compute the spectrum of an operator Let $C_{0}(\mathbb R)$ be the space of complex continuous valued functions on $\mathbb R$ which vanish at $\infty$ equipped with the sup-norm. Let $S:C_{0}(\mathbb R) \to C_{0}(\mathbb R)$ given by $(Sf)(t)=f(t+s)$ for a fixed $s\in \mathbb R$. 
 Prove that the spectrum of $\sigma(S)$ is given by the set $\sigma(S)=\{z\in \mathbb C ; ||z||=1\}$.
I proved that if $|\lambda|\ne 1$ then $\lambda \in p(T)$. I want to prove that if $|\lambda|=1$ then $\lambda \in \sigma(T)$
Since if $|\lambda|=1$ I proved that $\lambda I-T$ it's injective. It remains to prove that $\lambda I-T$ it's not surjective.
Thanks for the tips! But it remains to prove that if $||z||=1$ then $z\in \sigma(T)$. And I don't know how

If $0\le|\lambda|<1$, you can explicitly compute
$$
(\lambda I - T) ^{-1} = -\sum_{i=0}^{\infty} \lambda^{i}T^{-i-1}.
$$
You can verify this is, indeed, an inverse and it's continuous.
