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 $$ \operatorname{dist}\left(\sigma_{ph}(\Phi),(\sigma(\Phi)\setminus\sigma_{ph}(\Phi))\right)>0, $$ where $\sigma_{ph}$ stands for "peripheral spectrum".