Let $\mathcal{X}$ be a Banach space and let $T$ be a bounded linear operator on $\mathcal{X}$. We denote by $S(T)$ the set of all complex $\lambda \in \mathcal{C}$ such that $T$ does not have the single-valued extension property. In this paper it is shown that if $M_C$ is a $2\times 2$ upper triangular operator matrix acting on the Banach space $\mathcal{X}\oplus \mathcal{Y}$, then the passage from ${\sigma }_{LD}(A)\cup {\sigma }_{LD}(B)$ to ${\sigma }_{LD}(M_C)$ is accomplished by removing certain open subsets of ${\sigma }_d(A)\cap {\sigma }_{LD}(B)$ from the former, that is, there is the equality ${\sigma }_{LD}(A)\cup {\sigma }_{LD}(B)={\sigma }_{LD}(M_C)\cup \mathrm{\aleph },$ where $\mathrm{\aleph }$ is the union of certain of the holes in ${\sigma }_{LD}(M_C)$ which happen to be subsets of ${\sigma }_d(A)\cap {\sigma }_{LD}(B)$. Generalized Weyl's theorem and generalized Browder's theorem are liable to fail for $2\times 2$ operator matrices. In this paper, we also explore how generalized Weyl' theorem, generalized Browder's theorem, generalized a-Weyl's theorem and generalized a-Browder's theorem survive for $2\times 2$ upper triangular operator matrices on the Banach space.