According to the theory of relativity, time dilation occurs when an object is moving relative to another at speeds close to the speed of light. This means that observers in different frames of reference will perceive time to pass at different rates. However, in the scenario you described, where two astronauts are traveling at speeds close to the speed of light in opposite directions, they will not be able to directly observe each other's aging process.
The reason for this is that the speed of light acts as an absolute speed limit, and no information can travel faster than light. Therefore, the light signals carrying visual information between the two astronauts cannot reach each other because they are moving apart at speeds close to the speed of light. As a result, they cannot directly observe the effects of time dilation on each other.
However, from each astronaut's perspective, they will observe time dilation effects on everything else around them. Each astronaut will see external clocks and objects on the other astronaut's ship moving slower in time due to their relative motion. This is a consequence of the time dilation phenomenon predicted by the theory of relativity.
To summarize, while the astronauts will not be able to see each other age slowly directly, they will observe time dilation effects on everything else in their respective frames of reference.