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.