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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(T ) = π(T ), where E(T ) is the set of all eigenvalues of T which are isolated in the approximate point spectrum and π(T ) is the set of all left poles of T. Some applications are also given.

A note on the $a$-Browder’s and $a$-Weyl’s theorems