The Regularity and Uniform Positivity of the Range of Orthogonal Projections

Given two orthogonal projections P, \(Q\in \mathcal {B}(\mathcal {H})\) $Q\in \mathcal{B}(\mathcal{H})$ , let \(T=PQ\) $T=PQ$ and \(\mathcal {R}(T)^{\perp }\cap \mathcal {N}(T)=\{0\}\) $\mathcal{R}{(T)}^{\perp }\cap \mathcal{N}(T)=\{0\}$ , we characterize symmetries J such that \((JT)^{*}=JT\) $(JT)_{}^{\ast }=JT$ and \(JT\ge 0\) $JT\ge 0$ , where \((JT)^{*}=JT\) $(JT)_{}^{\ast }=JT$ if and only if \(PJ=JQ\) $PJ=JQ$ . Furthermore, we study the regularity and uniform positivity of \(\mathcal {R}(P)\) $\mathcal{R}(P)$ and \(\mathcal {R}(Q)\) $\mathcal{R}(Q)$ .