"The RC840T Electric Heating Relay is designed to control a line
voltage resistive load"
Source
So asking it to control a motor load, particularly one that starts as hard as the typical compressor, is probably a BAD idea. If the compressor is a fish-tank air pump, perhaps not a worry. If it's a shop air compressor, not a good idea at all. Airbrush compressor, probably yes unless it's abnormally huge.
Later in the specs it does claim it can handle a 0.8 HP inductive load (motor) on 120V so if the compressor motor is 3/4 HP or smaller, perhaps you're OK using it.
Unfortunately you have to get into quite big compressors before some of them have the sense to include an unloader so the motor can get up to speed before the pump starts pumping air. I don't think any that would plug into a 120V outlet have that feature.