Is N(T) supposed to be the null space of T, or the dimension of the null space?

If N(T) = null space of T, then you can get an example in two dimensions. Take .

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 . This is 1-1, so has zero-dimensional null space. But its adjoint is the backwards shift, which has one-dimensional kernel.