In this paper, we give necessary and sufficient conditions for a Banach space T to satisfy the generalized α-Browder’s theorem. We also prove that the spectral mapping theorem holds for the left Drazin invertible and for analytic functions on a neighborhood of σ(T). As applications, we show that if T^* is algebraically ωF(p,r,q) for each p,r>0 and q≥1, or if T^* is algebraically quasi-class A, then the generalized α-Weyl’s theorem hold for f(T), where f∈Hol(σ(T)), the space of functions analytic on an open neighborhoods of σ(T).

Weyl’s theorem, α-Weyl’s theorem, generalized α-Weyl’s theorem, α-Browder’s theorem, generalized α-Browder’s theorem, reduction -isoloid, reduced subspace