Let $Q$ be an idempotent in $L(H)$. Suppose there exists two sequences of unitaries $\{U_n\}$ and $\{V_n\}$ such that $X$ is the norm limit of $U_nQU_n^*+V_nQV_n^*$, and suppose that
$Q+Q^*\cong 2P\oplus \begin{pmatrix} 2I & R \\ R & 0 \end{pmatrix}\cong2P\oplus \{\begin{pmatrix} I & I \\ I & I\end{pmatrix}+ \begin{pmatrix} R & 0 \\ 0 & -R \end{pmatrix}\}$, where $A\cong B$ means that two operators $A$ and $B$ are unitarily equivalent , $R$ is a nonsingular operator on a Hilbert space $F$, and $I$ is the identity matrix.
Then (1)$\|(Q+Q^*)_{+}\|\geq\|(Q+Q^*)_{-}\|$. (2) $X+X^*$ cannot be negative definite.
How to check this conclusion?
My thought: For (1), since $Q+Q^*=(Q+Q^*)_{+}-(Q+Q^*)_{-}$ and we set $T=2P\oplus \{\begin{pmatrix} I & I \\ I & I\end{pmatrix}+ \begin{pmatrix} R & 0 \\ 0 & -R \end{pmatrix}\}$, $T_{+}=2P\oplus\begin{pmatrix} I & I \\ I & I \end{pmatrix}+0\oplus\begin{pmatrix} R & 0 \\ 0 & 0 \end{pmatrix}$ , $T_{-}=\begin{pmatrix} 0 & 0 \\ 0 & -R \end{pmatrix}$, we have $\|T_{+}\|\geq\|T_{-}\|$. As $(Q+Q^*)_{+}$ is unitarily equivalent to $T_{+}$, $(Q+Q^*)_{-}$ is unitarily equivalent to $T_{-}$,we have $\|(Q+Q^*)_{+}\|\geq\|(Q+Q^*)_{-}\|$.
For （2），if $\lim U_nQU_n^*+V_nQV_n^*$, then $X+X^*=\lim U_n(Q+Q^*)U_n^*+V_n(Q+Q^*)V_n^*$, We know that $Q+Q^*$ is not negative definite, how to derive $X+X^*$ is not negative definite ?
I'll appreciate it if anyone and give me some hint.