Method for operation of a gas turbine system with hydrogen-rich fuel gas, wherein the gas turbine system comprises - at least one flow branch point in an exhaust gas channel; - a first exhaust gas line from the flow branch point to a flue stack; - a second exhaust gas line from the flow branch point via a back-cooler to a compressor inlet for recirculation of exhaust gases (16, 24), wherein the at least one flow branch point is an actuator element for controlling the proportions of the exhaust gas mass flows in the first exhaust gas line and the second exhaust gas line, wherein the setpoint of the control loop is the residual oxygen content of the exhaust gas; - a third exhaust gas line from the second exhaust gas line downstream of the back-cooler and upstream of the compressor to at least one exhaust gas compressor (26, 27) which compresses a part of the recirculated exhaust gases to a pressure which is sufficiently high to introduce it into the hydrogen-rich fuel gas (5, 10) for fuel gas dilution by means of a device; - a first combustion chamber and a second combustion chamber; characterised in that the gas turbine system furthermore comprises: - a fourth exhaust gas line from the exhaust gas compressor to the second combustion chamber; - a fifth exhaust gas line from the exhaust gas compressor firstly to an inter-cooler and thereafter to a high-pressure exhaust gas compressor, and finally to the first combustion chamber; - a control valve in the fourth exhaust gas line and a control valve in the fifth exhaust gas line to control the respective dilution mass flow depending on a first fuel gas quantity for the first combustion chamber and a second fuel gas quantity for the second combustion chamber; - and a fuel gas distribution system which conducts the diluted fuel gases to the respective combustion chamber (4, 9); wherein the branched part of the recirculated exhaust gases controlled in the actuating element is cooled in the back-cooler, the first part of the exhaust gas cooled in the back-cooler is recirculated into an intake air of the compressor, and the second part of the exhaust gas cooled in the back-cooler is compressed in the exhaust gas compressor, and wherein the exhaust gas intended for fuel gas dilution in the second combustion chamber is mixed with the second fuel gas quantity, and the exhaust gas intended for fuel gas dilution in the first combustion chamber is further compressed via the high-pressure exhaust gas compressor with intermediate cooling and mixed with the first fuel gas quantity. - at least one flow branch point in an exhaust gas channel; - a first exhaust gas line from the flow branch point to a flue stack; - a second exhaust gas line from the flow branch point via a back-cooler to a compressor inlet for recirculation of exhaust gases (16, 24), wherein the at least one flow branch point is an actuator element for controlling the proportions of the exhaust gas mass flows in the first exhaust gas line and the second exhaust gas line, wherein the setpoint of the control loop is the residual oxygen content of the exhaust gas; - a third exhaust gas line from the second exhaust gas line downstream of the back-cooler and upstream of the compressor to at least one exhaust gas compressor (26, 27) which compresses a part of the recirculated exhaust gases to a pressure which is sufficiently high to introduce it into the hydrogen-rich fuel gas (5, 10) for fuel gas dilution by means of a device; - a first combustion chamber and a second combustion chamber; characterised in that the gas turbine system furthermore comprises: - a fourth exhaust gas line from the exhaust gas compressor to the second combustion chamber; - a fifth exhaust gas line from the exhaust gas compressor firstly to an inter-cooler and thereafter to a high-pressure exhaust gas compressor, and finally to the first combustion chamber; - a control valve in the fourth exhaust gas line and a control valve in the fifth exhaust gas line to control the respective dilution mass flow depending on a first fuel gas quantity for the first combustion chamber and a second fuel gas quantity for the second combustion chamber; - and a fuel gas distribution system which conducts the diluted fuel gases to the respective combustion chamber (4, 9);