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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some results related to the Corach-Porta-Recht inequality


Author: Ameur Seddik
Journal: Proc. Amer. Math. Soc. 129 (2001), 3009-3015
MSC (2000): Primary 47A30, 47B15
Published electronically: March 15, 2001
MathSciNet review: 1840106
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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\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 $P,Q$ 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 $\left( *\right) .$


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06041-5
PII: S 0002-9939(01)06041-5
Keywords: Operator-norm inequality, self-adjoint operator, positive operator.
Received by editor(s): February 29, 2000
Published electronically: March 15, 2001
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society