  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 \sigma(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 \in H(T), the space of
functions analytic on an open neighborhood of \sigma(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 \in H(\sigma(T)).
