Some results related to the Corach-Porta-Recht inequality

Let $L(H)$ be the algebra of all bounded operators on a complex Hilbert space $H$ and let $S$ be an invertible self-adjoint (or skew-symmetric) operator of $L(H)$. Corach-Porta-Recht proved that

The problem considered here is that of finding (i) some consequences of the Corach-Porta-Recht Inequality; (ii) a necessary condition (resp. necessary and sufficient condition, when $\sigma (P)=\sigma (Q))$ for the invertible positive operators $P,Q$ to satisfy the operator-norm inequality $\left\Vert PXP^{-1}+Q^{-1}XQ\right\Vert \geq 2\left\Vert X\right\Vert ,$ for all $X$ in $L(H)$; (iii) a necessary and sufficient condition for the invertible operator $S$in $L(H)$ to satisfy $\left( *\right) .$