ACR X TCASII TA RA WITH VFR SMA Y IN ARSA. EVASIVE ACTION TAKEN.

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ACR X CHKED ON FREQ. HE WAS CLRED FOR THE SJC RWY 30L ILS. SHORTLY THEREAFTER; VFR SMA Y INFORMED ME HE WISHED TO CROSS THE FINAL APCH COURSE AT 5500 FT. HE WAS BTWN 5 AND 10 MI SE OF SJC. TFC WAS ISSUED TO BOTH ACFT. SMA Y RPTED ACR X IN SIGHT AND WAS TOLD TO MAINTAIN VISUAL SEPARATION. I THEN INFORMED ACR X VISUAL SEPARATION WAS BEING APPLIED AND REISSUED THE TFC. TFC WAS ISSUED AGAIN AT APPROX 2 MI. I THEN OBSERVED SMA Y TURNING TO PARALLEL THE TFC. ACR X INFORMED ME HE WAS CLBING DUE TO AN RA. IN MY ESTIMATE; THE ACFT INVOLVED PASSED 1 MI LATERALLY AND 400 FT VERTICALLY. SUPPLEMENTAL INFO FROM ACN 255277: ON ILS 30L APCH TO SJC DSNDING ON THE GS; BAY TRACON ISSUED VFR TFC AT 10 O'CLOCK CONVERGING. THE TFC WAS NOT VISIBLE TO US IN REDUCED VISIBILITY OF 3 NM OR LESS IN HAZE AND SMOKE; BUT WE HAD A TCASII TARGET. THE CTLR ISSUED OUR ACFT AS TFC TO THE VFR ACFT; REFERRING TO OUR ACFT AS AN MLG. THE VFR ACFT SAID THAT HE HAD OUR ACFT IN SIGHT AND WAS INSTRUCTED BY THE CTLR TO KEEP OUR ACFT IN SIGHT. A TCASII TA WAS ACTIVATED AT ABOUT 3000 FT FOLLOWED ALMOST IMMEDIATELY BY A TCASII RA. A PULL-UP MANEUVER WAS INITIATED DURING WHICH THE VFR ACFT WAS OBSERVED VISUALLY AT 10 O'CLOCK; 1/4 TO 1/2 NM; LESS THAT 500 FT BELOW. DURING THE RA PULL-UP 600 FT BELOW WAS OBSERVED ON THE TCASII TARGET. A MISSED APCH WAS THEN EXECUTED BECAUSE OF OUR POS FAR ABOVE THE GS. AT NO TIME DID THE CTLR ISSUE ANY POSITIVE INSTRUCTIONS TO EITHER ACFT OTHER THAN 'KEEP THAT ACFT IN SIGHT' THAT WOULD HAVE PREVENTED THE 2 ACFT FROM FLYING SO CLOSE TOGETHER. SUPPLEMENTAL INFO FROM ACN 255211: WHILE XING THE SJC RWY 30 ILS APCH COURSE; ATC CALLED AIRLINE TFC AT OUR 2 O'CLOCK POS. WE SIGHTED THE TFC; RPTED TO APCH; AND TURNED TO PARALLEL THE APCH COURSE UNTIL THE TFC PASSED US. AT THAT POINT WE TURNED TO PASS BEHIND THE TFC. THE AIRLINER DIDN'T RPT VISUAL CONTACT AND THEIR TCASII SYS EVIDENTLY RPTED A CONFLICT. THE AIRLINER STARTED TO CLB AFTER IT HAD PASSED US. IT IS MY OPINION THAT THE TCASII SYS REQUIRED AN UNWARRANTED CLB RESPONSE ON THE PART OF THE AIRLINE CREW. THE CONDITIONS WERE VFR AND WE HAD RPTED THE TFC IN SIGHT AND HAD MANEUVERED TO AVOID ANY CONFLICT.