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 $\sigma_{LD}(A)\cup\sigma_{LD}(B)=\sigma_{LD}(M_C)\cup\aleph,$ where $\aleph$ 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.