What's the adjoint of a zero operator? Is that also a zero operator? If yes, how can you prove it?
We have $\langle T(v),w\rangle =\langle v,T^*(w)\rangle$. If $T(v)=0$, then what? 

If $T(v)=0$ $\forall v$, this implies $\langle v, T^\ast(w)\rangle=0$ $\forall v$.
In particular, you can choose $v=T^\ast(w)$ and find $\langle T^\ast(w), T^\ast(w)\rangle=0$ and thus $T^\ast(w)=0$.
