Let $T:[0,1]\to[0,1]$ be a continuous map, which is neither surjective nor injective. Put
$$
C([0,1])\ni f\mapsto \Phi(f):=f\circ T\in C([0,1]).
$$
Notice that, under the above conditions, $0\in\sigma(\Phi)$.

I'm searching for an example of $\Phi$ as above such that
$$
0<\operatorname{dist}\left(\sigma_{ph}(\Phi),(\sigma(\Phi)\setminus\sigma_{ph}(\Phi))\right)<1,
$$
where $\sigma_{ph}$ stands for "peripheral spectrum".