AN ACR HVY ACFT IN DSCNT RECEIVES A TCASII RA TO CLB AND ADVISES ATC. THE CTLR INSTRUCTED THE PLT NOT TO CLB DUE TO TFC ABOVE AND ISSUED A TURN INSTEAD. THE FLC COMMENCED A TURN AND SPOTTED THE TCASII TFC PASSING 2-3 MI AWAY IN A CLB.

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WE WERE DSNDING FROM FL350 TO FL240 APPROX 40 MI E OF PSP AT ATC DIRECTION. AT APPROX FL260 WE GOT A TA AT 12 O'CLOCK ON TCASII. THIS RAPIDLY BECAME A RED RA. WE ALSO GOT INSTRUCTION TO CLB FROM TCASII. THE FO WAS FLYING THE ACFT AND HE LEVELED OFF TO INITIATE A CLB. WE WERE DSNDING AT APPROX 2000 FPM. I TOLD ZLA WE HAD TCASII TFC AND WE WERE GOING TO CLB. ATC SAID DON'T CLB; TURN L TO 180 DEGS. WE TURNED L. ATC SAID WE HAVE TFC ABOVE US DON'T CLB. I SAID YOU'VE GOT US SANDWICHED; WHAT DO YOU WANT US TO DO? BY THIS TIME I COULD SEE THE TFC WAS AT 1 O'CLOCK ON TCASII; 400 FT BELOW US AND WAS GOING TO PASS OFF TO OUR R. THE FO SAW THE TFC AND REALIZED IT WOULD PASS US ON OUR R. IT APPEARED TO PASS BTWN 2-3 MI R AND 400 FT BELOW ON TCASII. A DIFFERENT CTLR CAME ON AND CLRED US DIRECT TO PSP. THE FLT ATTENDANTS AND PAX WERE A BIT ALARMED BY OUR ABRUPT BUT SMOOTH MANEUVERS. I MADE A PA AND EXPLAINED THAT WE HAD TO MANEUVER THE ACFT AGGRESSIVELY FOR TFC BUT THAT WE WERE NOT IN DANGER BECAUSE WE ACTED QUICKLY TO AVOID COMING TOO CLOSE TO ANOTHER ACFT. THE TFC WAS A B737.