Topological uniform descent and generalized property $(\omega)$

The generalized property $(\omega)$ is a variant of the generalized Weyl's Theorem. We establish necessary and sufficient conditions of generalized property $(\omega)$ for a bounded linear operator and its functional calculus, utilizing the spectral set induced by topological uniform descent. We further investigate the stability of this property under commuting compact perturbations. Finally, generalized property $(\omega)$ for algebraic operators is discussed.