I'm nor sure whether the following works:

Take an operator on a Banach space whose spectrum is $\{0\}$ but that has no eigenvectors, for example
$$
T(f)(x)=\int_x^1f(y)dy
$$ 
acting on $L^2([0,1])$.

Then its restriction to the dense subspace $\bigcap_n Im(T^n)$ should have the property you desire.