The Weyl’s Theorem and Its Perturbations in Semisimple Banach Algebra

Denote a semisimple Banach algebra with an identity e by A. This paper studies the Fredholm, Weyl and Browder spectral theories in a semisimple Banach algebra, and meanwhile considers the properties of the Fredholm element, the Weyl element and the Browder element. Further, for a ∈ A, we give the Weyl’s theorem and the Browder’s theorem for a, and characterize necessary and sufficient conditions that both a and f(a) satisfy the Weyl’s theorem or the Browder’s theorem, where f is a complex-valued function analytic on a neighborhood of σ(a). In addition, the perturbations of the Weyl’s theorem and the Browder’s theorem are investigated.