Generalized a-Weyl’s theorem and the single-valued extension property

Abstract

Let 𝑇 be a bounded linear operator acting on a Banach space X such that 𝑇 or 𝑇* has the SVEP. We prove that the spectral mapping theorem holds for the semi-essential approximate point spectrum 𝜎ₛSBF± + (𝑇); and we show that generalized a-Browder’s theorem holds for ƒ(𝑇) for every analytic function ƒ defined on an open neighbourhood 𝚄 of 𝜎(𝑇): Moreover, we give a necessary and sufficient condition for such 𝑇 to obey generalized a-Weyl’s theorem. An application is given for an important class of Banach space operators.