Annals of Functional Analysis

Singular value and‎ ‎arithmetic-geometric mean inequalities for operators

Hussien Albadawi

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‎A singular value inequality for sums and products of Hilbert space operators‎ ‎is given‎. ‎This inequality generalizes several recent singular value‎ ‎inequalities‎, ‎and includes that if $A$‎, ‎$B$‎, ‎and $X$ are positive operators‎ ‎on a complex Hilbert space $H$‎, ‎then ‎\begin{equation*}‎ ‎s_{j}\left( A^{^{1/2}}XB^{^{1/2}}\right) \leq \frac{1}{2}\left\Vert‎ ‎X\right\Vert \text{ }s_{j}\left( A+B\right) \text{, ‎\‎ ‎}j=1,2,\cdots\text{,}‎ ‎\end{equation*} ‎which is equivalent to‎ ‎ \begin{equation*}‎ ‎s_{j}\left( A^{^{1/2}}XA^{^{1/2}}-B^{^{1/2}}XB^{^{1/2}}\right) \leq‎ ‎\left\Vert X\right\Vert s_{j}\left( A\oplus B\right) \text{, ‎\ }j=1,2,\cdots ‎\text{.}‎ ‎\end{equation*}‎ ‎ Other singular value inequalities for sums and products of operators are‎ ‎presented‎. ‎Related arithmetic-geometric mean inequalities are also‎ ‎discussed‎.

Article information

Ann. Funct. Anal., Volume 3, Number 1 (2012), 10-18.

First available in Project Euclid: 12 May 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A30: Norms (inequalities, more than one norm, etc.)
Secondary: 15A18‎ ‎47A63‎ ‎47B10

Singular value ‎unitarily invariant norm ‎positive‎ ‎operator ‎arithmetic-geometric mean inequality


Albadawi, Hussien. Singular value and‎ ‎arithmetic-geometric mean inequalities for operators. Ann. Funct. Anal. 3 (2012), no. 1, 10--18. doi:10.15352/afa/1399900020.

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