Let $T(x_n)_{n \in \mathbb Z} = (x_{n+1})_{n \in \mathbb Z}$, proof that $\sigma(T) = S^1$.

You just need to note that, if $0<|\lambda|<1$, then $T-\lambda = T(Id - \lambda T^{⁻1})$ and $\|\lambda T^{-1}\|\leq |\lambda| <1$. Now, observe that $(Id - \lambda T^{⁻1})^{-1} = \sum_{n=0}^\infty (\lambda T^{-1})^n$ and: $$(T-\lambda)^{-1}= (Id-\lambda T^{-1})^{-1}T^{-1}.$$

It proves that $\sigma(T) \subset S^1$.

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Operator Theory

Functional Analysis

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