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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 $T\ge PTP$?

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  • $\begingroup$ Yes, if $P$ is an orthogonal projection. $\endgroup$ – DisintegratingByParts Apr 21 '16 at 14:53
  • $\begingroup$ @TrialAndError: no, it's not. $\endgroup$ – Martin Argerami Apr 21 '16 at 18:05
  • $\begingroup$ @MartinArgerami : Oops. $\endgroup$ – DisintegratingByParts Apr 21 '16 at 18:11
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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 $2\pm2\sqrt2$).

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