Need help because of battery light on problem

I have a 2004 Chevy Express Van with a 350 engine. It has only 35000 miles on it. Recently the battery light on the instrument panel comes on and stays on when the engine is hot and after the van has been restarted. The amp reading on the panel's ammeter reads only 1/2 of normal indicating that the alternator is not charging full strength. However after 10 minutes of running the van, the ammeter indicator all of a sudden rises back up to full strength and the engine battery light goes out at the same time. The battery light comes on only when the engine is hot and only after the van is restarted after stopping. It never comes on when the van is running. Can anyone comment if the alternator or something else is the problem?

Welcome to AF.
Alternator should be tested along with the battery and cables.
Most auto parts places will test them for you.
A good old volt meter will tell you if alt is charging good.