  For the class of Hardy spaces and standard weighted Bergman spaces of the
unit disk we prove that the spectrum of a generalized Ces\`aro operator $T_g$
is unchanged if the symbol $g$ is perturbed to $g+h$ by an analytic function
$h$ inducing a quasi-nilpotent operator $T_h$, i.e. spectrum of $T_h$ equals
$\{0\}$. We also show that any $T_g$ operator which can be approximated in the
operator norm by an operator $T_h$ with bounded symbol $h$ is quasi-nilpotent.
In the converse direction, we establish an equivalent condition for the
function $g \in$ BMOA to be in the BMOA-norm closure of $H^{\infty}$. This
condition turns out to be equivalent to quasi-nilpotency of the operator $T_g$
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 $T_{g}$ operators.
