If N(T) = null space of T, then you can get an example in two dimensions. Take $T=\begin{bmatrix}0&1\\0&0\end{bmatrix}$.
If N(T) means the nullity of T then you have to go to an infinite-dimensional space. The simplest example would be the unilateral shift on $\ell_2$. This is 1-1, so has zero-dimensional null space. But its adjoint is the backwards shift, which has one-dimensional kernel.