SMA X NON ADHERENCE TO ATC INSTRUCTIONS HAD NMAC WITH SMT Y ON APCH.

Narrative

SMA X CALLED FOR DEP ON RWY 16R AT LONG BEACH ARPT. AS LCL CTL 2; I INSTRUCTED SMA X TO TAXI INTO POS AND HOLD; AND SAY DIRECTION OF FLT. I CLRED SMA X FOR TKOF AND APPROVED A 'STRAIGHT OUT; L TURN AT THE SHORELINE (FOR ORANGE COUNTY).' I NOTICED SMA X HAD ACTUALLY TAKEN A SE HDG; TOWARD THE ILS RWY 30 FINAL. I ALSO OBSERVED AN SMT Y TURNING S OFF THE APCH; RIGHT INTO SMA X FLT PATH. I ISSUED TFC TO SMA X AS 'TFC 9 O'CLOCK; LESS THAN A MI; SBOUND SMT.' SMA X DID NOT RESPOND. SMA X WAS 1 MI FROM MY AIRSPACE BOUNDARY; AND HDG TOWARD THE SNA ARSA. I ASSUMED SHE HAD LEFT MY FREQ. NO OTHER TFC CALLS WERE MADE. SMA X ASKED IF SMT Y CALL WAS FOR HER. I RESPONDED AFFIRMATIVE. SMA X THEN INFORMED ME OF THE NMAC AND STATED SHE WOULD FILE A RPT. SHE ALSO INQUIRED AS TO WHO WAS WORKING SMT Y AND IF VISUAL CONTACT HAD BEEN MADE. SMT Y WAS UNDER THE CTL OF COAST TRACON. I BELIEVE I GAVE SMA X THE BEST SVC POSSIBLE AT THAT TIME. THE PLT; VFR IN VMC; HOWEVER; FEELS THAT I FAILED TO PROVIDE ADEQUATE SEPARATION AND INSISTS THE TFC CALL I ISSUED WAS TO ANOTHER ACFT (OR I USED THE WRONG CALL SIGN). (TAPE RECORDINGS PROVE THE CALL WAS CORRECT.)