Let T be a Banach space operator. In this paper we characterize a-Browder’s theorem for T by the localized single valued extension property. Also, we characterize a-Weyl’s theorem under the condition E a (T) = π a (T), where E a (T) is the set of all eigenvalues of T which are isolated in the approximate point spectrum and π a (T) is the set of all left poles of T. Some applications are also given.
Let X be a Banach space and T be a bounded linear operator on X. We denote by S(T) the set of all complex λ ∈ C such that T does not have the single-valued extension property at λ. In this note we prove equality up to S(T) between the left Drazin spectrum, the upper semi-B-Fredholm spectrum and the semi-essential approximate point spectrum. As applications, we investigate generalized Weyl’s theorem for operator matrices and multiplier operators.