A CE560 TCAS alerted an RA warning shortly after takeoff for traffic which ATC did not paint and the crew did not see visually. This may be a Null Bus fault.

Narrative

First Office was the flying pilot; we departed South. Shortly after takeoff as we were climbing through 2000ft MSL the RA warning alerted us to monitor vertical speed with a red indication on the positive climb side of the VSI; immediately followed by a descend RA. At the same time we were given a turn and a direct vector with a climb to a higher altitude (I think but not sure 16000ft) which we immediately declined and we informed ATC that we were unable because we were responding to an RA. As we were descending back down; the 'DON'T SINK' RA was triggered; as by now we had leveled off around 2000 ft. We reset the TCAS by going to STBY and back to TA/RA which eliminated all of the advisories and warnings. Once we confirmed visually and with the controller that there was no traffic around us; we then accepted a higher altitude and continued without any further complications. After talking to the controller; we were 100% positive that there was no traffic anywhere close to us and that it was simply a glitch in the equipment which was corrected by the recycle of the TCAS.