FALCON 900 CAPTAIN REPORTS TCAS RA AT 6000 FEET APPROACHING KLIDE INTERSECTION ON THE HYP 5 ARRIVAL TO SJC.

Narrative

I WAS THE PNF. WE WERE TALKING WITH NORCAL APCH; DSNDING TO 6000 FT MSL DIRECT KLIDE INTXN. APCH CALLED OUT TFC AT OUR 1-2 O'CLOCK POS AND INFORMED US THEY WERE NOT TALKING WITH THAT ACFT. WE LEVELED OFF AT 6000 FT MSL. OUR TCAS SHOWED THE OTHER ACFT WAS CLBING OUT OF 5500 FT MSL. NORCAL GAVE US A R TURN; WHICH WE COMPLIED WITH; BUT AS WE WERE IN OUR TURN THE OTHER ACFT TURNED L TOWARDS US. WE RECEIVED A TCAS RA AND RESPONDED TO THIS WITH AN IMMEDIATE CLB TO AN ALT OF APPROX 6800 FT MSL; AVOIDED THE TFC; AND DSNDED BACK TO 6000 FT MSL. WE LANDED AT SJC WITHOUT FURTHER INCIDENT. UNFORTUNATELY; THE HYP.HYP5 ARR INTO SJC IS A BUSY AREA WITH OTHER ACFT TRANSITIONING ALONG HWY 101; WITHOUT RECEIVING TA'S. IN RETROSPECT; WE SHOULD HAVE ASKED APCH TO STAY AT A HIGHER ALT AND GET A TURN EARLIER TO GIVE MORE AIRSPACE TO THE ACFT NOT SPEAKING WITH A CTLR. I INFORMED NORCAL THAT WE WERE CLBING TO AVOID THE TFC; BUT I'M NOT SURE IF I STATED TO THE CTLR THAT WE RECEIVED AN RA FROM OUR TCAS WHILE WE WERE CLBING.