Property $(\omega )$ and its compact perturbations

Abstract

Let ${\mathcal {H}}$ be an infinite dimensional complex Hilbert space and $\mathcal {B(H)}$ be the algebra of all bounded linear operators on ${\mathcal {H}}$. For $T\in \mathcal {B(H)}$, we say T has property $(\omega )$ if $\sigma _{a}(T){\setminus }\sigma _{aw}(T)=\pi _{00}(T)$ and is said to have property $(\omega _{1})$ if $\sigma _{a}(T){\setminus }\sigma _{aw}(T)\subseteq \pi _{00}(T)$, where $\sigma _a(T)$ and $\sigma _{aw}(T)$ denote the approximate point spectrum and the Weyl essential approximate point spectrum of T respectively, and $\pi _{00}(T)=\{\lambda \in iso\sigma (T): 0<dim N(T-\lambda I)<\infty \}$. In this paper, we focus on the characterization on the operators for which property $(\omega _{1})$ and property $(\omega )$ are stable under compact perturbations.