On local spectral properties of operator matrices

In this paper, we focus on a 2 × 2 operator matrix Tϵk as follows: Tϵk=(ACϵkDB), where ϵ_k is a positive sequence such that lim _k→∞ϵ_k= 0. We first explore how Tϵk has several local spectral properties such as the single-valued extension property, the property (β) , and decomposable. We next study the relationship between some spectra of Tϵk and spectra of its diagonal entries, and find some hypotheses by which Tϵk satisfies Weyl’s theorem and a-Weyl’s theorem. Finally, we give some conditions that such an operator matrix Tϵk has a nontrivial hyperinvariant subspace.