Criteria for the properties (ω) and $(W_E)$ under perturbations

Properties $(\omega)$ and $(W_{E})$ are the significant variants of Weyl's theorem. We present the necessary and sufficient conditions for the simultaneous existence of two properties of every generalized scalar operator. As an application, we prove the transmission of two properties from $T$ to $f(T)$ for any analytic function $f$ on a neighborhood of $\sigma(T)$ for the first time. As a consequence of the main theorem, we study the permanence of property $(\omega)$ and property $(W_{E})$ under finite or compact perturbations.