If $\text{Im}(S^*M)\subseteq \text{Im}(M)$, is $\text{Im}(SM)\subseteq \text{Im}(M)$?

Question

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 $$\text{Im}(SM)\subseteq \text{Im}(M)\;?$$

Answer 1

No: $F = \mathbb C^2$, $M= \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$, $S= \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$.