Yes, of course. Consider an instantaneous transmission. If the receiver is moving away from the transmitter then in the receiver's frame the reception will take place earlier than the transmission. If the receiver is moving towards the transmitter then in the receiver's frame the reception will take place after the reception. In both cases, the signal will be received no later than it's transmitted in the transmitter's frame. You can see this by considering how the receiver would stack up simultaneity surfaces (not axes!) that, in the frame of the transmitter, tilt in opposite directions in the two cases.
The situation with superluminal but not instantaneous messages is a bit messier to think about and describe, which is why I chose the simple case of an instantaneous transmission in my article. If you look at my diagram of a Lorentz transformation
you should be able to convince yourself that for any given subluminal receiver there will be some speeds of FTL communication that will appear to be future-directed and others that will appear past-directed. For example, a transmission from Q to P will appear to be past-directed to an observer in the blue frame, whereas one from P to an event in the vicinity of the blue "space'" label will be future-directed. Note that the second transmission is slower than the first one, even thought both are faster than light.
(I realised while writing this that I haven't read the earlier messages in the thread, so I'm sorry if the discussion solely concerns spacecraft moving away from Earth.)
From Physicist:
Psst: if the interlocutors are travelling towards each other, the time axis reaches into the future. By that statement I mean the spaceship's axis of simultaneity reaches into the Earth's future. From sharplue:
I realised while writing this that I haven't read the earlier messages in the thread, so I'm sorry if the discussion solely concerns spacecraft moving away from Earth.
Traveling away from the earth was the content of the examples given, yes. And I confess it's the only situation I'd given any thought to. When, years ago, I realized the paradoxes involved, I pretty much abandoned any additional thinking about FTL communications. Now I'll re-examine the matter. It's always good to correct one's thinking.