When $A\in \mathcal{L}(\mathbb{X})$ and $B\in \mathcal{L}(\mathbb{Y})$ are given, we denote by $M_C$ an operator acting on the Banach space $\mathbb{X}\oplus \mathbb{Y}$ of the form $M_C=\left(\begin{array}{cc}A& C\\ 0& B\end{array}\right)$. In this paper, first we prove that ${\sigma }_w(M_0)={\sigma }_w(M_C)\cup \{S(A^{\ast })\cap S(B)\}$ and ${\sigma }_{aw}(M_C)\subseteq {\sigma }_{aw}(M_0)\cup S_{+}^{\ast }(A)\cup S_{+}(B)$. 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 $\lambda \in {\sigma }_{aw}(M_0)\setminus {\sigma }_{S{F}_{+}}(A)$ and $A^{\ast }$ has SVEP at points $\mu \in {\sigma }_w(M_0)\setminus {\sigma }_{S{F}_{+}}(A)$, or (ii) $A^{\ast }$ has SVEP at points $\lambda \in {\sigma }_w(M_0)\setminus {\sigma }_{S{F}_{+}}(A)$ and $B^{\ast }$ has SVEP at points $\mu \in {\sigma }_w(M_0)\setminus {\sigma }_{S{F}_{+}}(B)$, then $M_C$ satisfies property $(w)$. Here, the hypothesis that points $\lambda \in E^0(M_C)$ are poles of $A$ is essential. We prove also that if $S(A^{\ast })\cup S(B^{\ast })$, points $\lambda \in E_a^0(M_C)$ 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.