For the class of Hardy spaces and standard weighted Bergman spaces of the unit disk, we prove that the spectrum of a generalized Cesàro operator is unchanged if the symbol is perturbed to by an analytic function inducing a quasi-nilpotent operator , that is, spectrum of equals . We also show that any operator that can be approximated in the operator norm by an operator with bounded symbol is quasi-nilpotent. In the converse direction, we establish an equivalent condition for the function to be in the BMOA norm closure of . This condition turns out to be equivalent to quasi-nilpotency of the operator on the Hardy spaces. This raises the question whether similar statement is true in the context of Bergman spaces and the Bloch space. Furthermore, we provide some general geometric properties of the spectrum of operators.