positive operator, projection, Hilbert space

Question

Let $T$ be a positive operator on a Hilbert space $H$. Let $P$ be a projection on $H$. Then, it is well-known that $PTP$ is also positive. My question is: whether $T\ge PTP$?

Answer 1

This is not true. Let $$ T=\begin{bmatrix}1&2\\2&4\end{bmatrix},\ \ P=\begin{bmatrix}1&0\\0&0\end{bmatrix}. $$ Then $T$ is positive (selfadjoint, with eigenvalues $0$ and $5$), but $$ T-PTP=\begin{bmatrix}0&2\\2&4\end{bmatrix} $$ is not positive (selfadjoint with eigenvalues $2\pm2\sqrt2$).