Generalized Browder’s and Weyl’s theorems for Banach space operators ✩

We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder’s theorem.We also prove that the spectral mapping theorem holds for the Drazin spectrum and for analytic functions on an open neighborhood of σ(T ). As applications, we show that if T is algebraically M-hyponormal, or if T is algebraically paranormal, then the generalized Weyl’s theorem holds for f (T ), where f ∈ H((T )), the space of functions analytic on an open neighborhood of σ(T ). We also show that if T is reduced by each of its eigenspaces, then the generalized Browder’s theorem holds for f (T ), for each f ∈ H(σ(T )). © 2007 Elsevier Inc. All rights reserved