ZJX Controller described a reported NMAC after one of the aircraft in question acknowledged sighting the subject traffic; but still responded to the TCAS RA commands.

Narrative

I was working the RADAR position at Sector 57/58 combined. I had no D-Side as no D-Side was warranted for the amount of traffic in the sector. Air Carrier X; a CRJ-700; departed from KDAB on a heading to the east and was vectored out to the west to intercept the airway at the ROYES intersection; climbing to 11;000 FT by Approach Control. A mode C intruder; VFR at 10;500 FT; was in the area of Air Carrier X. Communications were delivered to me somewhere between 7;000 and 8;000 FT where I issued the local altimeter and an 'expect higher altitude in two minutes' instruction. I then issued a traffic call of the intruder to Air Carrier X at 12 o'clock; 3 miles. Air Carrier X said he had traffic at his 11 o'clock. I then said the traffic was between his eleven and twelve o'clock unverified at 10;500 FT and type unknown; but that was probably him. Air Carrier X then seemed to confirm the traffic at 10;500 FT as he repeated the altitude and said; 'We have the Traffic.' At this point I felt it was no longer necessary to issue additional traffic calls or information as he said he had the traffic in sight. While a read back was occurring for a different airplane; Air Carrier X announced he was responding to an RA; but it was not audible to me because he was blocked. He called a second time with announcement of the RA; but it was unknown to me as he was blocked by frequency congestion. Daytona Approach called me to ask if I saw the VFR traffic and I confirmed; stating I had already issued the traffic call to Air Carrier X. When I came off the line; I saw Air Carrier X at 11;800 FT where I asked him to acknowledge level at 11;000 FT. He said negative. He also stated that he had tried to call 'like 6 times' that he was responding to a TCAS. I advised him to advise when he was complete with his maneuvers and say intentions. He then stated he was complete and through 1-1-thousand-6-hundred for 11;000 FT. After his eventual climb clearance; clear of other IFR traffic; he asked what airspace that guy was in. I asked him to say again as I wasn't quite sure what he had meant and he asked if I was speaking with the VFR traffic. I informed him I was not. He then stated that he would call that a NMAC and he could 'see him smiling.' This all happened outside of my airspace. I did have RADAR on Air Carrier X; but the both aircraft involved in this incident were outside of my stratum. I called the traffic to Air Carrier X and he acknowledged 'we have the traffic.' I did not issue any control instructions. I feel my response to the situation was appropriate based upon the pilot's acknowledgment of the traffic.