Let $\mathcal{L}$ be the operator defined on $\mathcal{C}(\mathbb{Z})$ (the space of continuous functions vanishing at infinity) given by $$\mathcal{L}f(k) = |k| \cdot (f(k+1)+f(k-1)-2f(k))$$
In order to show that this is the generator of a process, I need to verify the usual conditions. This was not a problem, but what is left now is to show that for all $\lambda>0$, the range of $(I-\lambda\mathcal{L})$ is $\mathcal{C}(\mathbb{Z})$. The author of the notes says this is clear, but I do not see it, after all $\mathcal{L}$ is unbounded. Does someone have a hint?
I'm guessing that $\ f\ $ "vanishing at infinity", which I originally took to be synonymous with $\ \lim_\limits{k\rightarrow\pm\infty} f(k)=0\ $, actually means "$\ f(k)\ne 0\ $ for only a finite number of $\ k\ $", and I think it's clear that if $\ f(k)\ $ is non-zero for only a finite number of $\ k\ $, then so is $\ (I-\lambda\mathcal{L})f\ $.