ACR FLIES A TCASII RA IN THE SJC CLASS C WITH NO RPTED OR SIGHTED TFC.

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WE WERE FLYING THE SJC ILS/VISUAL APCH TO RWY 30L IN GOOD VFR WX WITH VISIBILITY 10 MI PLUS. SEVERAL TCASII TARGETS TO EACH SIDE AS WE TRANSITIONED FROM GILRO TO KLYDE; 17 DME; CAPT WAS HAND FLYING. THERE WAS A B757 BEHIND US AND WE WERE ASKED TO KEEP THE SPD UP; SO I HELD 170 KTS UNTIL JUST BEFORE IRONN; THE FAF. APCHING IRONN; WE RECEIVED A TCASII TA AND OBSERVED A YELLOW TARGET SLIGHTLY BELOW AND ABOUT 1/2 MI BEHIND THE R WING. I INCREASED AIRSPD SLIGHTLY THINKING TO PUT THE TARGET BEHIND US WHILE THE FO QUERIED ATC. I AM NOT SURE; BUT BELIEVE WE HAD ALREADY BEEN HANDED OFF TO TWR BY THEN. ATC ADVISED NO TFC IN THAT AREA. I HELD AIRSPD FIGURING THAT ANY JETS WOULD BE KNOWN TO THEM. THE BANDIT WAS PROBABLY A LIGHT PLANE AND WOULD DROP BEHIND US AT 170 KTS. THE TARGET CHANGED TO A WHITE; NO THREAT. APCHING LMM; AS I FINISHED CONFIGURING AND SLOWING; THE TARGET RETURNED; FIRST AS A TA THEN AS AN RA. MONITOR VERT SPD WAS REPLACED BY A CLB COMMAND. THE TARGET WAS ABOUT 400 FT BELOW US AND TO THE R; WITHIN A HALF MI. I BEGAN A SLOW CLB WHILE FO LOOKED VISUALLY AND TOLD TWR WE WERE BEGINNING A CLB AND MIGHT GO MISSED APCH. NO TFC IN SIGHT OR ON RADAR. THEN THE TARGET TOTALLY DISAPPEARED. I MADE AN S-TURN TO THE L TO PUT DISTANCE BTWN US AND THE TARGET; AND TO LOSE ALT AS I REINITIATED THE VISUAL APCH. AS WE ANGLED BACK FROM THE L TOWARDS CTRLINE; THE FO HAD A GOOD VIEW BEHIND AND SAW NOTHING. WE LANDED WITHOUT FURTHER INCIDENT.