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arXiv · 1912.09718

The minus order for idempotents

Yuan Li·Jiajia Niu·Xiaoming Xu
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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 $Q \preceq P^{'} \}.$

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Yuan Li, Jiajia Niu, Xiaoming Xu. 2019-12-20. The minus order for idempotents. https://arxiv.org/abs/1912.09718

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