In this paper, we focus on a 2 × 2 operator matrix Tϵk as follows: Tϵk=(ACϵkDB), where ϵk is a positive sequence such that lim k→∞ϵk= 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.
- 2 × 2 operator matrices
- Hyperinvariant subspace
- The property (β)
- The single-valued extension property
- Weyl’s theorem