B757-200 FLT CREW SHUT DOWN LEFT ENGINE DURING ARR TO DUE TO RISING OIL TEMP AND EICAS FILTER BYPASS MSG.

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WHEN WE WERE ABOUT 140 NM N OF ZZZ2 AND IN OUR INITIAL DSCNT THROUGH THE MID OR HIGH 20'S; I NOTICED THE OIL TEMP ON THE L ENG WAS HIGHER THAN THE R ENG AND RISING. I NOTICED THAT AT ABOUT 167 DEGS C OR SO THE WHITE NUMERIC INDICATION FOR THE L ENG CHANGED FROM WHITE NUMERIC TO AMBER; BUT IT NEVER EXCEEDED 170 DEGS C (INITIALLY). THE THROTTLES WERE IN THE IDLE POS. I DID AN ENG COMPARISON WITH L SIDE AND R SIDE. AFTER COMPARING SIDES; EVERYTHING APPEARED NORMAL WITH THE EXCEPTION THAT THE L ENG OIL TEMP WAS HIGH. I DID NOTICE THAT THE L ENG WAS INDICATING 9 QUARTS OF OIL WHILE THE R ENG WAS INDICATING 11 QUARTS. OTHER THAN THAT; EVERYTHING ELSE SEEMED NORMAL AND WITHIN PARAMETERS. SINCE MY FO WAS THE PF; I TOLD HIM TO GO AHEAD AND BRING THE L THROTTLE OUT OF IDLE AND SEE IF THAT MADE A DIFFERENCE ON THE OIL TEMP AND IT DID; BUT EVERY TIME HE BROUGHT THE THROTTLE TO IDLE; THE OIL TEMP ON THE L ENG WOULD START TO RISE. I THEN REALIZED THAT WE COULD ACTUALLY CTL THE TEMP WITH THE L THROTTLE BY KEEPING IT OUT OF FLT IDLE. I TOLD THE FO TO KEEP THE L THROTTLE OUT OF IDLE THROUGHOUT THE DSCNT. ONCE WE REACHED 5000 FT AND HAD A VISUAL ON THE ARPT; ZZZZ TWR CLRED US FOR THE VISUAL APCH AND TOLD US TO REPORT ESTABLISHED ON A BASE LEG. SO WE SET OURSELVES UP FOR A L BASE; AT WHICH POINT WE GOT THE OIL FILTER BYPASS ON THE UPPER EICAS. AT THAT POINT I TOLD THE FO THAT I WOULD MONITOR THE L ENG CLOSELY AND ALSO TOLD HIM TO GO DIRECTLY TO THE RWY. ONCE WE WERE ESTABLISHED ON A MODIFIED L BASE AT AN ALT OF 2500 FT AND ABOUT 4-5 MI FROM THE ARPT; I THEN MADE A STATEMENT TO MY FO THAT IF THE SITUATION ON THE L ENG DETERIORATED; THAT I MIGHT HAVE TO SHUT THE ENG DOWN. AND SURE ENOUGH; ABOUT 3-5 MINS FROM TOUCHDOWN; I NOTICED THAT THE OIL TEMP WAS RISING RAPIDLY ON THE L ENG AND THE OIL QUANTITY AND OIL PRESSURE DECREASING RAPIDLY. I TOLD MY FO THAT WE WERE LOSING OIL QUANTITY AND OIL PRESSURE AND THAT I WAS SHUTTING DOWN THE L ENG. AT TIME OF THE SHUTDOWN; WE WERE ABOUT 2-3 MI FROM THE ARPT. RIGHT AFTER I SHUT THE ENG DOWN; I INFORMED THE TWR OF OUR SITUATION AND TOLD THEM TO ROLL OUT THE TRUCKS AS A PRECAUTION. ONCE ESTABLISHED ON FINAL; SINCE THE FO WAS PF AND DOING A NICE JOB; I ASKED HIM IF HE WAS OK WITH CONTINUING TO FLY THE ACFT. HE REPLIED THAT HE WAS FINE. ON FINAL; MY FO PUSHED THE SWITCH FOR THE GEAR OVERRIDE INSTEAD OF THE FLAP OVERRIDE WHICH WERE RIGHT NEXT TO EACH OTHER. OTHER THAN THAT; THE LNDG WAS UNEVENTFUL. AFTER LNDG; WE CLRED THE RWY AND I ASKED THE TWR IF THEY SAW ANYTHING UNUSUAL COMING OUT OF OUR L ENG (IE; SMOKE; FIRE; ETC). THEY REPLIED NO. SO I ELECTED TO TAXI TO OUR GATE.