For a normal operator $T$ acting on a Hilbert space it is easy to show that the kernel of $T$ coincides with the kernel of the adjoint $T^*$. Thus the norm-closures of the ranges $R(T)$ and $R(T^*)$ coincide. Using the spectral theorem for normal operators, it is easy to show that $R(T)=R(T^*)$.
Is it possible to give an elementary proof of this fact?