If $X$ is a separable Banach space and $(\epsilon\_n)\downarrow 0$, can we find a quasinilpotent, non-compact operator on $X$ such that $||T^n||^{1/n}<\epsilon\_n$ for all $n$? I suspect the answer is positive, but cannot come up with an example. 



Quasinilpotent , non-compact operators
On the Argyros-Haydon space every operator is a compact perturbation of a scalar multiple of the identity, and hence every quasinilpotent operator is compact.


