Let $T\colon \mathbb L^p[0,1]\to \mathbb L^p[0,1]$, $1<p<+\infty$, be a linear bounded operator such that $\operatorname{Im}(T)$ is contained in the space of continuous functions. It was shown in this thread that $T$ is a compact operator.
Some non trivial example of such $T$ can be given, for example kernel operators with appropriated conditions on the kernel. The question is whether there exists other example (maybe asking for a characterisation is too ambitious).