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\noindent\parbox{2.95cm}{\includegraphics*[keepaspectratio=true,scale=0.125]{AFA.jpg}}
\noindent\parbox{4.85in}{\hspace{0.1mm}\\[1.5cm]\noindent
\qquad Ann. Funct. Anal. 3 (2012), no. 1, 10--18\\
{\footnotesize \qquad \textsc{\textbf{$\mathscr{A}$}nnals of
\textbf{$\mathscr{F}$}unctional
\textbf{$\mathscr{A}$}nalysis}\\
\qquad ISSN: 2008-8752 (electronic)\\
\qquad URL:
\textcolor[rgb]{0.00,0.00,0.99}{www.emis.de/journals/AFA/}
}\\[.5in]}
\title[Singular values and AMG inequlalities]{Singular value and
arithmetic-geometric mean inequalities for operators}
\author[H. Albadawi]{Hussien Albadawi}
\address{Mathematics Program, Preparatory Year Deanship, King Faisal
University, Ahsaa, Saudi Arabia.}
\email{\textcolor[rgb]{0.00,0.00,0.84}{albadawi1@gmail.com}}
\dedicatory{{\rm Communicated by M. S. Moslehian}}
\subjclass[2010]{Primary 47A30; Secondary 15A18, 47A63, 47B10.}
\keywords{Singular value, unitarily invariant norm, positive
operator, arithmetic--geometric mean inequality.}
\date{Received: 15 October 2011; Accepted: 27 November 2011.}
\begin{abstract}
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.
\end{abstract}
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