# Generalized Ces\`aro operators: geometry of spectra and quasi-nilpotency

## TL;DR

This paper investigates the spectral properties and quasi-nilpotency of generalized Cesàro operators on Hardy and weighted Bergman spaces, revealing invariance under certain perturbations and characterizing their spectral geometry.

## Contribution

It establishes spectral invariance under perturbations by quasi-nilpotent operators and characterizes quasi-nilpotency via BMOA and bounded symbol approximations.

## Key findings

- Spectrum of $T_g$ is unchanged under specific perturbations.
- Operators approximable by bounded symbol operators are quasi-nilpotent.
- Quasi-nilpotency of $T_g$ relates to BMOA and $H^{\infty}$ closures.

## Abstract

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.