The generalized property (ω1) and generalized property (ω) for a bounded linear operator T defined on a Banach space are studied. A sufficient and necessary condition for an operator to nave the generalized property (ω1) is given by means of single-valued extension property(SVEP). It is proved that if T* has the SVEP at all points λσSBF－＋(T), then T satisfies the generalized property (ω) if and only if one of the following conditions is satisfied: (1) there exists n∈N such that H0(T-λ)=N［(T-λ)n］ for every λ∈E(T); (2) there exists n∈N such that R［(T-λ)n］ is closed for every λ∈E(T); (3) there exists n∈N such that K(T-λ)=R［(T-λ)n］ for every λ∈E(T); (4) there exists n∈N such that r(Tn) is discontinuous at every λ∈E(T); (5) des (T-λ)

generalized property (ω1); generalized property (ω); single valued extension property; spectrum