Some singular inequalities related to linear maps

Abstract

\begin{abstract}
\noindent
If $\left(
\begin{array}{cc}
A & X \\
X^{*} & B \\
\end{array}
\right)\in \mathbb{M}_2(\mathbb{M}_n)
$ is positive semidefinite, Lin \cite{BZ01} conjectured that \begin{eqnarray*}
% \nonumber to remove numbering (before each equation)
s_j(\Psi(X)) &\leq& s_j(\Psi(A)\sharp \Psi(B)),\;\; j=1,\ldots, n,
\end{eqnarray*}
where the linear map $\Psi: X\mapsto 2\tr (X)I_n-X$ and $s_j(\cdot)$ means the $j$-th largest singular value.\\
In this paper, we state that $$\left(
\begin{array}{cc}
\Psi(A) & \Psi(X) \\
\Psi(X^{*}) & \Psi(B) \\
\end{array}
\right)
$$ is PPT by using an approach which is different from that in \cite{BZ01} and prove the above singular value inequalities hold for the linear map $\Psi_1: X\mapsto (2n+1)\tr (X)I_n-X.$
\end{abstract}