ACR MLG WITH FAULTY INDICATION IN COCKPIT HAD INFLT SHUTDOWN.

Narrative

AFTER TKOF FROM MSY AND WHILE CLBING THROUGH APPROX 14000' MSL; THE RIGHT ENG START VALVE OPEN LIGHT ILLUMINATED. THIS SITUATION IF NOT IMMEDIATELY CORRECTED REQUIRES AN ENG SHUTDOWN. CORRECTIVE ACTION WAS UNSUCCESSFUL; SO THE REQUIRED CHKLISTS WERE INITIATED. I INSTRUCTED THE F/O TO ADVISE ATC THAT WE HAD AN ENG PROB AND WANTED TO RETURN TO MSY. WE WERE GIVEN IMMEDIATE CLRNC AND STARTED BACK FOR THE ARPT. THE NEXT FEW MINS WERE OCCUPIED WITH COMPLETING THE CHKLISTS; DSNDING AND ADVISING THE PAX OF THE PROB. I THEN INSTRUCTED THE F/O TO ADVISE COMPANY OF THE NATURE OF OUR PROB; THAT WE WERE RETURNING; AND THAT MAINT WOULD BE NEEDED. ATC ASKED IF WE WERE DECLARING AN EMER OR WANTED EQUIP STANDING BY. WE RESPONDED NO TO BOTH QUESTIONS. THE LNDG WAS UNEVENTFUL; AND THE PROB TURNED OUT TO BE A FALSE INDICATION. IN REVIEWING THE SITUATION AFTERWARDS; NEITHER THE F/O NOR I WERE CERTAIN THAT WE HAD SPECIFICALLY ADVISED ATC OF THE ENG SHUTDOWN. HOWEVER; WE BOTH FEEL THAT COMPANY WAS ADVISED ON THE SPECIFICS OF THE PROB AND THE FACT THAT AN ENG HAD BEEN SHUT DOWN. IN A REPEAT SITUATION; I WOULD TRY HARDER TO ENSURE THAT ATC KNEW THE ENG HAD BEEN SHUT DOWN; ALTHOUGH IN THIS CASE THERE WAS NO INTENTION TO KEEP ATC UNAWARE OF ANY OF THE FACTS.