Generalized Cesàro Operators: Geometry of Spectra and Quasi-Nilpotency

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 $T_g$ is unchanged if the symbol $g$ is perturbed to $g+h$ by an analytic function $h$ inducing a quasi-nilpotent operator $T_h$, that is, spectrum of $T_h$ equals $\{0\}$. We also show that any $T_g$ operator that 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 $g \in \textbf{BMOA}$ to be in the $\textbf{BMOA}$ 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.