positive operator, projection on Hilbert,$Q|T|Q \ge |QTQ|?$

Question

Let $T$ be an operator on a Hilbert space $H$. And $Q$ be a projection. Whether $$Q|T|Q \ge |QTQ|?$$

Obviously, if $T$ is positive, then $Q|T|Q = |QTQ|$. Also, there are some $T$ such that $QTQ=0$ but $Q|T|Q\ge 0$。

Answer 1

The inequality does not hold in general. Let $$ T=\begin{bmatrix}1&1\\0&0\end{bmatrix},\ \ Q_1=\begin{bmatrix}0&0\\0&1\end{bmatrix}, \ \ Q_2=\begin{bmatrix}1&0\\0&0\end{bmatrix}. $$ Note that $$ |T|=\left(\begin{bmatrix}1&0\\1&0\end{bmatrix}\begin{bmatrix}1&1\\0&0\end{bmatrix}\right)^{1/2}=\begin{bmatrix}1&1\\1&1\end{bmatrix}^{1/2}=\frac1{\sqrt2}\,\begin{bmatrix}1&1\\1&1\end{bmatrix}. $$ We have $Q_1TQ_1=0$, so $$ |Q_1TQ_1|\leq Q_1|T|Q_1. $$ And $$ Q_2|T|Q_2=\frac1{\sqrt 2}\,\begin{bmatrix}1&0\\0&0\end{bmatrix}\leq \begin{bmatrix}1&0\\0&0\end{bmatrix}=Q_2TQ_2=|Q_2TQ_2|. $$