  We investigate the Courr\`{e}ge theorem in the context of linear operators
$A$ that satisfy the positive maximum principle on a space of continuous
functions over a symmetric space. Applications are given to Feller--Markov
processes. We also introduce Gangolli operators, which satisfy the positive
maximum principle, and generalise the form associated with the generator of a
L\'{e}vy process on a symmetric space. When the space is compact, we show that
Gangolli operators are pseudo--differential operators having scalar symbols.
