6 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, and we obtain new necessary and sufficient conditions to guarantee that the spectral mapping theorem holds for the B-Weyl spectrum and for polynomials in T. We also prove that the spectral mapping theorem holds for the B-Browder 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)).