Is $PAPA^*P \leq AA^*$? I`m trying to figure out, if $PAPA^*P \leq AA^*$, where $A \in \mathfrak B (\mathcal H)$ and $P$ is some projector. It can be shown that $PAA^*P \nleq AA^*$ in general case. $\langle PAPA^*P x, x\rangle$ seems to be kind of a natural constriction of bilinear form $\langle AA^* x, x\rangle$ after projecting $B (\mathcal H)$ to $PB (\mathcal H)P$. That's my intuition, but I can't obtain anything more. I've tried to check explicitly, whether $\langle x , (AA^* - PAPA^*P) x\rangle $ is positive, but got nothing useful.

A slightly warped version of Ruy's example:
$$P = \frac12\begin{pmatrix} 1 & 1 \\ 1 & 1\end{pmatrix}, \quad A = \begin{pmatrix} 0 & 1 \\ 0 & 0\end{pmatrix}.$$
We have $$AA^* = \begin{pmatrix} 1 & 0 \\ 0 & 0\end{pmatrix}, \quad PAPA^*P = \frac18\begin{pmatrix} 1 & 1 \\ 1 & 1\end{pmatrix}$$
and the difference has determinant $-\frac18$.
