B737 ON DSCNT FOR SJC EXPERIENCED TCAS RA AT FL190 WITH UNRPTED TFC.

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ON DSCNT ON ROBIE ARR INTO SJC; XING RESTR OF 15 NM S OF SNS AT FL200. DURING THE DSCNT; FREQ CHANGE TO ANOTHER CTLR WHO GAVE US A DSCNT TO 15000 FT; LEAVING FL190; WE WERE CLRED DIRECT TO GILRO. WE WERE DSNDING THROUGH FL200 TOWARDS FL190. THE FO WAS BRINGING DIRECT TO GILRO UP TO THE TOP OF THE LEGS PAGE SO WE COULD GO DIRECT TO GILRO PASSING FL190. APCHING FL190 WE RECEIVED A TA FOR TFC ABOUT 10 O'CLOCK POS LOW; SEVERAL MI AWAY. I SLOWED THE RATE OF DSCNT TO LOOK FOR THE TFC. I COULDN'T LOCATE THE TFC VISUALLY. WE THEN RECEIVED AN RA CLB; WHICH I BEGAN TO CLB. THE CTLR THEN GAVE US A R TURN OF APPROX 20 DEGS. AT ABOUT FL200; THE TCAS GAVE A CLR OF CONFLICT. I FINALLY HAD A VISUAL CONTACT ON THE ACFT AT ABOUT 9 O'CLOCK POS SEVERAL MI AWAY. WE CONTINUED THE DSCNT TO AN UNEVENTFUL LNDG. THE FLT ATTENDANTS NOR ANY PAX QUESTIONED THE MANEUVER; SO THEY WERE APPARENTLY UNAWARE OF THE EVENT. I GUESS THE FIRST CTLR GAVE US THE DSCNT TO FL200 BECAUSE THE EVENTUAL TFC CONFLICT WAS CLBING TO FL190. I THINK THE NEXT CTLR FORGOT ABOUT THE FL190 ACFT WHEN HE GAVE US THE DSCNT TO 15000 FT AND THE TURN TO GILRO OR THOUGHT THERE WOULD BE ENOUGH LATERAL SEPARATION BECAUSE OF THE TURN. I KNOW THE SECOND CTLR GAVE US AN 'OUT OF FL190; CLRED DIRECT TO GILRO;' SO WE MUST HAVE HAD THE CLRNC BELOW FL200. IF WE DIDN'T HAVE A CLRNC BELOW FL200; THE SECOND CTLR NEVER QUESTIONED OUR DSCNT BELOW FL200 SO I DON'T THINK WE VIOLATED ANY CLRNCS. THE TCAS DID ITS JOB.