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.
We extend and generalize some results in local spectral theory for upper triangular operator matrices to upper triangular operator matrices with unbounded entries. Furthermore, we investigate the boundedness of the local resolvent function for operator matrices.
It is shown that the sum and the product of two commuting Banach space operators with Dunford’s property $\mathrm (C)$ have the single-valued extension property.