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.

令\（{\ mathcal {H}} \）为无穷维复希尔伯特空间，\（\ mathcal {B（H）} \）为\（{\ mathcal {H}}上所有有界线性算子的代数\）。对于\（T \ in \ mathcal {B（H）} \），我们说T具有\（（\ omega）\）属性，如果\（\ sigma _ {a}（T）{\ setminus} \ sigma _ { aw}（T）= \ pi _ {00}（T）\），如果\（\ sigma _ {a}（T）{\ setminus，则被称为具有属性\（（\ omega _ {1}）\）} \ sigma _ {aw}（T）\ subseteq \ pi _ {00}（T）\），其中\（\ sigma _a（T）\）和\（\ sigma _ {aw}（T）\）表示的近似点谱和Weyl本质近似点谱分别为T和\（\ pi _ {00}（T）= \ {\ lambda \ in iso \ sigma（T）：0 <dim N（T- \ lambda I）<\ infty \} \）。在本文中，我们专注于对在紧摄扰动下属性\（（\ omega _ {1}）\）和属性\（（\ omega）\）稳定的算子的刻画。