Let X be a
complex infinite dimensional Banach space. We use σ_a(T) andσ_{ea}(T)
, respectively, to denote the approximate point spectrum and the
essential approximate point spectrum of a bounded operator T onX
. Also, \pi _a(T) denotes the set <$>{\rm{iso}
σ_a(T)\backslash σ_{ea}(T)}<$>. An operator T onX
obeys the a-Browder's theorem provided that<$>σ_{ea}(T) =σ_a(T\,)\backslash π_a(T)<$>
. We investigate
connections between the Browder's theorems, the spectral mapping theorem and spectral
continuity.