minus order $P\preceq Q$ 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 $\{P^{'}: P^{'}$ is an orthogonal projection on $\mathcal{H}$ with $Q \preceq P^{'} \}.$