Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$. Let $M\in \mathcal{B}(F)^+$ (i.e. $M^*=M$ and $\langle Mx\;, \;x\rangle\geq 0$ for all $x\in F$).

I want to show that $\mathcal{B}^M(F)\subseteq \mathcal{B}^{M^{1/2}}(F)$, where $$\mathcal{B}^M(F)=\left\{S\in \mathcal{B}(F):\,\,\,\text{Im}(S^{*}M)\subseteq \text{Im}(M)\right\},$$ $$\mathcal{B}^{M^{1/2}}(F)=\left\{S\in \mathcal{B}(F):\,\,\,\text{Im}(S^{*}M^{1/2})\subseteq \text{Im}(M^{1/2})\right\}.$$ Thank you