The given operator is $$T:\ c_0(\mathbb{Z})\rightarrow c_0(\mathbb{Z})$$ $$x_n\rightarrow\frac{x_{n-1}+x_{n+1}}{2}$$ I am not sure how to find the spectrum. To find the point spectrum I suppose I need all $\lambda$ s.t. sequence defined by recurrent relation \begin{align} \lambda x_n&=\frac{x_{n-1}+x_{n+1}}{2}\\ x_{n+1}&=2\lambda x_{n}-x_{n-1} \end{align} is in $c_0$. So I solved this recurrent relation and obtained $$x_n=c_1\left(\lambda-\sqrt{\lambda^2-1}\right)^n+c_2\left(\lambda+\sqrt{\lambda^2-1}\right)^n$$ meaning that if I want $x_n\rightarrow0$ for $n\rightarrow+\infty$, I need either $|\lambda-\sqrt{\lambda^2-1}|<1$ if $c_1\neq0$, $|\lambda+\sqrt{\lambda^2-1}|<1$ if $c_2\ne0$ .
However I also need $x_n\rightarrow0$ for $n\rightarrow-\infty$ which gives me the conditions $|\lambda-\sqrt{\lambda^2-1}|>1$ if $c_1\neq0$, $|\lambda+\sqrt{\lambda^2-1}|>1$ if $c_2\neq0$.
So am I correct that the point spectrum is empty?
And what the complete spectrum? I believe I also need to find all $\lambda$ for which the operator $\lambda I-T$ is not onto $c_0(\mathbb{Z})$. So for any $(y_n)\in c_0$ I want to find $(x_n)$ s.t. $$\lambda x_n-\frac{x_{n-1}+x_{n+1}}{2}=y_n$$ which is again a recurrent relation, this time non-homogenous. But I have no idea how to show if the solution belongs to $c_0$.