Some results related to the Corach-Porta-Recht inequality
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- by Ameur Seddik PDF
- Proc. Amer. Math. Soc. 129 (2001), 3009-3015 Request permission
Abstract:
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 \begin{equation*} \forall X\in L(H),\;\left \| SXS^{-1}+S^{-1}XS\right \| \geq 2\left \| X\right \|.\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 $P,Q$ to satisfy the operator-norm inequality $\left \| PXP^{-1}+Q^{-1}XQ\right \| \geq 2\left \| X\right \| ,$ for all $X$ in $L(H)$; (iii) a necessary and sufficient condition for the invertible operator $S$ in $L(H)$ to satisfy $\left ( *\right ) .$
References
- G. Corach, H. Porta, and L. Recht, An operator inequality, Linear Algebra Appl. 142 (1990), 153–158. MR 1077981, DOI 10.1016/0024-3795(90)90263-C
- J. P. Williams, Finite operators, Proc. Amer. Math. Soc. 26 (1970), 129–136. MR 264445, DOI 10.1090/S0002-9939-1970-0264445-6
Additional Information
- Ameur Seddik
- Affiliation: Department of Mathematics, Faculty of Science, University of Batna, 05000 Batna, Algeria
- Address at time of publication: Department of Mathematics, Faculty of Science, University of Sana‘a, P.O. Box 14026, Sana‘a, Yemen
- Email: seddikameur@hotmail.com
- Received by editor(s): February 29, 2000
- Published electronically: March 15, 2001
- Communicated by: Joseph A. Ball
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 3009-3015
- MSC (2000): Primary 47A30, 47B15
- DOI: https://doi.org/10.1090/S0002-9939-01-06041-5
- MathSciNet review: 1840106