SINGULAR VALUE INEQUALITIES FOR HILBERT SPACE OPERATORS

Abstract

In this paper we show that if $A_i$, $B_i$, $X_i$  $\in$ $\mathcal{B}(\mathcal{H})$, where $X_i$ is compact operator  $i=1,2,\ldots, n$ and $f$, $g$ are non-negative  continuous functions on $[0,\infty)$ with $f(t)g(t)=t $ for all $t \in [0, \infty)$, also  $h$ is non-negative increasing function and operator convex on $[0,\infty)$, then
\begin{eqnarray} \nonumber
h \left( s_j \left(\sum_{i=1}^{n} \omega_i A^*_i X_i B_i \right)\right)  \leq s_j\left( \sum_{i=1}^{n} \omega_i h (A^*_i f(|X^*_i|)^2A_i) \oplus \sum_{i=1}^{n} \omega_i  h ( B^*_i g(|X_i|)^2 B_i) \right)
\end{eqnarray}
 for $j=1,2,\ldots$ and $\sum_{i=1}^{n} \omega_i = 1$.  
%Among other inequalities, we show that if $A$, $B$ and $X$ in $\mathcal{B}(\mathcal{H})$ such that $X$ is compact and $|A|\leq a$ and $|B|\leq b$ for some real numbers $a$ and $b$, then $s_j(AXB)\leq (\max\{a,b\})^2s_j(X\oplus X)$, for $j=1,2,\ldots$.  
 Also, applications of some inequalities are given.