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.
An operator T acting on a Banach space X possesses property (gw) if σa(T) \ σSBF− + (T) = E(T), where σa(T) is the approximate point spectrum of T, σSBF− + (T) is the essential semi-B-Fredholm spectrum of T and E(T) is the set of all isolated eigenvalues of T. In this paper we introduce and study two new properties (b) and (gb) in connection with Weyl type theorems, which are analogous respectively to Browder’s theorem and generalized Browder’s theorem. Among other, we prove that if T is a bounded linear operator acting on a Banach space X, then property (gw) holds for T if and only if property (gb) holds for T and E(T) = Π(T), where Π(T) is the set of all poles of the resolvent of T.