Evaluating difficult spectrum

Question

Can anyone see how to show the spectrum of the bounded linear operator $T$ on $l^1$ defined by $$T((\alpha_j)) = (\alpha_j - 2\alpha_{j+1} + \alpha_{j+2})$$ is the cardioid $$\{(r, θ) : 0 ≤ θ < 2π, \;0 ≤ r ≤ 2 + 2 \cosθ\}?$$

Answer 1

I now have the answer to this: Consider operator S defined by $S((α_j)) = (α_{j+1})$. Its spectrum is closed unit ball in R2 (fairly easy). Now use spectral mapping theorem, with $p(x)=(1−x)^2$, noting $p(S)=T$