A new criteria for two Weyl's type theorems

Abstract

Let $\mathcal{B(H)}$ be the collection of all bounded linear operators  on $\mathcal{H}$, where $\mathcal{H}$ is an infinite dimensional complex Hilbert space. For $T\in\mathcal{B(H)}$, we say property $(UW_\Pi)$( property $(\omega)$) holds for $T$ if $\sigma_{a}(T)\backslash\sigma_{ea}(T)=\Pi(T)(\pi_{00}(T))$, where $\sigma_{a}(T)$ and $\sigma_{ea}(T)$ denote the approximate point spectrum and the essential approximate point spectrum of $T$ respectively, also $\Pi(T)$ and $\pi_{00}(T)$ severally denote the set of all poles and all finite dimensional isolated eigenvalues. In this note, we introduce a new judgement method for bounded linear operators and their function calculus satisfying property $(UW_\Pi)$ and property $(\omega)$ together by the deformed property. Meanwhile, we investigate the relationships among property $(UW_\Pi)$, property $(\omega)$ and hypercyclic property.