Let $X$ be a compact Hausdorff space. A continuous map $f:X \to X$ defines a bounded linear operator $T_{f}$ on the Banach space $C(X)=\{\phi:X\to \mathbb{C} \mid \phi\; \text{is continuous}\}$ with $T_{f}(\phi)=\phi \circ f$.

Put $X=[0,1]$.

What is an example of a non constant map $f$ such that $T_{f}$ is a compact operator? What is an example of a map $f$ such that $T_{f}$ is a Fredholm operator of non-zero index?

For a linked MSE question see this MSE post.