When $A\in \mathscr{L}(\mathbb {X})$ and $B\in \mathscr{L}(\mathbb {Y})$ are given, we denote by $M_C$ an operator acting on the Banach space $\mathbb {X}\oplus \mathbb {Y}$ of the form $M_{C}=\left (\begin {array}{cccccccc} A & C \\ 0 & B \\ \end {array}\right )$. In this paper, first we prove that $\sigma_w(M_0) = \sigma_w(M_C)\cup\{S(A^∗) \cap S(B)\}$ and $\mathbf {\sigma}_{aw}(M_{C})\subseteq \mathbf {\sigma}_{aw}(M_{0})\cup S_{+}^{*}(A)\cup S_{+}(B)$. Also, we give the necessary and sufficient condition for $M_C$ to be obeys property $(w)$. Moreover, we explore how property $(w)$ survive for $2 \times 2$ upper triangular operator matrices $M_C$. In fact, we prove that if $A$ is polaroid on $E^{0}(M_{C})=\{\lambda \in \mathrm{iso\,}\sigma (M_{C}):0<\dim (M_{C}-\lambda )^{-1}\}$, $M_0$ satisfies property $(w)$, and $A$ and $B$ satisfy either the hypotheses (i) $A$ has SVEP at points $\mathbf {\lambda }\in \mathbf {\sigma }_{aw}(M_{0})\setminus \mathbf {\sigma }_{SF_{+}}(A)$ and $A^∗$ has SVEP at points $\mu \in \mathbf {\sigma }_{w}(M_{0})\setminus \mathbf {\sigma }_{SF_{+}}(A)$, or (ii) $A^∗$ has SVEP at points $\mathbf {\lambda }\in \mathbf {\sigma }_{w}(M_{0})\setminus \mathbf {\sigma }_{SF_{+}}(A)$ and $B^∗$ has SVEP at points $\mu \in \mathbf {\sigma }_{w}(M_{0})\setminus \mathbf {\sigma }_{SF_{+}}(B)$, then $M_C$ satisfies property $(w)$. Here, the hypothesis that points $λ \in E^0(M_C)$ are poles of $A$ is essential. We prove also that if $S(A^∗) \cup S(B^∗)$, points $\mathbf {\lambda }\in {E_{a}^{0}}(M_{C})$ are poles of $A$ and points $\mu \in {E_{a}^{0}}(B)$ are poles of $B$, then $M_C$ satisfies property $(w)$. Also, we give an example to illustrate our results.