Let $F$ be a complex Hilbert space and $\mathcal{B}(F)$ the algebra of all bounded linear operators defined on $F$.



> 
> Assume that
> 
> 
> * $M\in \mathcal{B}(F)^+$ (i.e. $\langle Mx\;, \;x\rangle\geq 0$ for all $x\in F$).
> * $S\in\mathcal{B}(F)$ such that $\text{Im}(S^\*M)\subseteq \text{Im}(M)$.
> 
> 
> It is true that
> $$\text{Im}(SM)\subseteq \text{Im}(M)\;?$$
> 
> 
> 



If $\text{Im}(S^*M)\subseteq \text{Im}(M)$, is $\text{Im}(SM)\subseteq \text{Im}(M)$?
No: $F = \mathbb C^2$,
$M= \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$, $S= \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$.


