$A:ell^2(mathbb Z) longmapsto ell^2(mathbb Z)$ A defined as following $A(x)_{n} = x_{n−1} − x_{n+1}$

Find A’s spectrum. $sigma(A)=sigma_c(A)=sigma_{ess}(A)=(-2i,2i)$

Operator is not self-adjoined, $A=S_r-S_l$ and $||A||=2$.

The space $ell^2(mathbb Z)$ is isomorphic with $L^2(mathbb T)$, via the identification $delta_nlongmapsto (tmapsto e^{inpi,t})$.

$S_r$ mapped to $M_z$, $S_l$ mapped to $M_z^{-1}$

So the spectrum of $S_r-S_l$ is the same as that of $M_{z^{-1} – z}$.

I don’t understand why $$sigma_c(M_f)=sigma(M_f)=overline{f(mathbb R)}.$$ Why for a multiplication operator, its spectrum is the closure of the range? And spectrum continuous?

$$sigma(S_r-S_l)=overline{{z-z^{-1}: zinmathbb T}}

=overline{{z -bar z: |z|=1}}=overline{{2 text{Im},z: |z|=1}}$$

How to prove from that that $sigma_c=(-2i;2i)$?