Leray-Schauder fixed point theorem
$D$ is closed and bounded, and $T$ compact, hence $K = \overline{T(D)} \subset D$ is compact. Hence the convex hull $\operatorname{co} K$ is totally bounded, and $C = \overline{\operatorname{co} K} \subset D$ is a compact convex nonempty set. The restriction $T\lvert_C \colon C \to C$ is continuous. By the Schauder fixed point theorem, $T\lvert_C$ has a fixed point in $C$.