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Some results related to the Corach-Porta-Recht inequality

Authors:	Ameur Seddik

Institution:	Department of Mathematics, Faculty of Science, University of Batna, 05000 Batna, Algeria

Abstract:	Let be the algebra of all bounded operators on a complex Hilbert space and let be an invertible self-adjoint (or skew-symmetric) operator of . Corach-Porta-Recht proved that $\begin{equation}\forall X\in L(H),\left\Vert SXS^{-1}+S^{-1}XS\right\Vert \geq 2\left\Vert X\right\Vert.\tag{$$ } \end{equation}$ 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 to satisfy the operator-norm inequality $\left\Vert PXP^{-1}+Q^{-1}XQ\right\Vert \geq 2\left\Vert X\right\Vert ,$ for all in ; (iii) a necessary and sufficient condition for the invertible operator in to satisfy $\left( \right) .$

Keywords:	Operator-norm inequality self-adjoint operator positive operator

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