In this paper we are concerned with the spectrum of the operator $T=D+T_{\mu}$ where D is a diagonal operator and $T_{\mu}=\displaystyle\sum_{i=1}^{\infty}\mu_{i}P_{i}$ is a compact and self-adjoint operator in the non-Archimedean Banach space $c_{0},$ where $\mu=(\mu_{i})_{i\in \mathbb{N}} \in c_{0} $ and for each $i\geq 1,$ $P_{i}=\frac{<.,y_{i}>}{<y_{i},y_{i}>}y_{i}$ is the normal projection defined by $(y_{i})_{i\in \mathbb{N}} \in c_{0}. $ Using Fredholm theory in the non-Archimedean setting and the concept of essential spectrum for linear operator, we compute the spectrum of $T.$