Generalized Browder's and Weyl's theorems for Banach space operators

Abstract

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 ) ) .