Judgement of two Weyl type theorems for bounded linear operators

Abstract

Let $H$ be an infinite dimensional separable complex Hilbert space and $B(H)$ the algebra of all linear bounded operators on $H$. $T\in B(H)$ is said to satisfy property $(UW_\Pi)$ if $\sigma_a(T)\backslash\sigma_{ea}(T)=\Pi(T)$, where $\sigma_a(T)$ and $\sigma_{ea}(T)$ denote the approximate point spectrum and the essential approximate point spectrum of $T$ respectively, $\Pi(T)$ denote the set of all poles of $T$. $T\in B(H)$ satisfy a-Weyl's theorem if $\sigma_a(T)\backslash \sigma_{ea}(T)=\pi_{00}^a(T)$, where $\pi_{00}^a(T)=\{\lambda\in\hbox{iso}\sigma_a(T): 0<n(T-\lambda I)<\infty\}$. In this paper, we give necessary and sufficient conditions for a bounded linear operator and its function calculus to satisfy both property $(UW_\Pi)$ and a-Weyl's theorem by topological uniform descent. In addition, the property $(UW_\Pi)$ and a-Weyl's theorem under perturbations are also discussed.