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