On local spectral properties of operator matrices

Abstract In this paper, we focus on a $2 \times 2$ 2 × operator matrix $T_{\epsilon _{k}}$ T ϵ k as follows: $$\begin{aligned} T_{\epsilon _{k}}= \begin{pmatrix} A & C \\ \epsilon _{k} D B\end{pmatrix}, \end{aligned}$$ = ( A C D B ) , where $\epsilon _{k}$ is positive sequence such that $\lim_{k\rightarrow \infty }\epsilon _{k}=0$ lim → ∞ 0 . We first explore how has several local spectral properties the single-valued extension property, property $(\beta )$ β , and decomposable. next study relationship between some spectra of its diagonal entries, find hypotheses by which satisfies Weyl’s theorem -Weyl’s theorem. Finally, give conditions an nontrivial hyperinvariant subspace.