There are different kinds of decomposition. The most rapid one is biological: bacteria and the body's own enzymes both begin to break down tissues within a few hours of death. These are the processes that we generally call rotting, and they produce the bad odors that we smell. Next are scavengers such as insects and other animals, but these probably won't be present in space, unless they hitched a ride in the spacecraft. Slower still would be heat degradation, where proteins break down. Chemical exposure will degrade what's left; bones will eventually decay in weathering, and oxygen will react with proteins and many other things. This won't happen in space either. Finally, very slow processes such as radiation would eventually break down molecules over millions of years. The levels of radiation in space are higher than on Earth but not so high that tissues would degrade very quickly.

I think there are three possibilities for decomposition:
1. Unless there's a source of heat nearby, the body will be quickly frozen, and decomposition will take thousands or even millions of years.
2. If there's heat, but no spacesuit or spacecraft, the body will very quickly dry out, because water evaporates extremely quickly in the vacuum of space. This will almost completely stop biological processes, and the lack of air will prevent weathering and chemical degradation.
3. If the astronaut died while wearing their spacesuit and the body stays warm, then decomposing of tissue will be almost as fast as on earth. But bones will last almost indefinitely. Of course, if the body happens to fall in to a planet, it will be destroyed by the extreme heat of re-entry into the atmosphere, like a meteorite.

I really don't know what effects radiation may have on a body decaying through bacteria but my guess is it wouldn't make much difference at the beginning but stop the decomposition process somewhere along the way. The astronaut's body would decompose from bacteria already inside his/her body (just like anyone else). On the other hand, the astronaut's suit and spacecraft should shield him/her from radiation. I hope this helps.

If he dies in his space suit, then yes, he would decompose, because he still has bacteria and such within his body that would decompose him. It would use up all of the oxygen, however, so the decomposition would become anaerobic very quickly, going from respiration to fermentation.

Radiation would not significantly affect the decomposition process. It would break up chemical bonds and such, but our deceased astronaut is more likely to collide with an object of some kind like a meteor before that would really break him up.

If our astronaut is not in his suit, then the vacuum would kill the bacteria decomposing him in fairly short order.