Let $X$,$Y$, and $Z$ with ($Y$ is separable) be a Hilbert space and let $F \in L(X,Z)$ and $G \in L(Y,Z)$ (Bounded linear maps). the following statement hold: $R(F) \subset R(G)$ if and only if there exists a positive constant $c$ such that: $\left\| {{F^*}x} \right\| \leqslant c\left\| {{G^*}x} \right\|$.
Now, let $A$ be an operator defined on some Hilbert space $V$, if we knew that $R(A^*)=W$, how to prove by using the previous statement that $R(A)=R(A^*)=W$ ? Thank you.