How to prove that both the subspaces are equal?

Question

Let $T$ be a bounded linear operator between Hilbert spaces. Then how to prove that $$R(T^*) = R((T^*T)^{1/2})$$ where $R(T)$ denotes range of operator and $T^*$ is adjoint of $T$.

Answer 1

Hint: The polar decomposition of the operator $T:\mathcal{H}_1\rightarrow\mathcal{H}_2$

$T=U(T^*T)^{\frac{1}{2}}$ with a partial isometry $U$ that maps $R((T^*T)^{\frac{1}{2}})$ to $R(T)$ implies

$U^*T=(T^*T)^{\frac{1}{2}}$ is a decomposition where $U^*$ is a partial isometry that maps $R(T)$ to $R((T^*T)^{\frac{1}{2}})$

Additionally: $T^*=(T^*T)^{\frac{1}{2}}U^*$.