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.