Some results related to the Corach-Porta-Recht inequality (Q2723499)

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 for all \(X\in L(H)\), NEWLINE\[NEWLINE\|SXS^{-1}+S^{-1}XS\|\geq 2\|X\|.\tag \(*\) NEWLINE\]NEWLINE In this paper, the author establishes an inequality similar to the Corach-Porta-Recht inequality for all invertible positive commuting operators under some conditions. The author also gives a necessary condition (resp. necessary and sufficient condition, when the spectra of \(P\) and \(Q\) are uniform) for the invertible positive operators \(P\), \(Q\) to satisfy the operator-norm inequality NEWLINE\[NEWLINE\|PXP^{-1}+Q^{-1}XQ\|\geq 2\|X\|NEWLINE\]NEWLINE for all \(X\in L(H)\). Moreover, the author gives a necessary and sufficient condition for the invertible operator \(S\in L(H)\) to satisfy \((\ast)\).