# functional analysis – Spectrum of Right shift minus Left shift operator on ℓ2(Z)

$$A:ell^2(mathbb Z) longmapsto ell^2(mathbb Z)$$ 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 $$sigma_c=(-2i;2i)$$?