LTT CAME VERY CLOSE TO WDB ON PARALLEL VISUAL APCHS TO LAX.

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LAX WAS CLR AND 7 MI VISIBILITY. WIND FROM THE W AT 7 KTS. THE INFLT VISIBILITY WAS APPROX 2-3 MI INTO THE SUN AND APPROX 5-7 MI AWAY FROM THE SUN. AS WE WERE GIVEN A BASE TURN; APCH INSTRUCTED A COMMUTER TO MAINTAIN 2500 FT; 180 KTS; AND TO KEEP US IN SIGHT; AND TO FOLLOW US TO THE ARPT -- HE WAS TO LAND 24L. THE COMMUTER PLT AT THE TIME SOUNDED CONFUSED AND WE WERE NOT SURE HE HAD US IN SIGHT. WE WERE THEN GIVEN A VECTOR HDG TO INTERCEPT THE LOC AND CLRED FOR AN ILS TO 24R TO MAINTAIN 2500 FT UNTIL ESTABLISHED AND 180 KTS TO THE OM. THE COMMUTER PLT REQUESTED 3000 FT TO STAY OUT OF OUR WAKE. HE WAS GIVEN CLRNC TO DO SO BUT TO KEEP US IN SIGHT. WE WERE ESTABLISHED ON THE APCH APPROX 1 MI OUTSIDE THE OM. THE COMMUTER CAME INTO VIEW OUT THE L DV WINDOW IN A DSNDING L BANK AHEAD AND APPROX 150 FT OFF OUR L WING; PASSING US. WE WERE AT 180 KTS WHEN HE WAS APPROX 500 FT AHEAD OF US AND LEVEL WITH US HE ABRUPTLY PULLED UP AND CLBED APPROX 250 FT ABOVE US AND SLOWED -- WE PASSED HIM AT THIS TIME. I FEEL THE COMMUTER PLT ACCEPTED THE VISUAL APCH WITHOUT SEEING OUR ACFT. HIS VOICE AND PHRASEOLOGY SOUNDED AS IF HE DID NOT HAVE THE SIT UNDER CTL AND PERHAPS SHOULD HAVE REQUESTED A VECTOR UNTIL HE HAD HIS TFC IN SIGHT TO FOLLOW. THE COMMUTER PLT'S ANSWERS TO THE CTLR WERE NOT POSITIVE AND SURE WHEN ASKED IF WE WERE IN SIGHT. I ALSO FEEL THE CTLR WAS NOT CONVINCED THE COMMUTER PLT COULD KEEP US IN SIGHT. SUPPLEMENTAL INFO FROM ACN 237403: WHILE EXECUTING A VISUAL APCH TO 24L AT LAX WE PULLED ALONG SIDE OF AN WDB ON FINAL TO 24R. INADVERTENTLY WE PASSED THE WDB WHILE ATTEMPTING TO MATCH SPD. WE CHOSE THE ALONG SIDE POS FOR WAKE TURB AVOIDANCE. THE CTLR MAY HAVE STATED FOR US NOT TO PASS THE WDB.