A gas turbine combustor and a method of operating the gas turbine combustor, with the gas turbine combustor comprising a combustion sleeve, a combustion sub-chamber disposed at an upstream side of the combustion sleeve and having an air and fuel supply system, and a combustion main chamber disposed on a downstream side of the sub-chamber and having an air and fuel supply system. The combustion sub-chamber and combustion main chamber are formed in such a manner that the start-up of the gas turbine is effected by the hot combustion gas generated in the sub-chamber. The combustion sub-chamber is provided with an inner sleeve protruding from a side wall of the sub-chamber such that a tip of the inner sleeve is located in the vicinity of a downstream end of the sub-chamber, and a fuel nozzle is provided in a tip of the inner sleeve for increasing the fuel-to-air ratio during an initiating period of an increase in the gas turbine rotation speed.