A turbomachine has a shaft mounted in a shaft housing, and a rotor in a rotor housing at one end of the shaft, for compression, expansion, or transport of a working gas. The shaft is mounted by hydrostatic, gas-lubricated bearings that have a connector to a compressed gas line that is connected with a pressure region of the turbomachine. A portion of the gas that is passed from the pressure region is applied to the bearing. The shaft is mounted so that the bearing effect is at least extensively independent of the rotational speed of the shaft, thereby guaranteeing support of the shaft during startup of the turbomachine, and good rigidity and low sensitivity to pressure surge stresses. The pressure in the connector to the compressed gas line is at least 5 bar greater than the pressure in the region of the ventilation connector.