I'm attempting the following exercise:
Consider the right shift $S$ on $l^2(\mathbb{Z})$, which to $x=\left(x_n\right)_{n \in \mathbb{Z}} \in l^2(\mathbb{Z})$ associates $S(x) \in l^2(\mathbb{Z})$ with components $S(x)_k=x_{k-1}$ for all $k \in \mathbb{Z}$. Define the discrete Laplacian $\Delta$ on $l^2(\mathbb{Z})$ as $(\Delta x)_k=x_{k-1}+x_{k+1}-2 x_k$. Show that $\Delta=S+S^*-2 I$, and prove that the spectrum of $\Delta$ is entirely continuous and consists of the interval $[-4,0]$.
I'm stuck on this problem. The Laplacian is clearly self adjoint and bounded, so the residual spectrum is empty. I know that the spectrum should be contained in the closed disk of radius $4$. Other than that, I do not know how to approach this problem. Could anyone help me with this?
I think this question is answered here: Spectrum of the "discrete Laplacian operator" but it uses operator function theory, which I haven't learned. Could someone help break that proof down to more elementary steps, or perhaps come up with a new one that can be understood by someone knowing only the very basics (definition of continuous spectra, self-adjoint operators have real spectra, etc..)?